H.Kleinert, PATH INTEGRALS September 13, 2013 (/home/kleinert/kleinert/books/pathis/pthic1.tex) Ay, call it holy ground, The soil where first they trod! F. D. Hemans (1793-1835), Landing of the Pilgrim Fathers 1 Fundamentals Path integrals deal with fluctuating line-like structures. These appear in nature in a variety of ways, for instance, as particle orbits in spacetime continua, as polymers in solutions, as vortex lines in superfluids, as defect lines in crystals and liquid crystals. Their fluctuations can be of quantum-mechanical, thermodynamic, or statistical origin. Path integrals are an ideal tool to describe these fluctuating line-like structures, thereby leading to a unified understanding of many quite different physical phenomena. In developing the formalism we shall repeatedly invoke well-known concepts of classical mechanics, quantum mechanics, and statistical mechanics, to be summarized in this chapter. In Section 1.13, we emphasize some important problems of operator quantum mechanics in spaces with curvature and torsion. These problems will be solved in Chapters 10 and 8 by means of path integrals.1 1.1 Classical Mechanics The orbits of a classical-mechanical system are described by a set of time-dependent generalized coordinates q1 (t), . . . , qN (t). A Lagrangian L(qi , q˙i , t) (1.1) depending on q1 , . . . , qN and the associated velocities q˙1 , . . . , q˙N governs the dynamics of the system. The dots denote the time derivative d/dt. The Lagrangian is at most a quadratic function of q˙i . The time integral A[qi ] = Z tb ta dt L(qi (t), q˙i (t), t) (1.2) of the Lagrangian along an arbitrary path qi (t) is called the action of this path. The path being actually chosen by the system as a function of time is called the classical path or the classical orbit qicl (t). It has the property of extremizing the action in comparison with all neighboring paths qi (t) = qicl (t) + δqi (t) 1 Readers familiar with the foundations may start directly with Section 1.13. 1 (1.3) 2 1 Fundamentals having the same endpoints q(tb ), q(ta ). To express this property formally, one introduces the variation of the action as the linear term in the Taylor expansion of A[qi ] in powers of δqi (t): δA[qi ] ≡ {A[qi + δqi ] − A[qi ]}lin term in δqi . (1.4) The extremal principle for the classical path is then δA[qi ] i. t)}lin dt ( ∂L ∂L δqi (t) + δ q˙i (t) ∂qi ∂ q˙i dt ( . q˙i (t) + δ q˙i (t).. q˙i (t). qi (t)=qicl (t) =0 (1.5) for all variations of the path around the classical path.6) Since the action is a time integral of a Lagrangian. Let us calculate the variation of A[qi ] explicitly: δA[qi ] = {A[qi + δqi ] − A[qi ]}lin = Z tb = Z tb = Z tb ta ta ta dt {L (qi (t) + δqi (t).e. (1. which vanish at the endpoints. t) − L (qi (t). which satisfy δqi (ta ) = δqi (tb ) = 0. the extremality property can be phrased in terms of differential equations. δqi (t) ≡ qi (t) − qicl (t). tb ∂L ∂L d ∂L δqi (t) + − δqi (t). . . ∂qi dt ∂ q˙i ∂ q˙i ta ) ) . (1. repeated indices are understood to be summed (Einstein’s summation convention). (1. Thus we find for the classical orbit qicl (t) the Euler-Lagrange equations: d ∂L ∂L = .7) The last expression arises from a partial integration of the δ q˙i term. (1. According to the general theory of Legendre transformations [1].10) ∂ q˙i H. Here. the natural variables which H depends on are no longer qi and q˙i . t). Kleinert. q˙i . PATH INTEGRALS . as in the entire text. the latter being defined by the N equations ∂ pi ≡ L(qi . q˙i . ∂ q˙i (1. due to (1.8) dt ∂ q˙i ∂qi There is an alternative formulation of classical dynamics which is based on a Legendre-transformed function of the Lagrangian called the Hamiltonian H≡ ∂L q˙i − L(qi . but qi and the generalized momenta pi .6).9) Its value at any time is equal to the energy of the system. t). The endpoint terms (surface or boundary terms) with the time t equal to ta and tb may be dropped. the action is the following functional of pi (t) and qi (t): A[pi .9).13) In terms of this Hamiltonian. t) . The result is inserted into (1. (1. the equations (1. t) − L (qi .10) have to be solved for q˙i by a velocity function q˙i = vi (pi . qi . t) . qi .14) This is the so-called canonical form of the action.11) This is possible provided the Hessian metric hij (qi .12) is nonsingular. qi ] = Z tb ta h i dt pi (t)q˙i (t) − H(pi (t). qi . qi . δqi (ta ) = δqi (tb ) = 0. the variation is δA[pi .1 Classical Mechanics In order to express the Hamiltonian H (pi .16) . qi (t).6)] whereas the momenta pi (t) are varied without restriction: qi (t) = qicl (t) + δqi (t). (1. t) ∂ q˙i ∂ q˙j (1.3). q˙i . t). q˙i . qi . (1. t) in terms of its proper variables pi . vi (pi . t) ≡ ∂2 L(qi . (1. qi ] = Z tb = Z tb ta ta " ∂H ∂H δpi − δqi dt δpi (t)q˙i (t) + pi (t)δ q˙i (t) − ∂pi ∂qi dt (" # " # ∂H ∂H q˙i (t) − δpi − p˙ i (t) + δqi ∂pi ∂qi ) # (1. In general. qi . t) .15) pi (t) = pcl i (t) + δpi (t). (1.3 1. The classical orbits are now speccl ified by pcl i (t). t) = pi vi (pi . leading to the Hamiltonian as a function of pi and qi : H (pi . qi (t). They extremize the action in comparison with all neighboring orbits in which the coordinates qi (t) are varied at fixed endpoints [see (1. tb . + pi (t)δqi (t). qi (t) must be solutions of the Hamilton equations of motion ∂H . we find that pcl i (t). ta cl Since this variation has to vanish for the classical orbits. .8) via (1. as can easily be verified.9) and (1. .17) These agree with the Euler-Lagrange equations (1. = ∂pi p˙ i = − q˙i (1.10). ∂qi ∂H . The 2N-dimensional space of all pi and qi is called the phase space. qj } = δij . (1. B} ≡ ∂A ∂B ∂B ∂A − . qi (t). O} + ∂t (1. qj } = 0.21) Jacobi identity.22) If two quantities have vanishing Poisson brackets. A function O(pi .19): d pi = {H.20) again with the Einstein summation convention for the repeated index i. pj } = 0.} called Poisson brackets: {A. moreover.4 1 Fundamentals An arbitrary function O(pi (t). A}} + {C. i. ∂pj ∂qj ∂pj ∂qj ∂qi ∂H ∂qi ∂qi ∂H ∂H − = . t) = p˙ i + q˙i + . due to (1. B}} = 0 antisymmetry. B} = − {B.. .23) By definition. {O. . (1. {pi . qi (t). The Poisson brackets have the obvious properties {A. ∂pi ∂qi ∂pi ∂qi (1. (1. Kleinert. since H commutes with itself. we may insert (1. is a constant of motion along the classical path. of the form H = H(pi . the energy is a constant of motion. H itself is often time-independent. {A. qi satisfy the Poisson brackets {pi . t) changes along an arbitrary path as follows: ∂O ∂O ∂O d O (pi (t).18) If the path coincides with a classical orbit.e. A} {A. {C.19) Here we have introduced the symbol {. the phase space variables pi . qi ) which has no explicit dependence on time and which. The original Hamilton equations are a special case of (1. C}} + {B..17) and find dO ∂H ∂O ∂O ∂H ∂O = − + dt ∂pi ∂qi ∂pi ∂qi ∂t ∂O . {B.25) Then. H. . ≡ {H. PATH INTEGRALS .19). . ∂pj ∂qj ∂pj ∂qj ∂pi (1. qi ). dt ∂pi ∂qi ∂t (1. H} = 0). In particular. commutes with H (i. they are said to commute. pi } = dt d qi = {H.24) {qi . qi } = dt ∂H ∂pi ∂pi ∂H ∂H − =− . . (1. .e. (1. dt ∂ Q˙ j ∂Qj (1. t (1. Q˙ j (t). t) .30) dt L fi (Qj (t).27) Otherwise Qi and qi could not both parametrize the same system. t) . Q˙ j . t) . this relation must be invertible. f˙i (Qj . the initial Lagrangian takes the form L′ Qj .26) Certainly. t) . t). (1. t (1. By performing variations δQj (t).5 1.28) In terms of Qi . t ≡ L fi (Qj . gives δA = Z tb = Z tb ta ta dt dt ∂L ˙ ∂L δfi + δ fi ∂qi ∂ q˙i ! ! ∂L . Let Qi be any other set of coordinates describing the system which is connected with qi by what is called a local 2 or point transformation qi = fi (Qj . t). f˙i (Qj (t). we find the equations of motion d ∂L′ ∂L′ − = 0.1 Classical Mechanics The Lagrangian formalism has the virtue of being independent of the particular choice of the coordinates qi . Therefore.29) and the action reads A = Z = Z tb ta tb ta dt L′ Qj (t). on the other hand. δ Q˙ j (t) in the first expression while keeping δQj (ta ) = δQj (tb ) = 0. t . fi must have a nonvanishing Jacobi determinant: det ∂fi ∂Qj ! 6= 0. (1. at least in some neighborhood of the classical path. to be of use. Qi = f −1 i (qj .31) The variation of the lower expression. . tb d ∂L ∂L δfi + − δfi . with δqi (ta ) = δqi (tb ) = 0. . 2 The word local means here at a specific time. then so is δfi .32) If δqi is arbitrary. . Hence the extremum of the action is determined equally well by the Euler-Lagrange equations for Qj (t) [as it was by those for qi (t)]. This terminology is of common use in field theory where local means. more generally. ta ∂qi dt ∂ q˙i ∂ q˙i (1. also δfi vanishes at the endpoints. Moreover. at a specific spacetime point . Qj (t) for which they hold. the Hamilton equations do not hold. They are characterized by the form invariance of the action. t). They will necessarily be linear in Q˙ j : ∂fi ∂fi ˙ Qj + .35) Qj = Qj (pi .33) In phase space. Qj . q˙i = f˙i (Qj . Qj . up to an arbitrary surface term. qi . with the inverse relations Pj = Pj (pi .34) qi = qi (Pj . qi . are referred to as canonical . Let them be related by pi = pi (Pj . qi . there exists also the possibility of performing local changes of the canonical coordinates pi . However. Qj . t)] = Z tb ta h dt Pj Q˙ j − H ′ (Pj . while the Euler-Lagrange equations maintain their form under any local change of coordinates. t). t). in general. t) . Qj (t). (1. for any transformed coordinates Pj (t). qi to new ones Pj .6 1 Fundamentals Note that the locality property is quite restrictive for the transformation of the generalized velocities q˙i (t). The local transformations pi (t). Qj . Z tb ta dt [pi q˙i − H(pi . t). t) = ∂Qj ∂t (1. (1. qi (t) → Pj (t). tb . Qj . t). + F (Pj . leading to Z tb ta ( ∂qi P j − pi ∂Qj ! dQj − pi ∂qi dPj ∂P!j ) ∂qi − H + pi − H dt ∂t ′ = . qi (Pj . ∂Qi ∂H ′ ˙ .36) . qi . qi (t) connecting the same endpoints (at least any in some neighborhood of the classical orbits). Qj . Qj . t) must be chosen in such a way that the equality of the action holds for any path pi (t).38) and subtracting it from the right-hand side. ta i (1. t). (1. where H ′ (Pj . Qj (t). Its relation with H(pi . Z tb ta dt ( pi ∂qi ˙ ∂qi ˙ ∂qi Pj + Qj + ∂Pj ∂Qj ∂t ! ) − H(pi (Pj . t) is some new Hamiltonian. Qi = ∂Pi (1.36) can be expressed differently by rewriting the integral on the left-hand side in terms of the new variables Pj (t). If such an invariance exists then a variation of this action yields for Pj (t) and Qj (t) the Hamilton equations of motion governed by H ′ : ′ ∂H P˙ i = − .37) The invariance (1. Qj . t) . t). tb . −F (Pj . Qj . t). (1.39) ta H. . Kleinert. PATH INTEGRALS . 46) . (1. and ∂qi ∂pi ∂(H ′ − H) ∂pi ∂qi − = . {Pk . Thus we conclude that the integrand on the left-hand side must be a total differential.7 1. qi and of the time t. (1. . As such it has to satisfy the standard Schwarz integrability conditions [2]. (1. (Qk . Ql ) = δkl . {Qk . After a little algebra involving the matrix of derivatives its inverse J= J −1 = ∂Pi /∂pj ∂Pi /∂qj ∂Qi /∂pj ∂Qi /∂qj ∂pi /∂Pj ∂pi /∂Qj ∂qi /∂Pj ∂qi /∂Qj and the symplectic unit matrix E= 0 −δij δij 0 ! . ∂t ∂Pl ∂t ∂Pl ∂Pl (1.42) Time-dependent coordinate transformations satisfying these equations are called symplectic. (1. (Pk . The right-hand side depends only on the endpoints.43) (1. Pl ) = 0.1 Classical Mechanics The integral is now a line integral along a curve in the (2N + 1)-dimensional space.41) ∂pi ∂qi ∂qi ∂pi ∂(H ′ − H) − = . ∂t ∂Ql ∂t ∂Ql ∂Ql The first three equations define the so-called Lagrange brackets in terms of which they are written as (Pk .40) = 0. ∂Pk ∂Ql ∂Pk ∂Ql ∂qi ∂pi ∂pi ∂qi − = 0. Ql ) = 0.44) .42) are equivalent to the Poisson brackets {Pk . Pl } = 0. these conditions are ∂pi ∂qi ∂qi ∂pi − = δkl . Explicitly. Ql } = δkl . ∂Pk ∂Pl ∂Pk ∂Pl ∂pi ∂qi ∂qi ∂pi − ∂Qk ∂Ql ∂Qk ∂Ql (1. consisting of the 2N-dimensional phase space variables pi .45) we find that the Lagrange brackets (1. Ql } = 0. according to which all second derivatives have to be independent of the sequence of differentiation. t space]. They can always be used to construct H ′(Pj .42) [and thus the Poisson brackets (1. (1.46) are therefore both necessary and sufficient for the transformation pi . Qj . (1.8 1 Fundamentals This follows from the fact that the 2N × 2N matrix formed from the Lagrange brackets −(Qi .46) for Pi . qi .49) The Poisson brackets (1. Qj (t). Once a system is described in terms of new canonical coordinates Pj . Qj .46) are equivalent to each other. ∂Pj ∂Qj ∂Pj ∂Qj (1. B}′ ≡ ∂A ∂B ∂B ∂A − . with the roles of pi . t) from H(pi . Qj exchanged [we could just as well have considered the integrand (1. t) becomes with (1. Pj } P≡ (1.50) j It is obvious that the process of canonical transformations is reflexive. qi .51) and the equation of motion for an arbitrary observable quantity O (Pj (t). dt (1. The other two equations (1. Hence L = P −1 . Qj to be canonical. Note that the Lagrange brackets (1. qi and Pj . It may be viewed just as well from the opposite side.37) ∂O dO ′ = {H ′ . t). Qj ) can be written as (E −1 J −1 E)T J −1 . The Lagrange brackets (1. Qj . (1. Qj ) L≡ (1. Pj } is equal to J(E −1 JE)T .48) {Qi .48). t). A canonical transformation preserves the volume in phase space. Qi have the same form as those in Eqs.42) and (1. O} + . qi → Pj . Pj ) (Pi .42) or Poisson brackets (1. Qj . Hence det (J) = ±1 and YZ [dpi dqi ] = i YZ [dPj dQj ] .47) (Pi .24) for the original phase space variables pi .39) as a complete differential in Pj . Qj } − {Pi . PATH INTEGRALS . while an analogous matrix formed from the Poisson brackets {Pi .52) dt ∂t H. we introduce the new Poisson brackets {A. so that (1. This follows from the fact that the matrix product J(E −1 JE)T is equal to the 2N × 2N unit matrix (1. Kleinert.46)] ensure pi q˙i − Pj Q˙ j to be a total differential of some function of Pj and Qj in the 2N-dimensional phase space: d pi q˙i − Pj Q˙ j = G(Pj . Pj ) −(Qi .41) relate the new Hamiltonian to the old one. Qj } − {Qi . = ∂Pj (1. Pj .56) A comparison of the two sides yields the equations for the canonical transformation pi = Qj ∂ F (qi . t). Pj . t) + F (qi . Pj . t) − H ′ (Pj .57) The second equation shows that the above relation between F (qi . t)q˙i + (qi .54) we could. t)P˙ j + (qi . t) + ∂ F (qi . t) . Pj . Qj . t) + Pj Qj . ∂qi ∂ F (qi . Pj .55) dt d PQ. dt j j defines F (qi . F (pi . This yields Z tb ta = n o dt pi q˙i + P˙ j Qj − [H(pi . Pj . t) to generate simple canonical transformations. Pj . Qj .58) Instead of (1. qi . Qj . A standard class of canonical transformations can be constructed by introducing a generating function F satisfying a relation of the type (1. Pj . t). of course. t) amounts to a Legendre transformation. = 0. t). t) .9 1. qi . Qj . The new Poisson brackets automatically guarantee the canonical commutation rules {Pi . Qj . t). but depending explicitly on half an old and half a new set of canonical coordinates. (1. The new Hamiltonian is H ′ (Pj . Qj . t). F (pi . also have chosen functions with other mixtures of arguments such as F (qi . t)] Z tb ta ( ) ∂F ∂F ∂F (qi . (1.54) One now considers the equation Z tb ta dt [pi q˙i − H(pi . for instance F = F (qi . (1. t) and F (qi . t). qi .19). and works out the derivatives. Qj . dt ∂qi ∂Pj ∂t (1. Qj }′ = 0. t) ≡ F (qi .1 Classical Mechanics by complete analogy with (1. Qj }′ {Pi .53) {Qi .36). . Qj . t)] = replaces Pj Q˙ j by −P˙ j Qj + Z tb ta " # d dt Pj Q˙ j − H ′ (Pj . t) = H(pi . Pj }′ = δij . Pj . ∂t (1. Pj . 65) H. Pj . qi . t). qi . Pj . PATH INTEGRALS . t). t) = pi (t)q˙i (t) − H(pi (t). t) = −H(pi . (1. the associated actions of these solutions form a function A(qi . the action changes as a function of the end positions (1.60) called the Hamilton-Jacobi equation. t) has the time derivative d A(qi (t). qi ] = pi (tb )δqi (tb ) − pi (ta )δqi (ta ).62) this implies ∂t A(qi . t).16) by δA[pi . To find cyclic coordinates we must search for a generating function F (qj . t) = −H(pi . (1. t) = −H(∂qi A(qi . A generating function which achieves this goal is supplied by the action functional (1.10 1 Fundamentals A particularly important canonical transformation arises by choosing a generating function F (qi . t). t).57). t). Kleinert.62).58) vanish identically. t). t): pi = ∂ A(qi . This leads to the following partial differential equation for F (qi .14) show that if a particle moves along a classical trajectory.64) If the momenta pi on the right-hand side are replaced according to (1. When following the classical solutions starting from a fixed initial point and running to all possible final points qi at a time t.63) Together with (1. t) is indeed seen to be a solution of the Hamilton-acobi differential equation: ∂t A(qi . qi (t). qi .61) From this we deduce immediately the first of the equations (1. Here and in the sequel we shall often use the short notations for partial derivatives ∂t ≡ ∂/∂t. t) which makes the transformed H ′ in (1. qi . Pj ) in such a way that it leads to time-independent momenta Pj ≡ αj . (1. Then all derivatives with respect to the coordinates vanish and the new momenta Pj are trivially constant. ∂qi ≡ ∂/∂qi . ∂qi (1. Pj . and the path is varied without keeping the endpoints fixed. Coordinates Qj with this property are called cyclic. t). Pj . t). (1. the function A(qi . A(qi . Expression (1.59) where the momentum variables in the Hamiltonian obey the first equation of (1. t): ∂t F (qi . Thus we seek a solution of the equation ∂ F (qi . ∂t (1. now for the generating function A(qi .57).62) Moreover. Pj .14). dt (1. t) = −H(∂qi F (qi . for instance S.fu-berlin. dt 2 (1.70) and the Christoffel symbol of the second kind3 ¯ µ ≡ g µσ Γ ¯ κνσ . Schouten. q) ˙ ≡ 1. . q¨µ + Γ (1.72) Then (1.69) ν For brevity. Kleinert. q) ˙ ≡ 1.2 Relativistic Mechanics in Curved Spacetime The classical action of a relativistic spinless point particle in a curved fourdimensional spacetime is usually written as an integral A = −Mc 2 Z dτ L(q. New York. gµν q¨ = 2 (1. Gauge Fields in Condensed Matter . World Scientific Publishing Co. q) ˙ 2L(q.8) reads " # d 1 1 gµν q˙ν = (∂µ gκλ ) q˙κ q˙λ . Springer. Vol. q) ˙ (1.11 1. in which case it coincides with the proper time of the particle.66) where τ is an arbitrary parameter of the trajectory. Wiley. Our notation follows J. 3 In many textbooks.69) can be written as Since the solutions of this equation minimize the length of a curve in spacetime. Singapore 1989. 1954. Γ 2 (1. For arbitrary τ . we have denoted partial derivatives ∂/∂q µ by ∂µ . (1. Berlin. II Stresses and Defects. pp. It can be chosen in the final trajectory to make L(q.67) If τ is the proper time where L(q. Gravitation and Cosmology. It will allow for a closer analogy with gauge fields in the construction of the Riemann tensor as a covariant curl of the Christoffel symbol in Chapter 10. they are called geodesics. dt L(q.. this simplifies to or 1 d (gµν q˙ν ) = (∂µ gκλ ) q˙κ q˙λ . At this point one introduces the Christoffel symbol ¯ λνµ ≡ 1 (∂λ gνµ + ∂ν gλµ − ∂µ gλν ). 744-1443 (http://www. See H.70) stand at the first position. the Euler-Lagrange equation (1.physik. Ricci Calculus. This partial derivative is supposed to apply only to the quantity right behind it.A.71) ¯ κλ µ q˙κ q˙λ = 0.68) 1 ∂µ gκλ − ∂λ gµκ q˙κ q˙λ . the upper index and the third index in (1. 1972.2 Relativistic Mechanics in Curved Spacetime 1.de/~kleinert/b2). q) ˙ = −Mc 2 Z q dτ gµν q˙µ (τ )q˙ν (τ ). Weinberg. Γ κν (1. pD ). Each scattering center. . t)|2 . (1. 1. called material waves. t) = eikx−iωt . all field strengths have to be added to the total field strength Ψ(x. one observes the number of particles arriving per unit time (see Fig. for example of electrons diffracted by a crystal. which would take place in classical mechanics. . The fact that waves cannot be squeezed into an arbitrarily small volume without increasing indefinitely their frequency and thus their energy.3. At the detector. If an electron beam of fixed momentum p passes through a crystal. .1 Bragg Reflections and Interference The most direct manifestation of the wave nature of small particles is seen in diffraction experiments on periodic structures.3 1 Fundamentals Quantum Mechanics Historically. is proportional to the number of electrons arriving at the detector. . In fact. becomes a source of a spherical wave with the spatial behavior eikR /R (with R ≡ |x − x′ | and k ≡ |k|) and the wavelength λ = 2π/k. It soon became clear that these phenomena reflect the fact that at a sufficiently short length scale. prevents the collapse of the electrons into the nucleus. |Ψ(x. The standard experiment where these rules can most simply be applied consists of an electron beam impinging vertically upon a flat screen with two parallel slits a with spacing d.1) .75) where k is the wave vector pointing into the direction of p and ω is the wave frequency. p2 . small material particles such as electrons behave like waves. xD ) (1. . (1. the extension of classical mechanics to quantum mechanics became necessary in order to understand the stability of atomic orbits and the discrete nature of atomic spectra.74) is associated with a plane wave. . They look very similar to the interference patterns of electromagnetic waves. say at x′ . A free particle moving with momentum p = (p1 . . The absolute square of the total field strength. whose field strength or wave function has the form Ψp (x. These are the well-known Bragg reflections. it is possible to use the same mathematical framework to explain these patterns as in electromagnetism. At a large distance R behind these.12 1. The discreteness of the atomic states of an electron are a manifestation of standing material waves in the atomic potential well. x2 . 1. it emerges along sharply peaked angles. .73) through a D-dimensional Euclidean space spanned by the Cartesian coordinate vectors x = (x1 . t). by analogy with the standing waves of electromagnetism in a cavity. . 2 1 1 1 dN ∝ |Ψ1 + Ψ2 |2 ≈ . . eik(R+ 2 d sin ϕ) + eik(R− 2 d sin ϕ) . . 2 , dt R (1.76) H. Kleinert, PATH INTEGRALS 13 1.3 Quantum Mechanics dN dt 2 1 1 . ∝ . eik(R+ 2 d sin ϕ) + eik(R− 2 d sin ϕ) . the wave function Ψ(x.1 Probability distribution of particle behind double slit. Its absolute square gives directly the probability density of the particle at the place x in space. where c is the light velocity .79) where the proportionality constant h ¯ is the same as in (1.3. R12 eikx Figure 1. It may be determined by an absorption process in which a light wave hits an electron and kicks it out of the surface of a metal. d3 x |Ψ(x. A similar relation holds between the energy and the frequency of the wave Ψ(x. The reason for this lies in the properties of electromagnetic waves. (1.e. the well-known photoelectric effect. t). Conventionally. On the one hand.78) h ¯≡ 2π (the number in parentheses indicating the experimental uncertainty of the last two digits before it).. t)|2 is the probability of finding the particle in the volume element d3 x around x.2 Matter Waves From the experimentally observed relation between the momentum and the size of the angular deflection ϕ of the diffracted beam of the particles.77). i. t) is normalized to describe a single particle. being proportional to the absolute square of the sum of the two complex field strengths.77) where h ¯ is the universal Planck constant whose dimension is equal to that of an action. From the threshold property of this effect one learns that an electromagnetic wave oscillating in time as e−iωt can transfer to the electron the energy E=h ¯ ω.0545919(80) × 10−27 erg sec (1. (1. where ϕ is the angle of deflection from the normal. their frequency ω and the wave vector k satisfy the relation ω/c = |k|. 1. h = 1. one deduces the relation between momentum and wave vector p=h ¯ k. and for massless particles such as photons Ep = c|p|. the photons. The limiting f (p) is concentrated at a specific mo¯ . a particle wave is some superposition of plane waves (1. t) ≡ ψ ∗ (x. t) ∇ ψ(x. i. All free particles of momentum p are described by a plane wave of wavelength λ = 2π/|k| = 2π¯h/|p|.84) With an appropriate choice of f (p) it is possible to prepare Ψ(x.82) The energy Ep depends on the momentum of the particle in the classical way. t) ∇ ψ(x. PATH INTEGRALS . 0) may be a function sharply centered around a space point x ¯.458 km/s. (1. Usually.. if the particle amplitude is spread out in space. both momentum and energy of a particle are not sharply defined as in the plane-wave function (1. Conversely.80). the current density of the particle probability j(x.77) and the constant h ¯ must be the same as in Eq. (1. the quanta of electromagnetic waves. In general. t)∇ψ(x. t) ∇ ψ(x. t) ≡ −i ↔ h ¯ ∗ ψ (x. Then f (p) is approximately a pure phase −ip¯ x/¯ h f (p) ∼ e . (1. t) 2m (1. With matter waves and photons sharing the same relations (1. it is suggestive to postulate also the relation (1. t)] ψ(x. t) ≡ ψ ∗ (x.and backward-derivatives ↔ → ← ψ ∗ (x. t) ∇ ψ(x. where ∇ is a short notation for the difference between forward. Thus. To avoid this.80) Z d3 p Ψ(x. and the wave contains all momenta with equal probability.80) where N is some normalization constant. say at t = 0. (1. t) − ψ ∗ (x. f (p) can be calculated via the integral f (p) = Z d3 x e−ipx/¯h Ψ(x. In a finite volume.77). this normalization makes the wave function vanish. Kleinert.79). with the explicit form Ψp (x. massive and massless ones. with mentum p the amplitude oscillating like Ψ(x. The common relation Ep = h ¯ ω for photons and matter waves is necessary to guarantee conservation of energy in quantum mechanics. energy and momentum are related by E/c = |p|.79) between energy and frequency to be universal for the waves of all particles. certainly satisfy (1. For example.81) ↔ is normalized in some convenient way. Ψ(x. t) − [∇ψ ∗ (x. In an infinite volume.e. 0). t) in any desired form at some initial time. its momentum distribution is confined to a small region. t) = N ei(px−Ep t)/¯h . t). H. t) = f (p)ei(px−Ep t)/¯h . (1. t) ∼ ei(¯px−Ep¯ t)/¯h .14 1 Fundamentals defined to be c ≡ 299 792. for nonrelativistic material particles of mass M it is Ep = p2 /2M.83) 3 (2π¯h) By the Fourier inversion theorem. On the other hand. The particle is found at each point in space with equal probability. the wave function is normalized to unity. for relativistic √ 2 ones Ep = c p + M 2 c2 . and thus the energy Ep .88) may be written as the differential equation [H(−i¯h∇) − i¯h∂t )]Ψ(x.3. If the wave is localized in a finite region of space while having at the same time a fairly well-defined average ¯ .83) to move with a velocity ¯ = ∂Ep¯ /∂ p ¯.91) ˆ is the differential operator where H ˆ ≡ H(−i¯h∇. H (1. p Eˆ = i¯h∂t (1.90) ˆ − i¯h∂t Ψ(x. The equation suggests that the motion of a particle with an arbitrary Hamiltonian H(p. (1. t).86) ¯. are given by H(p) = Ep = p2 . Eq.87) We may therefore derive the following identity for the wave field Ψp (x. (1.90) This is the Schr¨odinger equation for the wave function of a free particle. t) = 0. Then. H (1. v (1. This coincides with the velocity of a classical particle of momentum p 1.3 Quantum Mechanics In general.89) outside. (1. the width of Ψ(x. 0) in space and of f (p) in momentum space are inversely proportional to each other: ∆x ∆p ∼ h ¯.15 1. t) follows the straightforward generalization of (1.85) This is the content of Heisenberg’s principle of uncertainty. The classical Hamiltonian. t): Z d3 p f (p) [H(p) − Ep ] ei(px−Ep t)/¯h = 0. it is called a wave packet. The maximum in the associated probability momentum p density can be shown from (1. 2M (1.88) The arguments inside the brackets can be removed from the integral by observing that p and Ep inside the integral are equivalent to the differential operators ˆ = −i¯h∇. t) = 0.92) . x. x. 3 (2π¯h) (1.3 Schr¨ odinger Equation Suppose now that the particle is nonrelativistic and has a mass M. 4 We shall see in p→p Sections 1.. t) = d x [HΨ 3 Z ˆ 1 (x.93) The linear combinations of the solutions of the Schr¨odinger equation (1.5 i. Berlin. if it satisfies for arbitrary wave functions Ψ1 . H. xj ] = −i¯h. this is assured if and only if the Hamiltonian operator is Hermitian. 1932). Neumann.16 1 Fundamentals ˆ from the classical Hamiltonian H(p. Ψ2 the equality Z ˆ 2 (x. (1. t)|2 . d3 x Ψ∗2 (x. a Hilbert space. If the Hamiltonian does not depend explicitly on time. at each time t.15 that this simple correspondence principle holds only in Cartesian coordinates. t)HΨ (1. the Hilbert space can be spanned by the energy eigenstates ΨEn (x. x) = Z d3 x |Ψ(x. Kleinert.91) form. Since the square of the wave function. If Ψ(x.91). (1.97) 4 Our formulation of this principle is slightly stronger than the historical one used in the initial phase of quantum mechanics. where M is the reduced mass of the two particles.96) For a stable particle. ˆ t] = 0 = i¯h. (1. where ΨEn (x) are time-independent stationary states. the integral over the entire volume must be normalized to unity: H(p. Some quantum-mechanical operator subtleties will manifest themselves in this book as problems of path integration to be solved in Chapter 12. this normalization must remain the same at all times. Mathematische Grundlagen der Quantenmechanik . which solve the time-independent Schr¨odinger equation ˆ p. such as restrictions of the domains of definition. t) = e−iEn t/¯h ΨEn (x). x. The Schr¨odinger operators (1. The substitution rule for the momentum runs also under the name Jordan rule.13–1. If the Hamiltonian does not depend explicitly on time. |Ψ(x. t). 5 Problems arising from unboundedness or discontinuities of the Hamiltonian and other quantum-mechanical operators. t)]∗ Ψ1 (x. is interpreted as the probability density of a single particle in a finite volume.95) 2M r which governs the quantum mechanics of the hydrogen atom in the center-of-mass coordinate system of electron and proton. t) by the substitution The rule of obtaining H ˆ = −i¯h∇ will be referred to as the correspondence principle. which gave certain translation rules between classical and quantummechanical relations.91) is confirmed by experiment. (1. [E.89) of momentum and energy satisfy with x and t the so-called canonical commutation relations [ˆ pi .e. are ignored here since they are well understood. v. PATH INTEGRALS . The precise relationship between the two calls for further detailed investigations.94) H(ˆ The validity of the Schr¨odinger theory (1. Correspondingly we do not distinguish between Hermitian and selfadjoint operators (see J. x)ΨEn (x) = En ΨEn (x). most notably for the Coulomb Hamiltonian p2 e2 − . t) is to follow the Schr¨odinger equation (1. Springer. t)|2 = 1. unless otherwise stated. Before presenting this in Chapter 10 we shall avoid ambiguities by assuming H(p. there seem to be. H (1. ˆ is not Hermitian. t) − d3 x Ψ∗2 (x. An operator H † ˆ : with its Hermitian-adjoint H ˆ =H ˆ †. t)Ψ1 (x.91). if it coincides for all wave functions Ψ1 (x.101) This is always the case for nonrelativistic particles in Cartesian coordinates x. If the ordering problem is caused by the geometry of the space in which the particle moves. t) x Ψ∗2 (x. several Hermitian operators which. t)|2 = 1. Ψ2 (x.100) ˆ 2 (x. t). in the above examˆ and two x ˆ operators [for instance ple. If p and x appear in one and the same term of H. t). t)]∗ Ψ1 (x. With the Schr¨odinger equation (1.98) ˆ is Hermitian.17 1. H Since p be a Hermitian operator if it is a sum of a kinetic and a potential energy: H(p. d3 x [HΨ This also implies the time independence of the normalization integral R 3 d x |Ψ(x. the corresponˆ Then dence principle does not lead to a unique quantum-mechanical operator H. t) to have the standard form (1. if H † norm changes with time: any eigenstate of (H − H )/i has this property. this time change vanˆ is Hermitian: ishes indeed as long as H i¯h d dt = Z Z d3 x Ψ∗2 (x. t) + V (x. This is referred to as the operator-ordering problem of quantum mechanics which has plagued many researchers in the past. They all correspond to the same classical αˆ p2 x p2 x2 . t)]∗ Ψ1 (x. t)HΨ Z (1. (1. in principle. t) = 0. t)Ψ1 (x.3 Quantum Mechanics ˆ † of the operator H. t). t). Z 3 d x Ψ∗2 (x. t) = T (p. At first sight it appears as though only a comparison with experiment could select the correct operator ordering. ˆ will automatically ˆ = −i¯h∇ and x are themselves Hermitian operators. x.101). one can always find an eigenstate of H ˆ whose Conversely. x. . t) ˆ 1 (x. ˆ which satThe left-hand side defines the Hermitian-adjoint H isfies the identity Z d 3 ˆ † Ψ1 (x. there exists a surprisingly simple geometric principle which specifies the ordering in the physically correct way.99) Let us calculate the time change of the integral over two arbitrary wave functions. for instance as p2 x2 . t) d3 x [HΨ (1. can be constructed from the product of two p ˆ 2 +β x ˆ2p ˆ 2 +γ p ˆx ˆ 2p ˆ with α+β+γ = 1]. t)H ≡ Z ˆ 2 (x. 105) where dS are the directed infinitesimal surface elements. t) (1.103) ∂t ρ(x. An obvious contact with the usual vector notation may be established. 3. t) = − Z S dS · j(x. . N. t) as a subscript: Ψ(x. (1.105) over the entire space and assuming the currents to vanish at spatial infinity. The configuration space variable x plays the role of a continuous “index” of these vectors. xν .106) H(pν . The Hamiltonian has the form N X p2ν + V (xν . t) 2m (1. t) = ν=1 2Mν where the arguments pν . t). More general dynamical systems with N particles in Euclidean space are parametrized in terms of 3N Cartesian coordinates xν (ν = 1. By extending the integral (1. we recover the conservation of the total probability (1. . t) satisfies the N-particle Schr¨odinger equation ( − 1. Kleinert. t) may be considered as a vector in an infinite-dimensional complex vector space called Hilbert space.18 1. t) = i¯h∂t Ψ(xν . This equation states that the probability in a volume decreases by the same amount by which probability leaves the surface via the current j(x. (1. xν with ν = 1. 2. (1. . PATH INTEGRALS . one finds Z V d3 x ∂t ρ(x. .102) with the probability density ρ(x. . the wave function Ψ(x. by writing the argument x of Ψ(x.3. t) ≡ −i ↔ h ¯ ψ(x. (1. .96) is a consequence of a more general local conservation law linking the current density of the particle probability j(x. . in which a D-dimensional vector v is given in terms of its components vi with a subscript i = 1. (1. xν in H and V stand for all pν ’s. t). t) 2Mν ν #) Ψ(xν .107) Dirac’s Bra-Ket Formalism Mathematically speaking.108) H. . t) = −∇ · j(x.104) via the relation By integrating this current conservation law over a volume V enclosed by a surface S. t) = ψ ∗ (x. t). and using Green’s theorem. t). .96). t) ∇ ψ(x.4 N X ν=1 " h ¯2 ∂x 2 + V (xν . N). . D. . t) = − Z V d3 x ∇ · j(x.4 1 Fundamentals Particle Current Conservation The conservation of the total probability (1. t) ≡ Ψx (t). The wave function Ψ(xν . t). t)ψ(x. . . . . 1. except in a cube of size ǫ3 centered around xn . n1.4 Dirac’s Bra-Ket Formalism The usual norm of a complex vector is defined by |v|2 = X vi∗ vi . . . . t). 2. The normalization condition (1. .96) requires that the wave functions have the norm |Ψ| = 1. b.114) a In the space of wave functions (1. j. (1. 7 An orthogonality relation implies usually a unit norm and is thus really an orthonormality relation but this name is rarely used. using for the initial components i. .113) i which in a finite-dimensional vector space implies the completeness relation X bi a∗ bj a = δ ij . hn (x) = (1.109) Z (1. .112) i The latter equation is a consequence of the orthogonality relation 7 X ′ ′ bi a∗ bi a = δ aa . . but physicists prefer to shorten the notation by distinguishing the different components via different types of subscripts.2. 3.1 Basis Transformations In a vector space. . ±2. It may be constructed in the following fashion: Imagine the continuum of space points to be coarse-grained into a cubic lattice of mesh size ǫ.110) i The continuous version of this is 2 |Ψ| = Z d 3 x Ψ∗x (t)Ψx (t) = d3 x Ψ∗ (x. i. c. n2 . in terms of which6 vi = X bi a va . . ( √ 1/ ǫ3 |xi − xn i | ≤ ǫ/2.e..116) 0 otherwise.108) there exists a special set of basis functions called local basis functions of particular importance..111) a with the components va given by the scalar products va ≡ X bi a∗ vi . . (1. there are many possible choices of orthonormal basis vectors bi a labeled by a = 1. at positions xn = (n1 . (1.3 = 0. (1.4. and for the b-transformed components a. i = 1.e. n3 )ǫ. ±1. (1. P (b) Mathematicians would expand more precisely vi = a bi a va . (1. 6 .19 1. t)Ψ(x. for each component xi of x. that they are unit vectors in the Hilbert space. D. i. k. .115) Let hn (x) be a function that vanishes everywhere in space. 20 1 Fundamentals These functions are certainly orthonormal: Z ′ ′ (1. if Ψ(x. (1. The same convergence of discrete approximations is found in any scalar product. (1.119) in the limit ǫ → 0. there are many possible orthonormal basis functions f a (x) in the Hilbert space which satisfy the orthonormality relation Z ′ ′ d3 x f a (x)∗ f a (x) = δ aa . to re-expand Ψ(x. In fact. (1.121) shows that the components are related to each other by ˜ b (t) = Ψ X Z a d3 x f˜b (x)∗ f a (x) Ψa (t).122) Suppose we use another orthonormal basis f˜b (x) with the orthonormality relation Z ′ ′ d3 x f˜b (x)∗ f˜b (x) = δ bb . (1. t).118) will always converge to Ψ(x. t) varies. t) ≈ √ ǫ3 Ψ(xn . t) = f a (x)Ψa (t). (1.119) It provides an excellent approximation to the true wave function Ψ(x. t). the integral over the sum (1.120) in terms of which we can expand X Ψ(x. Consider now the expansion Ψ(x. t) = X n with the coefficients Ψn (t) = Z d3 x hn (x)∗ Ψ(x. t) is integrable.123) ˜ b (t).121) a with the coefficients Ψa (t) = Z d3 x f a (x)∗ Ψ(x. t). as long as the mesh size ǫ is much smaller than the scale over which Ψ(x. The functions hn (x) may therefore be used as an approximate basis in the same way as the previous basis functions f a (x). t) = (1.124) d3 x f˜b (x)∗ Ψ(x. In general. They can all be calculated with arbitrary accuracy knowing the discrete components of the type (1. t). PATH INTEGRALS . t). X b f˜b (x)f˜b (x′ )∗ = δ (3) (x − x′ ). Kleinert. and thus in any observable probability amplitudes.126) H.125) X b with the components ˜ b (t) = Ψ Z Inserting (1. with any desired accuracy depending on the choice of ǫ.118) d3 x hn (x)∗ hn (x) = δ nn . f˜b (x)Ψ (1.117) hn (x)Ψn (t) (1. (1. g b (x). (1. It may be referred to as completeness relation `a la Dirac. Dirac called the formal objects ha| and |a′ i.126) takes the form h˜b|Ψ(t)i = X a h˜b|aiha|Ψ(t)i. (1.117) in bracket notation as hxn |xn′ i ≡ Z ′ d3 x hn (x)∗ hn (x) = δnn′ . respectively.116). The components Ψn (t) may be considered as the scalar products √ Ψn (t) ≡ hxn |Ψ(t)i ≈ ǫ3 Ψ(xn .133) (1.129) The right-hand side of this equation may be formally viewed as a result of inserting the abstract relation X |aiha| = 1 (1. (1. ′ ′ d x f˜b (x)∗ f˜b (x) = δ bb . t). the relation (1. In the bracket notation. the orthonormality of the basis f a (x) and g b (x) may be expressed as follows: ′ ha|a i = h˜b|˜b′ i = Z Z ′ ′ d3 x f a (x)∗ f a (x) = δ aa . Since the scalar products are written in the form of brackets ha|a′ i.127) In this notation.131) Since this expansion is only possible if the functions f b (x) form a complete basis. (1. (1.21 1. abstract way of stating the completeness of the basis functions.134) . Ψ The transformation formula (1.132) 3 In the same spirit we introduce abstract bra and ket vectors associated with the basis functions hn (x) of Eq. the components of the state vector Ψ(x.125) are Ψa (t) = ha|Ψ(t)i.130) a between h˜b| and |Ψ(t)i on the left-hand side: h˜b|Ψ(t)i = h˜b|1|Ψ(t)i = X a h˜b|aiha|Ψ(t)i.128) ˜ b (t) = h˜b|Ψ(t)i. from which the brackets are composed. denoting them by hxn | and |xn i.2 Bracket Notation It is useful to write the scalar products between two wave functions occurring in the above basis transformations in the so-called bracket notation as h˜b|ai ≡ Z d3 x f˜b (x)∗ f a (x). and writing the orthogonality relation (1.4 Dirac’s Bra-Ket Formalism 1. t) in (1.130) is alternative. (1. bra and ket. (1.122). respectively. (1.4. (1.130).140) The completeness relation can be used to write ha|Ψ(t)i ≈ X ha|xn ihxn |Ψ(t)i ≈ X ǫ3 ha|xihx|Ψ(t)i. ǫ3 (1. finer and finer sums are eventually replaced by integrals. t).138) n The approximate sign turns into an equality sign in the limit of zero mesh size. (1.137) The completeness of the basis hn (x) may therefore be expressed via the abstract relation X |xn ihxn | ≈ 1. The same thing is done here.136) This is.4.139) where xn are the lattice points closest to x. x and xn coincide and we have hx|Ψ(t)i ≡ Ψ(x. In the limit ǫ → 0. of course.22 1 Fundamentals Changes of basis vectors. We define new continuous scalar products 1 hx|Ψ(t)i ≈ √ hxn |Ψ(t)i. ǫ → 0.135) Also the inverse relation is true: ha|Ψ(t)i = X n ha|xn ihxn |Ψ(t)i. can be performed according to the rules developed above by inserting a completeness relation a` la Dirac of the type (1. t). the right-hand side is equal to Ψ(xn .134). (1. 1. With (1. (1. for instance from |xn i to the states |ai. Thus we may expand Ψn (t) = hxn |Ψ(t)i = X a hxn |aiha|Ψ(t)i. just an approximation to the integral Z d3 x hn (x)∗ hx|Ψ(t)i.3 Continuum Limit In ordinary calculus. (1. n n which in the limit ǫ → 0 becomes ha|Ψ(t)i = Z . . (1. . PATH INTEGRALS .142) This may be viewed as the result of inserting the formal completeness relation of the limiting local bra and ket basis vectors hx| and |xi.143) H. (1.141) x=xn d3 x ha|xihx|Ψ(t)i. Kleinert. Z d3 x |xihx| = 1. (1. (1.145) a resulting in the expansion |Ψ(t)i = X a |aiha|Ψ(t)i. t) = hx|Ψ(t)i plays the role of an xth component of the state vector |Ψ(t)i in the local basis |xi.150) |xi ≈ √ |xn i . the expansion ha|Ψ(t)i = Z d3 x ha|xihx|Ψ(t)i (1. it has become customary to deal with an arbitrary physical state vector in a basis-independent way and denote it by a ket vector |Ψ(t)i. in turn. from the left to obtain the expansion formula (1.146) This can be multiplied with any bra vector.129). (1.148) and leads to the expansion |Ψ(t)i = Z d3 x |xihx|Ψ(t)i. d3 x f˜b (x)∗ f a (x) (1. In fact. A vector can be described equally well in bra or in ket form. (1.144) may be viewed as a re-expansion of a component of |Ψ(t)i in one basis. ǫ3 with xn being the lattice points closest to x. (1.151) .138) reads Z d3 x |xihx| = 1. into those of another basis. This vector may be specified in any convenient basis by multiplying it with the corresponding completeness relation X |aiha| = 1.149) in which the wave function Ψ(x. (1. In order to express all these transformation properties in a most compact notation. To apply the above formalism consistently. |xi.4 Dirac’s Bra-Ket Formalism evaluated between the vectors ha| and |Ψ(t)i. 1 (1. With the limiting local basis. we observe that the scalar products ha|˜bi = h˜b|ai = Z Z d3 x f a (x)∗ f˜b (x). the wave functions can be treated as components of the state vectors |Ψ(t)i with respect to the local basis |xi in the same way as any other set of components in an arbitrary basis |ai. |ai. just as in (1. say hb|. is the limit of the discrete basis vectors |xn i.147) The continuum version of the completeness relation (1.23 1. This.131): hb|Ψ(t)i = X a hb|aiha|Ψ(t)i. one inconsistency with some fundamental postulates of quantum mechanics: When introducing state vectors. For x 6= x′ . the norm was required to be unity in order to permit a proper probability interpretation of single-particle states. one admits all states which can be obtained by a limiting sequence from properly normalized state vectors. (1. Kleinert. when expanding a ket vector as |Ψ(t)i = X |aiha|Ψ(t)i. (1.153) hΨ(t)| = X hΨ(t)|aiha|. n n ǫ3 ǫ3 (1. the states are orthogonal. (1. The scalar product between states hx|x′ i is not a proper function.96) by admitting the limiting states |xi to the physical Hilbert space. the limit ǫ → 0 is infinite. In fact.154) a or a bra vector as a a multiplication of the first equation with the bra hx| and of the second with the ket |xi produces equations which are complex-conjugate to each other. PATH INTEGRALS . ǫ3 n′ Therefore. it is useful to weaken the requirement of normalizability (1. however.157) The right-hand side vanishes everywhere.156) δnn′ = 1. In fact. it is so large that its volume integral is unity: Z d3 x′ δ (3) (x − x′ ) = 1. (1. the scalar product between two different states hx| and |x′ i is hx|x′ i ≈ 1 1 ′i = hx |x δnn′ .158) At x = x′ .159) H. on the other hand. Thus the δ-function satisfies δ (3) (x − x′ ) = 0 for x 6= x′ . It is denoted by the symbol δ (3) (x − x′ ) and called Dirac δ-function: hx|x′ i ≡ δ (3) (x − x′ ). the limiting state |xi is not a properly normalizable vector in the Hilbert space.152) Therefore. For x = x′ .24 1 Fundamentals satisfy the identity h˜b|ai ≡ ha|˜bi∗ . approached in such a way that X 1 ǫ3 (1. (1. 1.4.155) where xn and xn′ are the lattice points closest to x and x′ . As far as the vectors |xi are concerned there is.4 Generalized Functions Dirac’s bra-ket formalism is elegant and easy to handle. (1. except in the infinitely small box of width ǫ around x ≈ x′ . For the sake of elegance. The limiting states |xi introduced above do not satisfy this requirement. 155) to the δ-function are proper functions. hx|ˆ x ≡ xhx|. although they are not normalizable. (1. the Schr¨odinger equation can be expressed in a basis-independent way as an operator equation ˆ ˆ . If we want to achieve this for the path integral formulation of quantum mechanics.159) implies the following property of the δ function: 1 (3) δ (3) (a(x − x′ )) = δ (x − x′ ).4 Dirac’s Bra-Ket Formalism Obviously. the scalar product hx|x′ i behaves just like the states |xi themselves: Both are ǫ → 0 -limits of properly defined mathematical objects. (1. t)|Ψ(t)i = i¯h∂t |Ψ(t)i. the role of the test functions is played by the wave packets Ψ(x. In this respect. An important requirement of quantum mechanics is coordinate invariance. In quantum mechanics. there exists no proper function that can satisfy both requirements. H|Ψ(t)i ≡ H(ˆ p. and thus all distributions. In mathematics.159).162) Test functions are arbitrarily often differentiable functions with a sufficiently fast falloff at spatial infinity.5 Schr¨ odinger Equation in Dirac Notation In terms of the bra-ket notation. They form a linear space.160) |a| In one dimension. By admitting the generalized states |xi to the Hilbert space.164) (1. Only the finite-ǫ approximation in (1. one calls the δ-function a generalized function or a distribution. x (1. however.8. we must set up a definite extension of the existing theory of distributions. that physics forces us to go beyond these rules. (1.4.1 we shall find.158) and (1.165) . this leads to the more general relation δ(f (x)) = X i 1 |f ′ (xi )| δ(x − xi ).163) to be supplemented by the following specifications of the canonical operators: hx|ˆ p ≡ −i¯h∇hx|.161) where xi are the simple zeros of f (x). 1. There exist a rich body of mathematical literature on distributions [3]. (1. we also admit the scalar products hx|x′ i to the space of wave functions. t). x] ≡ Z d3 x δ (3) (x − x′ )f (x′ ) = f (x). This space is restricted in an essential way in comparison with ordinary functions: products of δ-functions or any other distributions remain undefined. which specifies uniquely integrals over products of distributions. (1.25 1. (1. Note that the integral Eq. It defines a linear functional of arbitrary smooth test functions f (x) which yields its value at any desired place x: δ[f . In Section 10. (1.101). such an operator is called Hermitian if all its matrix elements have this property. (1. for instance with the local basis vector |x′ i: hx|ˆ p|x′ i = −i¯h∇hx|x′ i = −i¯h∇δ (3) (x − x′ ). t.168) ˆ and x ˆ are Hermitian matrices in any basis. p ha|ˆ p|a′ i = ha′ |ˆ p|ai∗ . The most general basis-independent operator that can be constructed in the ˆ.169)–(1. PATH INTEGRALS . (1. H|E (1. x. x (1. generalized Hilbert space spanned by the states |xi is some function of p ˆ ≡ O(ˆ ˆ . ha|H|a (1. H The stationary states in Eq.171) in the basis-independent form ˆ = p ˆ †.26 1 Fundamentals Any matrix element can be obtained from these equations by multiplication from the right with an arbitrary ket vector. x ˆ . Obviously.175) H. x (1.97) of a ˆ †(t) implies the equality of the matrix elements Hermitian-adjoint operator O ∗ ˆ †(t)|a′ i ≡ ha′ |O(t)|ai ˆ ha|O . O(t) p. (1.172) In general.170) ˆ ′ i = ha′ |H|ai ˆ ∗.163) with the bra vector hx| from the left: ˆ . (1.174) ˆ = H ˆ †.167) The original differential form of the Schr¨odinger equation (1. Kleinert. (1. p ˆ = x ˆ†.173) Thus we can rephrase Eqs. (1. t)hx|Ψ(t)i hx|H(ˆ p. and satisfy the time-independent operator equation ˆ n i = En |En i. the definition (1. In the basis-independent Dirac notation.171) and so is the Hamiltonian as long as it has the form (1. t).169) ha|ˆ x|a′ i = ha′ |ˆ x|ai∗ . x = i¯h∂t hx|Ψ(t)i.166) hx|ˆ x|x′ i = xhx|x′ i = xδ (3) (x − x′ ). (1. t)|Ψ(t)i = H(−i¯h∇.94) have a Dirac ket representation |En i.91) follows by multiplying the basis-independent Schr¨odinger equation (1. L mi = 0. p (1.185) |pi ≈ L3 |pm i. Its eigenstates are given by the eigenvalue Let us now look at the momentum p equation ˆ |pi = p|pi. (2π¯h)3 (1. m3 ).179) Then we adjust the factor in front of exp (ipm x/¯h) to achieve unit normalization 1 hx|pm i = √ exp (ipm x/¯h) .4 Dirac’s Bra-Ket Formalism 1.75) to describe free particles of momentum p. .178) Up to a normalization factor. L3 (1. (1. In order for the states |pi to have a finite norm. . the sum over the discrete momenta pm can be approximated by an integral over the momentum space [4]. the system must be confined to a finite volume. the states |pm i may be used to define a continuum of basis vectors with an improper normalization √ (1. this is just a plane wave introduced before in Eq. .181) The states |pm i are complete: X m |pm ihpm | = 1. .164).184) In this limit. m2 .177) The solution is hx|pi ∝ eipx/¯h . Assuming periodic boundary conditions. say a cubic box of length L and volume L3 .180) and the discrete states |pm i satisfy Z d3 x |hx|pm i|2 = 1. (1. X m ≈ Z d3 pL3 . the momenta are discrete with values pm = 2π¯h (m1 . (1.183) If the box is very large. we find the differential equation hx|ˆ p|pi = −i¯h∂x hx|pi = phx|pi. t) = hx|Ψ(t)i = X m hx|pm ihpm |Ψ(t)i.27 1. ±2.6 Momentum States ˆ . . (1. (1.4.176) By multiplying this with hx| from the left and using (1. (1.182) We may use this relation and the matrix elements hx|pm i to expand any wave function within the box as Ψ(x. ±1. (1. t).186).186): hp|p′ i = = Z Z d3 x hp|xihx|p′ i ′ (1.192) between the momentum states on the left-hand side of the orthogonality relation (1.28 1 Fundamentals √ in the same way as |xn i was used in (1.84).188) with the momentum eigenfunctions hx|pi = eipx/¯h . Kleinert. with the identification hp|Ψ(t)i = f (p)e−iEp t/¯h . The amplitudes hp|Ψ(t)i are referred to as momentum space wave functions. t) = d3 p hx|pihp|Ψ(t)i. t) in (1. PATH INTEGRALS .190) The bra-ket formalism accommodates naturally the technique of Fourier transforms.191) d3 x e−ipx/¯h Ψ(x.150) to define |xi ∼ (1/ ǫ3 )|xn i. (1. (2π¯h)3 Z (1. we obtain the Fourier representation of the δ-function (1. (1.187) such that the expansion (1. Their completeness relation reads Z d3 p |pihp| = 1. The momentum states |pi satisfy the orthogonality relation hp|p′ i = (2π¯h)3 δ (3) (p − p′ ). (2π¯h)3 (1. (1. By inserting the completeness relation Z d3 x|xihx| = 1 (1.183) becomes Ψ(x. H. (1.189) This coincides precisely with the Fourier decomposition introduced above in the description of a general particle wave Ψ(x.83).186) with δ (3) (p−p′ ) being again the Dirac δ-function. The Fourier inversion formula is found by simply inserting into hp|Ψ(t)i a R 3 completeness relation d x|xihx| = 1 which yields hp|Ψ(t)i = = Z Z d3 x hp|xihx|Ψ(t)i (1.193) d3 x e−i(p−p )x/¯h = (2π¯h)3 δ (3) (p − p′ ). (1. + eN ·2πiµ + c. (1. 2πiµ 1−e sin πµ This function is well known in wave optics (see Fig. (1.29 1. It has large peaks at .160): ∞ X 2π¯h 2π¯hm (p−p′ )a .c. These are precisely the Fourier coefficients on the left-hand side. n=−N 1 − e2πiµ(N +1) = −1 + + c. .198) −m = δ p−p′ − hp|xn ihxn |p i = δ 2π¯h a m=−∞ a n=−∞ m=−∞ ∞ X ′ ∞ X ! ! In order to prove the Poisson formula (1. the sum over n on the left-hand side of (1.194) restricted to a sum over equally spaced points xn = na: N X n=−N |xn ihxn |. = . 1 − e2πiµ = 1+ ! (1. we observe that the sum s(µ) ≡ side is periodic in µ with a unit period and has m δ(µ − m) on the right-hand P 2πiµn the Fourier series s(µ) = ∞ .196) n=−N For N → ∞ we can perform the sum with the help of Poisson’s summation formula ∞ X e2πiµn = n=−∞ ∞ X m=−∞ δ(µ − m). (1.197) Identifying µ with (p − p′ )a/2π¯h.197) yields P N X e2πiµn = 1 + e2πiµ + e2·2πiµ + . . we find using Eq. Z dx |xihx| = 1. The Fourier coefficients are given by n=−∞ sn e R 1/2 −2πiµn sn = −1/2 dµ s(µ)e ≡ 1.7 Incompleteness and Poisson’s Summation Formula For many physical applications it is important to find out what happens to the completeness relation (1.199) sin πµ(2N + 1) e2πiµ − e2πiµ(N +1) + c.148) if one restrict the integral so a subset of positions. (1.197).4). For a finite N.c. Most relevant will be the one-dimensional integral.4. It determines the diffraction pattern of light behind a grating with 2N + 1 slits. we obtain N X n=−N hp|xn ihxn |p′ i = N X n=−N hp|xn ihxn |p′ i = N X ′ ei(p−p )na/¯h (1.195) Taking this sum between momentum eigenstates |pi. 2.c.4 Dirac’s Bra-Ket Formalism 1. and N − 1 small maxima between each pair of neighboring peaks. sin πµ sin πµ −n/2N (1. .197) in the limit N → ∞. N − 1. In this limit. .2 Relevant function N n=−N e limit N → ∞. Kleinert. Inserting µ = (p − p′ )a/2π¯h into (1. . PATH INTEGRALS . at ν = (1 + 4k)/2(2N + 1) for k = 1. . P µ = 0. N − 1. . . for N → ∞ at fixed n/N.200) Let us see how the right-hand side of (1.201) Keeping a fixed ratio n/N ≪ 1.202) + π −πn x 2πN −πn Z n/2N Z dµ Z where we have used the integral formula Z ∞ −∞ dx sin x = π.30 1 Fundamentals 2πiµn in Poisson’s summation formula. ±3. ±1. . the area under each large peak can be calculated by an integral over the central large peak plus a number n of small maxima next to it: Z n/2N −n/2N dµ sin πµ(2N + 1) Z n/2N sin 2πµN cos πµ+cos 2πµN sin πµ = dµ .203) H. we may replace in the integrand sin πµ by πµ and cos πµ by 1. Then the integral becomes. . we obtain N X n=−N hp|xn ihxn |p′ i = sin (p − p′ )a(2N + 1)/2¯h . . x (1. In the Figure 1. µ is squeezed to the integer values. ±2. sin (p − p′ )a/2¯h (1. . There are zeros at ν = (1 + 2k)/(2N + 1) for k = 1.199). n/2N sin 2πµN sin πµ(2N + 1) N →∞ n/2N dµ dµ cos 2πµN − −−→ + sin πµ πµ −n/2N −n/2N −n/2N Z πn Z πn N →∞ 1 N →∞ sin x 1 − −−→ dx dx cos x − −−→ 1. (1.199) turns into the right-hand side of (1. . (1. i.205).208) . This will be done for the Hamiltonian operator in Chapter 8. Consider an arbitrary smooth function f (µ) which possesses a convergent sum ∞ X f (m). There exists another useful way of expressing Poisson’s formula. Among the eigenvectors. that there is no ordering problem of the type discussed in context with the Hamiltonian (1. Important examples for such an A are p and x themselves.205) n=−∞ The auxiliary sum over n squeezes µ to the integer numbers. converting them into observable operators.206) This replacement rule is the extension of the correspondence principle for the Hamiltonian operator (1. Quantum-mechanically.197) and thus (1. Let A = A(p. It is obtained by simply replacing the variables p and x in ˆ and x ˆ: A by the corresponding operators p ˆ ). It must be assumed that the replacement leads to a unique Hermitian operator. there will be an observable operator associated with each such quantity. and the angular momentum L = x × p.8 If there are ambiguities. 1. the Hamiltonian H(p. there is always a choice of orthonormal vectors |ai fulfilling the completeness relation X a 8 |aiha| = 1.e.197) implies that the sum can be rewritten as an integral and an auxiliary sum: ∞ X m=−∞ f (m) = Z ∞ −∞ dµ ∞ X e2πiµn f (µ). unless it can be specified by means of simple geometric principles. the naive correspondence principle is insufficient to determine the observable operator. x (1. Aˆ ≡ A(ˆ p.. Then the correct ordering must be decided by comparison with experiment. we find indeed (1.204) m=−∞ Then Poisson’s formula (1. x). (1.207) can be used as a basis to span the Hilbert space.5 Observables Changes of basis vectors are an important tool in analyzing the physically observable content of a wave vector. (1.92) to more general functions in phase space. Note that this is true for the angular momentum L = x × p. x) be an arbitrary time-independent real function of the phase space variables p and x. it has the useful property that the set of all eigenvectors |ai obtained by solving the equation ˆ = a|ai A|ai (1.5 Observables In the limit N → ∞.31 1.101). Once an observable operator Aˆ is Hermitian. t) itself is an example of this interpretation. operator x The expectation value of the observable operator (1.32 1 Fundamentals The vectors |ai can be used to extract physical information concerning the observable A from arbitrary state vector |Ψ(t)i. [ˆ xi . t) = hx|Ψ(t)i. (1. but the components satisfy the canonical commutation rules [ˆ pi . due to the properties of Fourier analysis. PATH INTEGRALS .206) in the state |Ψ(t)i is defined as the matrix element ˆ hΨ(t)|A|Ψ(t)i ≡ 1. (1.83).212) Uncertainty Relation We have seen before [see the discussion after (1.215) This follows from the expansion |Ψ(t)i = X a |aiha|Ψ(t)i. (1. The wave function Ψ(x. pˆj ] = 0.213) From the Hilbert space point of view this uncertainty relation can be shown to be ˆ and p ˆ do not commute with each a consequence of the fact that the operators x other.214) In general.210) yield the probability amplitude for measuring the eigenvalue a for the observable quantity A.209) a The components ha|Ψ(t)i (1. (1.. A|ai (1. x)hx|Ψ(t)i. x ˆj ] = 0.84)] that the amplitudes in real space and those in momentum space have widths inversely proportional to each other. if an observable operator Aˆ is measured to have a sharp value a in one state.e.216) H. |Ψ(x. Kleinert. its momentum space wave function has a width ∆p given by ∆x ∆p ∼ h ¯. this state must be an eigenstate of Aˆ with an eigenvalue a: ˆ = a|ai. (1. [ˆ pi . If a wave packet is localized in real space with a width ∆x.211) it gives the probability amplitude for measuring the eigenvalues x of the position ˆ . If we write it as Ψ(x. x ˆj ] = −i¯hδij . (1. t)|2 is the probability density in x-space. i.1 Z d3 xhΨ(t)|xiA(−i¯h∇.5. (1. For this we expand this vector in the basis |ai: X |Ψ(t)i = |aiha|Ψ(t)i. (1. 1. p. t) = hx1 |Ψ(t)ihΨ(t)|x2 i. t) (2π¯h)3 (1. The ˆ requirement implies that the states |ai are also eigenstates of B. If this probability is sharply focused at a specific value of a.m |En iρnm (t)hEm | = X n. (1. −i¯h∇)hx|Ψ(t)i.m |En ihEn |Ψ(t)ihΨ(t)|Em ihEm |. p ˆ )|Ψ(t)i = tr [f (x. p ρ(t)] = Z d3 xhΨ(t)|xif (x. If this is true for all |ai. can be measured sharply in each of these states. say B. the Wigner function W (X. ˆ B (1. x2 . (1.5 Observables in which |ha|Ψ(t)i|2 is the probability to measure an arbitrary eigenvalue a. [A. (1. ˆ we may ask under what circumstances Given the set of all eigenstates |ai of A.221) ˆ ) can be calculated from the trace The expectation value of any function f (x. t) ≡ Z d3 ∆x ip∆x/¯h e ρ(X + ∆x/2.33 1. |Ψ(t)i = Pn |En ihEn |Ψ(t)i. the state necessarily coincides with |ai.223) Wigner showed that the Fourier transform of the density matrix. ˆ another observable. it can be shown that a vanishing commutator is also sufficient for two observable operators to be simultaneously diagonalizable and thus to allow for simultaneous sharp measurements. (1.5.222) If we decompose the states |Ψ(t)i into stationary eigenstates |En i of the Hamiltonian ˆ [recall (1. (1.2 Density Matrix and Wigner Function An important object for calculating observable properties of a quantum-mechanical system is the quantum mechanical density operator associated with a pure state ρˆ(t) ≡ |Ψ(t)ihΨ(t)|. p ˆ )ˆ hΨ(t)|f (x. ˆ B|ai = ba |ai.219) Conversely.175)].217) with some a-dependent eigenvalue ba .218) ˆ necessarily commute: the operators Aˆ and B ˆ B] ˆ = 0. X − ∆x/2. then the density matrix has operator H the expansion ρˆ(t) ≡ X n.224) .220) and the associated density matrix associated with a pure state ρ(x1 . ˆ A|ai ˆ = ba a|ai = aba |ai = AˆB|ai. . . h ¯ (2π¯h)3 (1. . xN |x′1 . . t) in powers of q. for a single particle of mass M in a potential V (x). . all other formulas given above can be generalized to N-body state vectors.227) where F (X) ≡ −∇X V (X) is the force associated with the potential V (X). .229) (1. Kleinert. . (1. the basisindependent Schr¨odinger equation (1. . . hx1 . p. 1. . . xN |ˆ pν = −i¯h∂xν hx1 . . t)|Ψ(t)i = i¯h∂t |Ψ(t)i. . p − q. (1. . xN | = 1. t) d3 ∆x V (X − ∆x/2)eiq∆x/¯h .230) The Schr¨odinger equation for N particles (1. Z and define d3 x1 · · · d3 xN |x1 .225) ∂t + v · ∇X W (X. . . which we rewrite in front of the exponential eiq∆x/¯h as powers of −i¯h∇q . we may expand W (X. . v≡ p . . xN ihx1 . . xN |. . In the same way. t) = −F (X)∇p W (X.226) In the limit h ¯ → 0. . If H(pν . .5. hx1 . . . xN |. x (1. . p. (1. p. . . 1. . . .228) We may introduce a complete local basis |x1 .228) by multiplying it from the left with the bra vectors hx1 . . and V (X − ∆x/2) in powers of ∆x. v ≡ M where Z 2 Z d3 q Wt (X. xN |. t).163) can be integrated to find the wave function |Ψ(t)i at any time tb from the state at any other time ta ˆ |Ψ(tb )i = e−i(tb −ta )H/¯h |Ψ(ta )i. . .6 Time Evolution Operator If the Hamiltonian operator possesses no explicit time dependence. . xN i with the properties hx1 . M (1. p. H(ˆ pν . xν . .107) follows from (1.34 1 Fundamentals satisfies. xN . . xN |ˆ xν = xν hx1 . t) is the Hamiltonian. t) ≡ W (X. the Schr¨odinger equation becomes ˆ ν . x′N i = δ (3) (x1 − x′1 ) · · · δ (3) (xN − x′N ).3 Generalization to Many Particles All this development can be extended to systems of N distinguishable mass points with Cartesian coordinates x1 . and perform the integral over q to obtain the classical Liouville equation for the probability density of the particle in phase space ∂t + v · ∇X W (X. t) = Wt (X. t). p − q. . . . Then we perform the integral over ∆x to obtain (2π¯h)3 δ (3) (q). . .231) H. . the Wigner-Liouville equation p . p. . . PATH INTEGRALS . N + 1.232) is called the time evolution operator . ta ) = e−i(tb −ta )H/¯h (1. ta ) = ei(tb −ta )Hˆ † /¯h U (1. . The solution may be found iteratively: For tb > ta . U Indeed. i¯h∂tb U(t (1. the time interval is sliced into a large number N + 1 of small pieces of thickness ǫ with ǫ ≡ (tb − ta )/(N + 1).235) ˆ † (tb . (1. ta ).6 Time Evolution Operator The operator ˆ Uˆ (tb . x (1.35 1. ta ).234) As an exponential of i times a Hermitian operator. slicing once at each time tn = ta + nǫ for n = 0. ˆ . ta ) = H ˆ U(t ˆ b .236) ˆ = ei(tb −ta )H/¯h = Uˆ −1 (tb . . .233) Its inverse is obtained by interchanging the order of tb and ta : ˆ Uˆ −1 (tb .163) is somewhat more involved. . (1.163) to relate the wave function in each slice approximately to the previous one: |Ψ(ta + ǫ)i ≈ |Ψ(ta + 2ǫ)i ≈ . the integration of the Schr¨odinger equation If H(ˆ p. Uˆ is a unitary operator satisfying ˆ † = Uˆ −1 . We then use the Schr¨odinger equation (1. t) depends explicitly on time. . i 1− h ¯ Z ta +ǫ i h ¯ Z ta +2ǫ i 1− h ¯ Z ta +(N +1)ǫ |Ψ(ta + (N + 1)ǫ)i ≈ 1− ta . ta ) ≡ ei(tb −ta )H/¯h = Uˆ (ta .. tb ). It satisfies the differential equation ˆ b . E . ˆ dt H(t) . . ta ) ≈ 1 − h ¯ tb Z tN dt′N +1 ˆ ′ ) ×···× 1− i H(t N +1 h ¯ Z t1 ta dt′1 ˆ ′) . dt′1 H(t 3 2 1 (1.239) . ta +N ǫ (1.238) By multiplying out the product and going to the limit N → ∞ we find the series i Uˆ (tb . ta +ǫ ˆ dt H(t) |Ψ(ta + ǫ)i. ta ) = 1 − h ¯ Z −i + h ¯ tb ta dt′1 3 Z tb ta ˆ ′ ) + −i H(t 1 h ¯ dt′3 Z t′3 ta dt′2 Z 2 Z t′2 ta tb ta dt′2 Z t′2 ta ˆ ′ )H(t ˆ ′) dt′1 H(t 2 1 ˆ ′ )H(t ˆ ′ )H(t ˆ ′) + ..Ψ(ta ) . H(t 1 (1. . From the combination of these equations we extract the evolution operator as a product i Uˆ (tb .237) ! ˆ dt H(t) |Ψ(ta + Nǫ)i. tb ] in the (t1 . It is useful to introduce a time-ordering operator which. > ti1 .241) can be pulled out in front of the Tˆ operator. the Neumann-Liouville expansion can be rewritten in a more compact way. . . O (1. . . ˆ n (tn ) · · · O ˆ 1 (t1 ). .242) Any c-number factors in (1. Note that each integral has the time arguments in the Hamilton operators ordered causally: Operators with later times stand to left of those with earlier times.240) reorders the times successively.243). With this formal operator. 1.241) where tin . so that tin > tin−1 > . PATH INTEGRALS . More explicitly we define ˆ i1 (ti1 ). (1. t2 ∈ [ta . for instance.2). when applied to an arbitrary product of operators. t2 ) plane (see Fig. (1. . .239) Z tb ta dt2 Z t2 ˆ 2 )H(t ˆ 1 ).245) H.244) ta t2 we see that the two expressions coincide except for the order of the operators.243) The integration covers the triangle above the diagonal in the square t1 . This can be corrected with the use of a time-ordering operator Tˆ . ˆ n (tn ) · · · O ˆ 1 (t1 )) ≡ O ˆ in (tin ) · · · O Tˆ(O (1. The expression Tˆ Z tb ta dt2 Z tb t2 ˆ 2 )H(t ˆ 1) dt1 H(t (1. (2A. Take. . Kleinert.36 1 Fundamentals tb t2 ta ta t1 tb Figure 1. An interesting modification of this is the so-called Magnus expansion to be derived in Eq. the third term in (1.25). . ti1 are the times tn . dt1 H(t ta (1. t1 relabeled in the causal order.3 Illustration of time-ordering procedure in Eq. By comparing this with the missing integral over the lower triangle Z tb Z tb ˆ 2 )H(t ˆ 1) dt2 dt1 H(t (1. known as the Neumann-Liouville expansion or Dyson series. . 239) as Z Z tb Z tb 1 ˆ tb ˆ n )H(t ˆ n−1 ) · · · H(t ˆ 1) T dtn dtn−1 · · · dt1 H(t n! ta ta ta Z 1 = Tˆ n! "Z tb ta (1. Since the time arguments are properly ordered. and we recover the previous result (1.249) (1. the time-ordering operation is superfluous. n! h ¯ ta The right-hand side of Tˆ contains simply the power series expansion of the exponential so that we can write Z i tb ˆ ˆ ˆ U (tb .247) Apart from the dummy integration variables t2 ↔ t1 .243). ta ) = T exp − dt H(t) . as tb Z ta dt1 Z t1 ta ˆ 1 )H(t ˆ 2 ). the integrations now run over the full square in the t1 .232).37 1. (1. we may rewrite the nth-order term of (1.253) .+ + ..250) #n ˆ dt H(t) .. t′ ) δ H(t h ¯ ta A simple application for this relation is given in Appendix 1A. ˆ ˆ Note that a small variation δ H(t) of H(t) changes Uˆ (tb . ta ).6 Time Evolution Operator is equal to (1. The time evolution operator Uˆ (tb .251) n tb 1 −i ˆ Tˆ dt H(t) +. 2 ta Similarly. after interchanging the order of integration.248) 2 ta ta On the right-hand side.243) can trivially be multiplied with the time-ordering operator. Z Z Z t′ ta ˆ dt H(t) ) (1. (1. t2 plane so that the two integrals can be factorized into 2 tb 1ˆ ˆ T dt H(t) .246) or. ta ) = 1 − i Tˆ U(t h ¯ Z tb ta ˆ + 1 −i dt H(t) 2! h ¯ 2 Tˆ Z tb ta ˆ dt H(t) 2 (1. .243) can alternatively be written as Z Z tb 1 ˆ tb ˆ 2 )H(t ˆ 1 )..243) since it may be rewritten as tb Z Z dt2 ta tb t2 ˆ 1 )H(t ˆ 2) dt1 H(t (1. ta ) has therefore the series expansion ˆ b . this double integral coincides with (1. ta ) by n Z ( tb i tb ˆ b . =− dt′ Uˆ (tb . dt2 H(t (1.. T dt2 dt1 H(t (1. The conclusion of this discussion is that (1. the If H integral can be done trivially.252) h ¯ ta ˆ does not depend on the time. ta ) = − i ˆ ˆ ′ ) Tˆ exp − i δ U(t dt H(t) δ H(t dt′ Tˆ exp − h ¯ ta h ¯ t′ h ¯ Z tb i ˆ ′ ) U(t ˆ ′ . (1. we must have ˆ b . tb )..252) for the time evolution operator Uˆ (tb . ta ) = Uˆ (tb . ta ) from the later time ta to the earlier time tb . H. tb ): ˆ (ta .260) the unitarity of the operator Uˆ (tb .254) This composition law makes the operators Uˆ a representation of the abelian group of time translations.257) the order of succession is inverted by multiplying both sides by Uˆ −1 (ta . however. (1. We may. for tb later than ta . ta ) given by (1. Kleinert. i.258) The operator on the right-hand side is defined to be the time evolution operator ˆ U(tb . tb )−1 . the corresponding operators Uˆ are related by ˆ ′ . tb )−1 |Ψ(ta )i.254) is trivial. For time-independent Hamiltonians with Uˆ (tb . t′ )U(t t′ ∈ (ta .232). (1. ta ) = Uˆ (tb . Uˆ (tb . (1. tb ) = e−i(ta −tb )H/¯h .256) Indeed. ta ). ta ) has some important properties: By construction. If the Hamiltonian is independent of time. it follows from the simple manipulation valid for tb > ta : i Tˆ exp − h ¯ " Z tb t′ ˆ dt Tˆ exp − i H(t) h ¯ i = Tˆ exp − h ¯ i = Tˆ exp − h ¯ tb Z t′ Z tb ta Z t′ ta ! ˆ dt H(t) ˆ dt exp − i H(t) h ¯ Z t′ ta !# ˆ dt H(t) (1. define Uˆ (tb .7 1 Fundamentals Properties of the Time Evolution Operator ˆ b . ta ) ≡ Uˆ (ta .259) tb < ta . ta ) . ta ) also for the anticausal (or advanced ) case where tb lies before ta . |Ψ(tb )i = U tb < ta . when considering two states at successive times ˆ a . U(t a) Fundamental composition law If two time translations are performed successively. U(t (1. with the time evolution operator being ˆ Uˆ (ta .38 1.252).e. |Ψ(ta )i = U(t (1. the proof of (1.255) ˆ dt .254). In the general case (1. PATH INTEGRALS . H(t) b) Unitarity The expression (1. tb )|Ψ(tb )i. To be consistent with the above composition law (1. ta ) is obvious: −1 Uˆ † (tb . ta > tb . ta ) was derived only for the causal (or retarded ) time arguments. 252)] ˆ b . = Tˆ O (1. The classical equations hold for the Heisenberg operators . whose indices can be partly continuous. tb ) ≡ Uˆ (tb . ta ) for tb < ta has a representation just like (1. with an obvious definition analogous to (1. tb ). ta )|Ψ(ta )i. (1.260).267) Heisenberg Picture of Quantum Mechanics ˆ ta ) may be used to give a different formuThe unitary time evolution operator U(t. (1.228) implies that the operator Uˆ (tb .263) With Uˆ (ta . ta )−1 . but later it became clear that these matrices had to be functional matrix elements of operators. c) Schr¨odinger equation for Uˆ (tb . ta ) = H −1 −1 ˆ i¯h∂t Uˆ (t.264) the Schr¨odinger equation (1.265) (1. U tb > ta . ta ) = Tˆ exp U(t i h ¯ Z tb ta ˆ dt . with the initial condition 1. ˆ b . ta ) = Uˆ (ta .8 Heisenberg Picture of Quantum Mechanics Let us verify this property for a general time-dependent Hamiltonian.241). except for a reversed time order of its arguments. i¯h∂t Uˆ (t. Heisenberg postulated that they are matrices. lation of quantum mechanics bearing the closest resemblance to classical mechanics.252).266) (1. this proves the unitarity relation (1. This operator satisfies the relation h ˆ 1 (t1 )O ˆ 2 (t2 ) Tˆ O i† ˆ 2† (t2 )O ˆ 1† (t1 ) . Originally. a direct solution of the Schr¨odinger equation (1. qHi (t).163) for the state vector shows that the operator Uˆ (tb . ta ) = −Uˆ (t.39 1. to be denoted by pHi (t). One writes this in the form [compare (1.261) where Tˆ denotes the time-antiordering operator.8 Uˆ (ta . H(t) (1.262) with an obvious generalization to the product of n operators. is in a way more closely related to classical mechanics than the Schr¨odinger formulation. ta ) Since the operator Uˆ (tb . We can therefore conclude right away that ˆ † (tb . ta ) satisfies the corresponding equations ˆ Uˆ (t. ta ) rules the relation between arbitrary wave functions at different times. ta ).242). |Ψ(tb )i = U(t (1. in general. There. (1. ta ) H. ta ) = 1. This formulation. Many classical equations remain valid by simply replacing the canonical variables pi (t) and qi (t) in phase space by Heisenberg operators. called the Heisenberg picture of quantum mechanics. The Hamilton equations of motion (1. d ˆ i ˆ ˆ ∂ ˆ O H = [H OH . rather than qi . xˆ.19) goes over into the matrix commutator equation for the Heisenberg operator namely. In this case we shall always use the notation xi for the position variable. 0)|ΨH a i. (1.40 1 Fundamentals and as long as the canonical commutation rules (1.275) H.4. the operator O(t) can be specified in terms of its functional matrix elements ˆ Oab (t) ≡ hΨa (t)|O(t)|Ψ b (t)i. xˆH (t)] = 0.24).268) [ˆ xH (t).23) turn into the Heisenberg equations where i i hˆ d pˆH (t) = HH . [ˆ pH (t). (1. xˆH (t)] = −i¯h. t). (1. the equation of motion for an arbitrary observable function O(pi (t). The corresponding Heisenberg operators are xˆHi (t). t) derived in (1. the canonical equal-time commutation rules are [ˆ pH (t). recalling the fundamental Poisson brackets (1. where a runs through discrete and continuous indices. pˆH (t) . Similarly. The relation between Schr¨odinger’s and Heisenberg’s picture is supplied by the ˆ be an arbitrary observable in the Schr¨odinger detime evolution operator. Kleinert. dt h ¯ i i hˆ d HH . (1. |ΨH a i. xˆH (t).268) are a special case of this rule. According to Heisenberg.274) ˆ 0) to go to a new time-independent basis We can now use the unitary operator U(t. Let O scription ˆ ≡ O(ˆ O(t) p. PATH INTEGRALS .269) ˆ H ≡ H(ˆ H pH (t). qi (t) must be Cartesian coordinates. OH ] + dt h ¯ ∂t These rules are referred to as Heisenberg’s correspondence principle. t) (1. x ˆH (t).271) is the Hamiltonian in the Heisenberg picture. xi (t). as in Section 1. (1. classical equations involving Poisson brackets remain valid if the Poisson brackets are replaced by i/¯h times the matrix commutators at equal times. The canonical commutation relations (1. In addition. pˆH (t)] = 0. (1.93) are respected at any given time. xˆH (t) = dt h ¯ (1.272) H . defined by |Ψa (t)i ≡ Uˆ (t.270) ˆ H (t) ≡ O(ˆ O pH (t).273) If the states |Ψa (t)i are an arbitrary complete set of solutions of the Schr¨odinger ˆ equation. Suppressing the subscripts i. xˆH (t) . t). ta )O(t) U (t. Due to the absence of such terms in (1. we obtain ! d ˆ i h ˆ −1 ˆ ˆ ˆ i ˆ −1 ∂ ˆ OH (t) = O(t) Uˆ . t) . U H U.277) At the time t = 0. 0). Consider the time ˆ H (t). we transform the Schr¨odinger operators of the canonical coordinates pˆ and xˆ into the time-dependent canonical Heisenberg operators pˆH (t) and xˆH (t) via ˆ 0). ta ) O(t) dt ! ! ∂ ˆ d ˆ −1 −1 ˆ ˆ ˆ ˆ + U (t. The Heisenberg matrices OH (t)ab are then obtained from the Heisenberg operators ˆ H (t) by sandwiching O ˆ H (t) between the time-independent basis vectors |ΨH a i: O ˆ H (t)|ΨH b i. ta )O(t) ˆ Uˆ (t. ta ) U (t. where the right-hand side would have contained two more terms from the time dependence of the wave functions. OH + U dt h ¯ ∂t (1. xˆH (t) ≡ Uˆ (t. ta ) .280) it is possible to study the equation of motion of the Heisenberg matrices independently of the basis by considering directly the Heisenberg operators. ta ) (1. ta ) + U (t. xˆH (t).280) This is in contrast to the Schr¨odinger representation (1.41 1. ta ) + Uˆ (t. It is straightforward to verify that they do indeed satisfy the rules of Heisenberg’s correspondence principle.274). derivative of an arbitrary observable O ! d ˆ OH (t) = dt d ˆ −1 ˆ Uˆ (t. ta ) U (1. d ˆ d OH (t)ab ≡ hΨH a | O H (t)|ΨH b i. ta ) O(t) U (t. pˆH (t) ≡ Uˆ (t. dt ∂t Using (1.282) . ta )−1 O(ˆ O p. ta )O(t) O(t) Uˆ (t. 0)−1 pˆ U(t. respectively. ta ) Uˆ (t. ta ). the Heisenberg operators pˆH (t) and xˆH (t) coincide with the timeindependent Schr¨odinger operators pˆ and xˆ. ta ) U (t. t)Uˆ (t. ta ) Uˆ (t. ta ) U ˆ (t.276) (1. 0)−1 xˆ Uˆ (t.281) dt ! h i d ∂ −1 −1 −1 ˆ Uˆ (t. ∂t dt which can be rearranged as " ! # d ˆ −1 ˆ −1 (t. xˆ. OH (t)ab ≡ hΨH a |O (1. (1. An arbitrary observable ˆ O(t) is transformed into the associated Heisenberg operator as ˆ H (t) ≡ Uˆ (t.8 Heisenberg Picture of Quantum Mechanics Simultaneously.279) Note that the time dependence of these matrix elements is now completely due to the time dependence of the operators.278) ≡ O (ˆ pH (t).265). dt dt (1. ta ) ˆ + Uˆ (t. Kleinert. ta ).283) H By sandwiching this equation between the complete time-independent basis states |Ψa i in the Hilbert space.287) i¯h∂tb UˆI (tb . ta ) satisfies the equation of motion where VˆI (t) ≡ eiH0 t/¯h Vˆ e−iH0 t/¯h (1. In this case it is useful to describe the system in Dirac’s interaction picture. (1.42 1 Fundamentals After inserting (1. It is governed by the time evolution operator UˆI (tb . these equations reduce.275) of the operator H ˆ The operator UI (tb .290) H. states (1. |ψI (tb )i = U (1. of course. 1. and Vˆ is an interaction potential which perturbs H these solutions slightly.278). PATH INTEGRALS . ta ) = 1 − i U h ¯ Z tb ta dtVI (t)UˆI (t.9 Interaction Picture and Perturbation Expansion For some physical systems. H (1.288) If Vˆ = 0. For the phase space variables pH (t). and reads (1. (1.284) ˆ 0 is a so-called free Hamiltonian operator for which the Schr¨odinger equation where H ˆ 0 |ψ(t)i = i¯h∂t |ψ(t)i can be solved.285) Their time evolution comes entirely from the interaction potential Vˆ . and that the Heisenberg matrices obey the same Hamilton equations as the classical observables. ta ) ≡ eiH0 tb /¯h e−iH(tb −ta )/¯h e−iH0 ta /¯h . xH (t) themselves.289) is the potential in the interaction picture. This equation of motion can be turned into an integral equation ˆI (tb . We remove the time evolution of the unperturbed Schr¨odinger solutions and define the states ˆ |ψI (t)i ≡ eiH0 t/¯h |ψ(t)i. the states |ψI (tb )i are time-independent and coincide with the Heisenberg ˆ 0. ta )|ψI (ta )i. to the Hamilton equations of motion (1.269). we find the equation of motion for the Heisenberg operator: i i hˆ ˆ ∂ ˆ d ˆ OH (t) = O HH .286) ˆI (tb . OH (t) + dt h ¯ ∂t ! (t). Thus we have shown that Heisenberg’s matrix quantum mechanics is completely equivalent to Schr¨odinger’s quantum mechanics. ta ) = VI (tb )UˆI (tb . the Hamiltonian operator can be split into two contributions ˆ =H ˆ 0 + Vˆ . it holds for the matrices and turns into the Heisenberg equation of motion. (1. (1. ta ). .294) e−iH(tb −ta )/¯h = e−iH0 (tb −ta )/¯h − h ¯ ta Note that the lowest-order correction agrees with the previous formula (1. this reads Z tb ˆ ˆ ˆI (tb .293) + − h ¯ ta ta This expansion is seen to be the recursive solution of the integral equation i Z tb ˆ ˆ dt e−iH0 (tb −t)/¯h V e−iH(t−ta )/¯h .43 1. (1. . ta ). ta ) = 1 − dt eiH0 t/¯h V e−iH0 t/¯h h ¯ ta Z Z t i 2 tb ′ ˆ ˆ ˆ ′ dt dt′ eiH0 t/¯h V e−iH0 (t−t )/¯h V e−iH0 t /¯h + . the right-hand side cannot be evaluated with the help Lie’s expansion formula.253) Another way of writing the expansion (1.9 Interaction Picture and Perturbation Expansion Inserting Eq.293) is i Z tb ˆ ˆ dt eiH0 t/¯h V e−iH0 t/¯h eiHta /¯h . The proper expression of the right-hand side is referred to as the Campbell-Baker-Hausdorff expansion A simple consequence of Hadamard’s lemma is a variation formula for a timeˆ dependent operator A(t): ˆ A(t) δe = Z 0 1 ˆ ˆ dte(1−t)A δ Aˆ etA . U dt eiH0 t/¯h V e−iH0 t/¯h U (1. . (1.297) by setting B − eT A to ˆ This is. also known as Hadamard’s lemma ˆ −tA e t2 ˆ ˆ ˆ ˆ tA ˆ ˆ ˆ ˆ Be = B − t[A. (1.296) Due to the time-ordering operator. this can also be rewritten as i Z tb ˆ ˆ −iH(tb −ta )/¯ h −iH0 (tb −ta )/¯ h e =e − dt e−iH0 (tb −t)/¯h V e−iH0 (t−ta )/¯h h ¯ ta 2 Z t Z t b i ′ ˆ ˆ ˆ ′ dt dt′ e−iH0 (tb −t)/¯h V e−iH0 (t−t )/¯h V e−iH0 (t −ta )/¯h + . h ¯ ta This may be recorded as a mathematical operator formula e−iH(tb −ta )/¯h = e−iH0 tb /¯h Tˆ exp − RT ˆ ˆ eT (A+B) = Tˆe 0 ˆ ˆ ˆ tA dte(T −t)A Be ˆ RT = eT A Tˆe 0 ˆ ˆ ˆ tA dte−tA Be . of course. just another way of expressing Eq.253). lowest order in δ A. . (1. . ta ) in powers of the interaction potential: i tb ˆ ˆ UˆI (tb . . [A. . ta ) = 1 − i ˆI (t. B] + [A. + − h ¯ ta ta Z (1. (1. (1. 2! (1.295) (1. the evaluation is considerably more involved and relegated to Appendix 2A. .291) h ¯ ta This equation can be iterated to find a perturbation expansion for the operator UˆI (tb .286) and multiplying the equation ˆ ˆ from the left by e−iH0 tb /¯h and from the right by eiH0 ta /¯h .298) ˆ A) ˆ ˆ ˆ = δ Aˆ and expanding eT (A+δ which follows from (1.292) Inserting on the left-hand side the operator (1.289). . B]] + . .297) Due to Tˆ. Both Heaviside functions have the property that their derivative yields Dirac’s δ-function ∂t Θ(t) = δ(t).10 Time Evolution Amplitude In the subsequent development. xb .304) 0 for t ≤ 0. ta ) is given by (1.259).265). ta )|xa i. ta ). U(t 0. an important role will be played by the matrix elements of the time evolution operator in the localized basis states. To abbreviate the case distinction in (1.304) only by the value at tb = ta : 1 for t ≥ 0. (xb tb |xa ta )R ≡ Θ(tb − ta )(xb tb |xa ta ).302).44 1 Fundamentals 1. the propagator is simply ˆ b − ta )/¯h]|xa i. The functional matrix (xb tb |xa ta ) is also called the propagator of the system. (xb tb |xa ta ) = hxb | exp[−iH(t (1.302) and the associated causal time evolution amplitude or retarded time evolution amplitude ˆ R (tb . (1.306) 0 for t < 0. Kleinert. Θ(t) ≡ (1. ta ). (xb tb |xa ta ) ≡ hxb |U(t (1.303) Since this differs from (1.299) only for tb < ta . tb ) − i¯h∂tb ] (xb tb |xa ta ) = 0. we shall often omit the superscript R. (1. it is convenient to use the Heaviside function defined by 1 for t > 0. ta )|xa i. the propagator satisfies the Schr¨odinger equation [H(−i¯h∂xb . R Θ (t) ≡ (1.300) Due to the operator equations (1. and since all formulas in the subsequent text will be used only for tb > ta . ta ) ≡ Θ(tb − ta )Uˆ (tb . For a system with a time-independent Hamiltonian operator where Uˆ (tb . tb < ta .1 H. PATH INTEGRALS . It is therefore customary to introduce the so-called causal time evolution operator or retarded time evolution operator :9 ˆR U (tb .307) 9 Compare this with the retarded Green functions to be introduced in Section 18. only the propagators from earlier to later times will be relevant. ˆ b .301) In the quantum mechanics of nonrelativistic particles.305) There exists also another Heaviside function which differs from (1. (1. (xb tb |xa ta )R ≡ hxb |U (1. and write U R (tb . (1.299) They are referred to as time evolution amplitudes. ta ) ≡ ( ˆ b . tb ≥ ta . infinitely many possible definitions for the Heaviside function depending on the value assigned to the function at the origin. due to (1. The integral f˜(E) ≡ Z ∞ 0 dt f (t)eiEt/¯h (1. (1. Since the contour encloses no singularities. 2π¯h (1. The integral representation is undefined for t = 0 and there are. Θl (t). If the Hamiltonian does not depend on time. The Heaviside function Θ(t) itself is the simplest retarded function.313) Usually.310) is an analytic function in the upper half of the complex energy plane.267). All three distributions Θr (t).45 1.10 Time Evolution Amplitude If it is not important which Θ-function is used we shall ignore the superscript. (1.306) and (1. This analyticity property is necessary and sufficient to produce a factor Θ(t) when inverting the Fourier transform via the energy integral f (t) ≡ Z ∞ −∞ dE ˜ f (E)e−iEt/¯h . (1. it can be contracted to a point.162). the contour of integration may be closed by an infinite semicircle in the upper half-plane at no extra cost. (1.314) . Functions f (t) with this property have a characteristic Fourier transform. The retarded propagator satisfies the Schr¨odinger equation h i H(−i¯h∂xb . The retarded propagator vanishes for t < 0. tb )R − i¯h∂tb (xb tb |xa ta )R = −i¯hδ(tb − ta )δ (3) (xb − xa ). the difference in the value at the origin does not matter since the Heaviside function appears only in integrals accompanied by some smooth function f (t). 0 for t < 0. This makes the Heaviside function a distribution with respect to smooth test functions ¯ f (t) as defined in Eq. in fact. xb . which is equal to 1/2 at the origin: 1 for t > 0.309) and the initial condition hxb ta |xa ta i = hxb |xa i.311) For t < 0. with a Fourier representation containing just a single pole just below the origin of the complex energy plane: Z ∞ dE i Θ(t) = e−iEt .308) The nonzero right-hand side arises from the extra term −i¯h [∂tb Θ(tb − ta )] hxb tb |xa ta i = −i¯hδ(tb − ta )hxb tb |xa ta i = −i¯hδ(tb − ta )hxb ta |xa ta i (1. ¯ 1 Θ(t) ≡ 2 for t = 0.304). the propagator depends only on the time difference t = tb − ta . (1. and Θ(t) define the same linear functional of the test functions by the integral Θ[f ] = Z dt Θ(t − t′ )f (t′ ).312) −∞ 2π E + iη where η is an infinitesimally small positive number. yielding f (t) = 0. A special role is played by the average of the Heaviside functions (1. for later use.319) which is the Fourier transform of the retarded time evolution operator (1.316) Fixed-Energy Amplitude The Fourier-transform of the retarded time evolution amplitude (1.162). that one knows all solutions |ψn i of the equation ˆ n i = En |ψn i.317) is called the fixed-energy amplitude. for for for (1. dtb eiE(tb −ta )/¯h U(t (1. integrals over products of distribution and thus give rise to an important extension of the ¯ − t′ ) plays theory of distributions in Chapter 10. (1. ǫ(t − t′ ) ≡ Θ(t − t′ ) − Θ(t′ − t) = Θ(t (1. (1. the Heaviside function Θ(t the main role. H|ψ (1.321) These satisfy the completeness relation X n |ψn ihψn | = 1. t = t′ .303) (xb |xa )E = Z ∞ −∞ iE(tb −ta )/¯ h dtb e R (xb tb |xb ta ) = Z ∞ ta dtb eiE(tb −ta )/¯h (xb tb |xa ta ) (1. In this.302): ˆ R(E) = Z ∞ −∞ iE(tb −ta )/¯ h dtb e Uˆ R (tb . we insert here Eq.315) which is a step function jumping at the origin from −1 to 1 as follows: 1. i.322) H. (1. ta ) = Z ∞ ta ˆ b . in addition.. If the Hamiltonian does not depend on time.320) Let us suppose that the time-independent Schr¨odinger equation is completely solved. path integrals will specify. Kleinert. As announced after Eq. While discussing the concept of distributions let us introduce.46 1 Fundamentals and this is an element in the linear space of all distributions.318) of the so-called resolvent operator ˆ R(E) = i¯h . ˆ + iη E−H (1. the closely related distribution ¯ − t′ ) − Θ(t ¯ ′ − t). t < t′ . ta ).11 1 ′ ǫ(t − t ) = 0 −1 t > t′ .e.300) and find that the fixed-energy amplitudes are matrix elements ˆ (xb |xa )E = hxb |R(E)|x ai (1. PATH INTEGRALS . An iη-prescription will appear in another context in Section 2.326). the poles in the lower half-plane give. When doing the Fourier integral (1.324) being the wave functions associated with the eigenstates |ψn i.325) The fixed-energy amplitude (1. there is no pole inside the closed contour making the propagator vanish. E − En + iη E − En − iη (1. which is recovered from it by the inverse Fourier transformation Z ∞ dE −iE(tb −ta )/¯h (xb ta |xa ta ) = e (xb |xa )E . which are thus placed by an infinitesimal piece below the real energy axis.317).327) This so-called iη-prescription ensures the causality of the Fourier representation (1. For tb > ta .11 Fixed-Energy Amplitude which can be inserted on the right-hand side of (1.326) h −∞ 2π¯ The small iη-shift in the energy E in (1. (1.47 1.300) between the Dirac brackets leading to the spectral representation (xb tb |xa ta ) = X n ψn (xb )ψn∗ (xa ) exp [−iEn (tb − ta )/¯h] . which lies in the upper half-plane for tb < ta and in the lower half-plane for tb > ta . (1. the residues at the poles of (1. the exponential eiE(tb −ta )/¯h makes it always possible to close the integration contour along the energy axis by an infinite semicircle in the complex energy plane. there is a cut in the complex energy plane which may be thought of as a closely spaced sequence of poles. going to zero at infinite time: e−i(En −iη)t/¯h → 0.328) .317) contains as much information on the system as the time evolution amplitude. In general. For a system with a continuum of energy eigenvalues.325) render directly the products of eigenfunctions (barring degeneracies which must be discussed separately). E − En + iη (1. via Cauchy’s residue theorem. (1. the wave functions are recovered from the discontinuity of the amplitudes (xb |xa )E across the cut.326).3.323) with ψn (x) = hx|ψn i (1.325) may be thought of as being attached to each of the energies En . If the eigenstates are nondegenerate. using the formula disc i¯h E − En ! ≡ i¯h i¯h − = 2π¯hδ(E − En ). on the other hand. we obtain (xb |xa )E = X ψn (xb )ψn∗ (xa )Rn (E) = n X ψn (xb )ψn∗ (xa ) n i¯h . Then the exponential behavior of the wave functions is slightly damped.323) of the propagator. The iη-prescription guarantees that for tb < ta . the spectral representation (1. Applying the Fourier transform (1. The energy integral over the discontinuity of the fixed-energy amplitude (1.331) In general. The cut will mostly be omitted. 2π¯h n (1.187) into Eq. (1. Kleinert. H. for brevity. the system possesses also a continuous spectrum. (1.330) The completeness relation reflects the following property of the resolvent operator: Z ∞ −∞ dE ˆ disc R(E) = ˆ1.329) where P indicates that the principal value of the integral has to be taken. and (1. we obtain for the time evolution amplitude of a free particle the Fourier representation (xb tb |xa ta ) = Z p2 i dD p p(x − x ) − exp (tb − ta ) b a (2π¯h)D h ¯ 2M ( " #) .12 Free-Particle Amplitudes ˆ =p ˆ 2 /2M. The eigenfunctions are (1.300).325) (xb |xa )E reproduces the completeness relation (1.333) The momentum integrals can easily be done. PATH INTEGRALS . valid inside integrals over E: 1 P = ∓ iπδ(E − En ). It is the beginning of an expansion in powers of η > 0: 1/(x ± iη) = P/x ∓ iπδ(x) + η [πδ ′ (x) ± idx P/x] + O(η 2 ). Inserting the completeness relation (1. 1. First we perform a quadratic completion in the exponent and rewrite it as p(xb − xa ) − 1 xb − xa 1 1 2 p− p (tb − ta ) = 2M 2M M tb − ta 2 (tb − ta ) − M (xb − xa )2 . E − En ± iη E − En (1. Z ∞ −∞ X dE disc (xb |xa )E = ψn (xb )ψn∗ (xa ) = hxb |xa i = δ (D) (xb − xa ). in which case the completeness relation contains a spectral integral and (1.322) taken between the local states hxb | and |xa i. 2π¯h (1. 2 tb − ta (1.322) has the form X n |ψn ihψn | + Z dν |ψν ihψν | = 1.189) with energies E(p) = p2 /2M.332) The continuum causes a branch cut along in the complex energy plane.334) 10 This is often referred to as Sochocki’s formula.330) includes an integral over the discontinuity along the cut.48 1 Fundamentals Here we have employed the relation10 . the spectrum is conFor a free particle with a Hamiltonian operator H tinuous. (1. the integral is certainly convergent yielding (1.337). At α = ±ia + η with a > 0 and < infinitesimal η > 0. h ¯ 2 tb − ta " # (1.335) i p′ 2 d D p′ exp − (tb − ta ) . (2π¯h)D h ¯ 2M (1. exp h ¯ 2 tb − ta " # (1.340) so that the full time evolution amplitude of a free massive point particle is 1 (xb tb |xa ta ) = q D 2πi¯h(tb − ta )/M i M (xb − xa )2 .341) . Note that differentiation of Eq. On the boundaries. an extra integrand p2n produces a factor (i/a)n .338) 2 α −∞ 2π by continuing α analytically from positive values into the right complex half-plane.338) with respect to α yields the more general Gaussian integral formula Z ∞ −∞ 1 (2n − 1)!! dp α √ p2n exp − p2 = √ 2 α αn 2π Re α > 0. 2 −∞ 2π |a| 1/ i.337). See Appendix 1B. and the amplitude (1. we obtain 1 F (tb − ta ) = q D.337) √ Here the square root i denotes the phase factor eiπ/4 : This follows from the Gauss formula Z ∞ dp α 2 1 √ exp − p = √ . this is straightforward. As long as Re α > 0.. Since the Fresnel formula is a special analytically continued case of the Gauss formula. For odd powers p2n+1 . one has to be careful. Applying this formula to (1.336).339) where (2n − 1)!! is defined as the product (2n − 1) · (2n − 3) · · · 1. 2πi¯h(tb − ta )/M (1. we shall in the sequel always speak of Gaussian integrations and use Fresnel’s name only if the imaginary nature of the quadratic exponent is to be emphasized. √i. Re α > 0. i.12 Free-Particle Amplitudes Then we replace the integration variables by the shifted momenta p′ = p − (xb − xa )/(tb − ta )M.333) becomes i M (xb − xa )2 (xb tb |xa ta ) = F (tb − ta ) exp . (1. (1. the integral vanishes. (1. on the positive and negative imaginary axes. But the integral also converges for η = 0.49 1. In the Fresnel formula (1. as can easily be seen by substituting x2 = z. (1.e.336) where F (tb − ta ) is the integral over the shifted momenta F (tb − ta ) ≡ Z ( ) This can be performed using the Fresnel integral formula ( √ Z ∞ 1 dp a 2 a > 0. √ exp i p = q a < 0. 2. New York. 1965.341): (xb |xa )E = Z 1 ∞ d(tb − ta ) q D 2πi¯h(tb − ta )/M 0 M (xb −xa )2 i E(tb − ta ) + exp h ¯ 2 tb − ta " ( #) . (1. and Products.432..471.50 1 Fundamentals In the limit tb → ta . Formulas 3.347) where Kν (z) is the modified Bessel function which satifies Kν (z) = K−ν (z)12 The result is 2M κD−2 KD/2−1 (κR) . Academic Press. Formula 8. we set κ≡ q −2ME/¯h2 . Kleinert.342) Inserting Eq.349) 11 I. PATH INTEGRALS . and 8.344) where we have inserted a damping factor e−η(tb −ta ) into the integral to ensure convergence at large tb − ta .11. The simplest modified Bessel function is13 K1/2 (z) = K−1/2 (z) = r π −z e . Abramowitz and I.333) into (1. Dover.10.343) Performing the time integration yields ∞ Z ( Z Z (xb |xa )E = " !#) dD p i¯h exp [ip(xb − xa )] . 2z (1. Gradshteyn and I. Ryzhik.346) and perform the integral with the help of the formula11 Z 0 ∞ ν−1 −iγt+iβ/t dt t e β =2 γ !ν/2 −iνπ/2 e q K−ν (2 βγ).M. New York. Stegun. (1. H. (1.471. Series. 3. implying the following limiting formula for the δ-function δ (D) (xb − xa ) = 1 lim tb −ta →0 q 2πi¯h(tb − ta )/M D i M (xb − xa )2 exp .486. D 2 (2π¯h) E − p /2M + iη (1.16 13 M. Table of Integrals.S. For a more explicit result it is more convenient to calculate the Fourier transform (1. exp d(tb − ta ) (xb |xa )E = b a b a (2π¯h)D h ¯ 2M 0 (1. h ¯ 2 tb − ta " # (1. we have for the fixed-energy amplitude the integral representation p2 i dD p p(x − x ) + (t − t ) E − .6 12 ibid.317). (1. Formula 10. the left-hand side becomes the scalar product hxb |xa i = δ (D) (xb − xa ).17. (1.348) (xb |xa )E = −i h ¯ (2π)D/2 (κR)D/2−1 where R ≡ |xb − xa |. Handbook of Mathematical Functions. 1980.345) For E < 0. 2. Formulas 3.355) The relation16 π iνπ/2 (1) ie Hν (z) (1. to find (1) Mπ k D−2 HD/2−1 (kR) (xb |xa )E = . (D−1)/2 (D−1)/2 h ¯ (2πi) (kR) For D = 1 and 3.359) .7. 15 Formula 9. −i K0 (κR). 16 ibid.4)..407. the amplitude (1.11 and 8.358) (1.356) 2 connects the two formulas with each other when continuing the energy from negative to positive values.348) is finite for all D ≤ 2. which will be needed later (for example in Subsection 4. (1..350) −i h ¯ κ h ¯ π h ¯ 2πR At R = 0.. For E > 0 we set q (1.471..9.354) where Hν(1) (z) is the Hankel function.357) (1.451. (1. Formulas 8. (1.6. h ¯ (2π)D/2 (kR)D/2−1 (1. 3.3. the amplitudes M1 M 1 −κR M 1 −κR e .421. −i e .9. Hν(1) (z) ≈ e 2z πz shows that the fixed-energy amplitude behaves for E < 0 like 1 1 M e−κR/¯h . Formula 8. (1.352) (x|x)E = −i h ¯ (4π)D/2 This result can be continued analytically to D > 2. these asymptotic expressions hold for all R. the asymptotic behavior17 Kν (−iz) = s π −z 2 i(z−νπ/2−π/4) Kν (z) ≈ e .351) 2 2 to obtain 2M κD−2 Γ(1 − D/2). ibid. r 14 ibid. (1.51 1. which replaces κ by e−iπ/2 k = −ik.6 and 8.451. (xb |xa )E ≈ −i κD−2 h ¯ (2π)(D−1)/2 (κR)(D−1)/2 and for E > 0 like M D−2 1 1 (xb |xa )E ≈ k eikR/¯h .353) k ≡ 2ME/¯h2 and use the formula15 Z 0 ∞ ν−1 iγt+iβ/t dtt e β = iπ γ !ν/2 (1) q e−iνπ/2 H−ν (2 βγ). For large distances. 17 ibid. where we can use the small-argument behavior of the associated Bessel function14 −ν 1 z Kν (z) = K−ν (z) ≈ Γ(ν) for Re ν > 0.12 Free-Particle Amplitudes so that we find for D = 1.1. etc. . .). For the Cartesian coordinates we shall use Latin indices alternatively as sub. (1. In the original definition of generalized coordinates in Eq. Then N = D and a variable change from xi to qj in the Schr¨odinger equation leads to the correct quantum mechanics.365) goes over into ¯2 ˆ0 = − h H ∆. this was unnecessary since transformation properties were ignored.363) and yields the curvilinear transform of the Cartesian quantum-mechanical momentum operators pˆi = −i¯h∂i = −i¯hei µ (q)∂µ .364) The free-particle Hamiltonian operator h ¯2 2 1 2 ˆ ˆ ˆ =− ∇ p H0 = T = 2M 2M (1. ei µ ej µ = δ i j . . PATH INTEGRALS . (1.362) Then. (1.360) is inverted to ∂i = ei µ (q)∂µ (1.13 1 Fundamentals Quantum Mechanics of General Lagrangian Systems An extension of the quantum-mechanical formalism to systems described by a set of completely general Lagrange coordinates q1 .367) H.1).366) (1. . . Kleinert. It will be useful to label the curvilinear coordinates by Greek superscripts and write q µ instead of qj . N) being merely a curvilinear reparametrization of a D-dimensional Euclidean space are the above correspondence rules sufficient to quantize the system. qN is not straightforward. Let ei µ (q) = ∂q µ /∂xi be the inverse matrix (assuming it exists) called the reciprocal D-ad .361) with the transformation matrix called basis D-ad (in 3 dimensions triad. (1. Only in the special case of qi (i = 1. 2M where ∆ is the Laplacian expressed in curvilinear coordinates: ∆ = ∂i2 = eiµ ∂µ ei ν ∂ν = eiµ ei ν ∂µ ∂ν + (eiµ ∂µ ei ν )∂ν . .360) ei µ (q) ≡ ∂µ xi (q) (1. (1. The coordinate transformation xi = xi (q µ ) implies the relation between the derivatives ∂µ ≡ ∂/∂q µ and ∂i ≡ ∂/∂xi : ∂µ = ei µ (q)∂i .or superscripts. This will help when we write all ensuing equations in a form that is manifestly covariant under coordinate transformations. in 4 dimensions tetrad. . (1. satisfying with ei µ the orthogonality and completeness relations ei µ ei ν = δµ ν .52 1. . 368) g µν (q) = eiµ (q)ei ν (q). (1. Using this determinant. (1.379) .369) we have furthermore Γµ µν = −∂µ g µν − Γµ νµ . (1.378) With the inverse metric (1.369) its inverse defined by g µν gνλ = δ µ λ .370) Then the Laplacian takes the form ∆ = g µν (q)∂µ ∂ν − Γµ µν (q)∂ν .376) is the determinant of the metric tensor. (1. (1. we form the quantity 1 Γµ ≡ g −1/2 (∂µ g 1/2 ) = g λκ (∂µ gλκ ) 2 (1.372) The reason why (1.371) with Γµ λν being defined as the contraction Γµ λν ≡ g λκ Γµκ ν . and the so-called affine connection Γµν λ (q) = −ei ν (q)∂µ ei λ (q) = ei λ (q)∂µ ei ν (q).373) becomes in curvilinear coordinates ds2 = ∂x ∂x µ ν dq dq = gµν (q)dq µ dq ν .377) and see that it is equal to the once-contracted connection Γµ = Γµλ λ .368) is called a metric tensor is obvious: An infinitesimal square distance between two points in the original Cartesian coordinates ds2 ≡ dx2 (1. (1.374) The infinitesimal volume element dD x is given by dD x = √ g dD q. (1.375) where g(q) ≡ det (gµν (q)) (1.13 Quantum Mechanics of General Lagrangian Systems 53 At this point one introduces the metric tensor gµν (q) ≡ eiµ (q)ei ν (q). ∂q µ ∂q ν (1.1. (1. t) ≡ − 2M # " (1. p 2M (1. H. g (1. Together with (1.. (1. t).54 1 Fundamentals We now take advantage of the fact that the derivatives ∂µ . q) ˙ = M gµν (q)q˙µ q˙ν − V (q). t).e. Kleinert. It is important to realize that this Schr¨odinger equation would not be obtained by a straightforward application of the canonical formalism to the coordinatetransformed version of the Cartesian Lagrangian ˙ = L(x.383) where V (q) is short for V (x(q)). Hψ(q. x) = 1 2 ˆ + V (x). t). transforms into Z √ (1. Γµν λ = Γνµ λ and hence Γµ νµ = Γν . t) = i¯h∂t ψ(q.380) which allows the Laplace operator ∆ to be rewritten in the more compact form 1 √ ∆ = √ ∂µ g µν g∂ν .385) With the velocities transforming as x˙ i = ei µ (q)q˙µ . t)ψ1 (q. The scalar product of two wave functions R D d xψ2∗ (x. 18 (1.387) Up to a factor M.12)]: Hµν (q) = Mgµν (q).12)–(11. 2 (1. (11. PATH INTEGRALS .384) dD q g ψ2∗ (q. g (1.382) the Schr¨odinger equation in curvilinear coordinates becomes h ¯2 ˆ ∆ + V (q) ψ(q. ∂ν applied to the coordinate transformation xi (q) commute causing Γµν λ to be symmetric in µν.388) More details will be given later in Eqs. the metric is equal to the Hessian metric of the system. 2 (1. t)ψ1 (x.377) we find the rotation 1 √ Γµ µν = − √ (∂µ g µν g).381) This expression is called the Laplace-Beltrami operator . which depends here only on q µ [recall (1.386) the Lagrangian becomes L(q. x) M 2 x˙ − V (x).18 Thus we have shown that for a Hamiltonian in a Euclidean space H(ˆ p. i. which determines the transition amplitudes of the system.18). t)[−i¯h∂µ g 1/4 Ψ1 (q. qˆµ = q µ . t)]∗ Ψ1 (q. qˆν ] = −i¯hδµ ν . (1.395) 2M in which the momentum operators have been arranged symmetrically around the inverse metric to achieve Hermiticity.13 Quantum Mechanics of General Lagrangian Systems The canonical momenta are pµ ≡ ∂L = Mgµν q˙ν . (1. An important constraint is provided by the required Hermiticity of the Hamiltonian operator. This operator. (1. this can also be rewritten as pˆµ = −i¯h(∂µ + 21 Γµ ). In terms of the quantity (1. [ˆ pµ . (1.394) When trying to turn this expression into a Hamiltonian operator. t)] = Z = Z d3 q g 1/4 Ψ∗2 (q. We may.387).55 1. we encounter the operator-ordering problem discussed in connection with Eq. define the canonical Hamiltonian operator as ˆ can ≡ 1 pˆµ gµν (q)ˆ H pν + V (q). for instance. The correspondence principle requires replacing the momenta pµ by the momentum operators pˆµ . qˆν ] = 0. is not equal to the . however. t). but this is not sufficient for a unique specification.268): [ˆ pµ . 2M (1.377). t)] √ d3 q g [−i¯hg −1/4 ∂µ g 1/4 Ψ2 (q.392) as is easily verified by partial integration.101).389) is simply H = pµ q˙µ − L = 1 gµν (q)pµ pν + V (q). (1. (1. µ ∂ q˙ (1.393) Consider now the classical Hamiltonian associated with the Lagrangian (1.389) The associated quantum-mechanical momentum operators pˆµ have to be Hermitian in the scalar product (1. [ˆ q µ .390) An obvious solution is pˆµ = −i¯hg −1/4 ∂µ g 1/4 . but only z = 1/4 produces a Hermitian momentum operator: Z 3 dq √ g Ψ∗2 (q.391) The commutation rules are true for −i¯hg −z ∂µ g z with any power z. pˆν ] = 0. which by (1. but it does not specify the position of these operators with respect to the coordinates q µ contained in the inverse metric g µν (q).384) and must satisfy the canonical commutation rules (1. t)[−i¯hg −1/4 ∂µ g 1/4 Ψ1 (q. r r (1.403) The Laplace-Beltrami operator becomes The canonical Laplacian.383). (1.381) in (1.397) The correct Hamiltonian operator could be obtained by suitably distributing pairs of dummy factors of g 1/4 and g −1/4 symmetrically between the canonical operators [5]: ˆ = 1 g −1/4 pˆµ g 1/4 g µν (q)g 1/4 pˆν g −1/4 + V (q). (1.399) Since the infinitesimal square distance is ds2 = dr 2 + r 2 dϕ2 .395). (1. q 2 = ϕ: x1 = r cos ϕ. r 4r r ∆can = (∂r + 1/2r)2 + (1.56 1 Fundamentals correct Schr¨odinger operator in (1. H 2M (1.401) !µν . The kinetic term contains what we may call the canonical Laplacian ∆can = (∂µ + 12 Γµ ) g µν (q) (∂ν + 21 Γν ).396) It differs from the Laplace-Beltrami operator (1. the metric reads 1 0 0 r2 gµν = ! . (1. 4r 2 (1.397) is therefore ∆can − ∆ = − 1 . (1.383) by ∆ − ∆can = − 21 ∂µ (g µν Γν ) − 14 g µν Γν Γµ . Unfortunately.400) µν It has a determinant g = r2 and an inverse g µν = 1 0 0 r −2 (1. on the other hand.402) 1 1 ∆ = ∂r r∂r + 2 ∂ϕ 2 . The simplest system exhibiting the breakdown of the canonical quantization rules is a free particle in a plane described by radial coordinates q 1 = r.394) as (1.398) This operator has the same classical limit (1. Kleinert. the correspondence principle does not specify how the classical factors have to be ordered before being replaced by operators.404) The discrepancy (1. reads 1 2 ∂ϕ r2 1 1 1 = ∂r 2 + ∂r − 2 + 2 ∂ϕ 2 . x2 = r sin ϕ. PATH INTEGRALS .405) H. For symmetry reasons. the group of motion is the Euclidean group containing translations and rotations. The generators of translations and rotations are the momenta and the angular momenta.. The replacement rule will be referred to as the group correspondence principle. and a specification of the order is required to obtain the correct result. respectively. (1. severe difficulties in quantizing such a system. According to the group correspondence principle. for instance on the angular momenta. Only the need to introduce dummy g 1/4 . The canonical commutation rules in Euclidean space may be viewed as a special case of the commutation rules between group generators.14 Particle on the Surface of a Sphere For a particle moving on the surface of a sphere of radius r with coordinates x1 = r sin θ cos ϕ. In a Cartesian coordinate frame. of the Lie algebra of the group. The brackets containing two group generators specify the structure of the group.e. With the canonical quantization rules being unreliable in curvilinear coordinates there are.and g −1/4 -factors creates such problems. the commutators between the generators have to be used for quantization rather than the canonical commutators between positions and momenta. If the Lagrangian coordinates qi do not merely reparametrize a Euclidean space but specify the points of a general geometry. in a Euclidean space. In systems whose energy depends on generators of the group of motion other than those of translations. This is why the literature contains many proposals for handling this problem [6]. i. The prime examples for such systems are a particle on the surface of a sphere or a spinning top whose quantization will now be discussed. those containing a generator and a coordinate specify the defining representation of the group in configuration space. the correspondence principle must then be imposed not on the Poisson brackets of the canonical variables p and q. the commutation rules between group generators and coordinates lead to the canonical quantization rules. Thus. These systems have the common property that their Hamiltonian can be expressed in terms of the generators of a group of motion in the general coordinate frame. The replacement of these brackets by commutation rules constitutes the proper generalization of the canonical quantization from Cartesian to non-Cartesian coordinates.57 1. It is called group quantization. and this appears to be the deeper reason why the canonical rules are correct. 1. the Poisson brackets between the generators and the coordinates are to be replaced by commutation rules. a large class of non-Cartesian systems allows for a unique quantummechanical description on completely different grounds. x2 = r sin θ sin ϕ. at first sight. but on those of the group generators and the coordinates.14 Particle on the Surface of a Sphere Note that this discrepancy arises even though there is no apparent ordering problem in the naively quantized canonical expression pˆµ gµν (q) pˆν in (1.406) . Fortunately.404). x3 = r cos θ. we cannot proceed as above and derive the Laplace-Beltrami operator by a coordinate transformation of a Cartesian Laplacian. PATH INTEGRALS . (1.409) According to the canonical quantization rules. leading once more to the same quantum mechanics as the one described here.410) sin θ But as explained in the previous section. these momentum operators are not expected to give the correct Hamiltonian operator when inserted into the Hamiltonian (1.407) The canonical momenta are ˙ pθ = Mr 2 θ.268): [ˆ pi .413) whose components satisfy the commutation rules of the Lie algebra of the rotation group ˆi. ˙ (1. (1. They are introduced and applied in Sections 10.414) Note that there is no factor-ordering problem since the xˆi ’s and the pˆi ’s appear with different indices in each Lˆk . x ˆj ] = −i¯hδi j . pϕ = Mr 2 sin2 θ ϕ.409). 2 (1. The angular momentum L =x×p (1. k cyclic). L ˆ j ] = i¯hL ˆk [L (i. j.408) and the classical Hamiltonian is given by 1 1 2 H= p2θ + p . An important property of the angular momentum 19 There exist. x ˆj ] = 0. (1. there exists no proper coordinate transformation from the surface of the sphere to Cartesian coordinates19 such that a particle on a sphere cannot be treated via the safe Cartesian quantization rules (1. [ˆ xi . H. pˆϕ = −i¯h∂ϕ . the momenta should become operators 1 pˆθ = −i¯h 1/2 ∂θ sin1/2 θ. 2 2Mr sin2 θ ϕ (1. certain infinitesimal nonholonomic coordinate transformations which are multivalued and can be used to transform infinitesimal distances in a curved space into those in a flat one. Kleinert.412) can be quantized uniquely in Cartesian coordinates and becomes an operator ˆ =x ˆ×p ˆ L (1.58 1 Fundamentals the Lagrangian reads L= Mr 2 ˙2 (θ + sin2 θ ϕ˙ 2 ).411) The only help comes from the group properties of the motion on the surface of the sphere. however. [ˆ pi . Moreover.2 and Appendix 10A. pˆj ] = 0. L ˆ 3 = −i¯h∂ϕ . We re-express the classical Hamiltonian (1. 2 2 2Mr 2Mr sin θ sin θ (1. for instance They can be chosen to diagonalize simultaneously one component of L ˆ the third one. L (1.421) The spherical harmonics are orthonormal with respect to the rotation-invariant scalar product Z 0 π dθ sin θ Z 2π 0 ∗ dϕ Ylm (θ. L3 . x3 = r cos θ.422) . The result is the Hamiltonian operator: H= ˆ = H h ¯2 1 1 ˆ2 1 2 L =− ∂θ (sin θ ∂θ ) + 2 ∂ϕ . (1.14 Particle on the Surface of a Sphere operator is its homogeneity in x.419) ˆ 2 are well known.418) 2 2Mr and replace the angular momenta by the operators (1. = l (1 − z ) l+m 2 l! dx (1.420) with Plm (z) being the associated Legendre polynomials Plm (z) l+m 1 2 m/2 d (z 2 − 1)l .409) in terms of the classical operators L angular momenta L1 = Mr 2 − sin ϕ θ˙ − sin θ cos θ cos ϕ ϕ˙ .417) as 1 L2 .415) the radial coordinate cancels making the angular momentum a differential operator involving only the angles θ. (1. x2 = r sin θ sin ϕ. L3 = Mr 2 sin2 θ ϕ˙ (1. ϕ) = δll′ δmm′ .416).59 1. The eigenfunctions diagonalizing the rotation-invariant operator L ˆ i . ϕ: ˆ1 = L i¯h (sin ϕ ∂θ + cot θ cos ϕ ∂ϕ ) . ϕ)Yl′m′ (θ. in which case they are equal to the spherical harmonics Ylm (θ. (1. It has the consequence that when going from Cartesian to spherical coordinates x1 = r sin θ cos ϕ.416) There is then a natural way of quantizing the system which makes use of these ˆ i . ˆ 2 = −i¯h (cos ϕ ∂θ − cot θ sin ϕ ∂ϕ ) . (1. L2 = Mr 2 cos ϕ θ˙ − sin θ cos θ sin ϕ ϕ˙ . ϕ) = (−1)m " 2l + 1 (l − m)! 4π (l + m)! #1/2 Plm (cos θ)eimϕ . In the symmetric case in which two moments of inertia coincide. the only dynamical degrees of freedom are the orientations in space.60 1 Fundamentals Two important lessons can be learned from this group quantization.15 1 0 0 sin2 θ 2 ! . (1. however. (1.410) into (1. The correct result would.427) where Lξ . arise by distributing dummy factors g −1/4 = r −1 sin−1/2 θ.409). Lζ are the components of the orbital angular momentum in the directions of the principal body axes with Iξ . First. they commute with the constraints of the rigid body. 2Iξ 2Iζ (1. They can uniquely be specified by the rotation matrix which brings the body from some standard orientation to the actual one. The classical angular momentum of an aggregate of mass points is given by X L= xν × pν .426) Spinning Top For a spinning top. Since rotations with the commutation rules (1. it is written as H= 1 1 (Lξ 2 + Lη 2 ) + Lζ 2 . g 1/4 = r sin1/2 θ (1. (1. L x (1. Iζ being the corresponding moments of inertia.429) ν ˆ i .e.424) 2M where ∆ is the Laplace-Beltrami operator associated with the metric gµν = r i. PATH INTEGRALS . Lη . the correct Hamiltonian operator is equal to h ¯2 ˆ H=− ∆. just as in the case of polar coordinates. Second. θ θ r 2 sin θ sin2 θ ϕ (1.428) ν where the sum over ν runs over all mass points. the optimal starting point is again not the classical Lagrangian but the Hamiltonian expressed in terms of the classical angular momenta.419) does not agree with the canonically quantized one which would be obtained by inserting Eqs. Kleinert. (1. Iη ≡ Iξ . If the center of mass of the rigid body is placed at the origin. the correct Hamiltonian operator (1. (1.414) between the components L do not change the distances between the mass points.. We may choose the standard orientation H. ∆= 1.425) 1 1 1 ∂ (sin θ∂ ) + ∂2 .423) between the canonical momentum operators as observed earlier in Eq. The angular momentum possesses a unique operator X ˆ= ˆν × p ˆν.398). 430) where R3 (α). γ ′ ) ≡ e−iα L3 e−iβ L2 e−iγ L3 . γ) has the following derivatives −i¯h∂α R−1 = R−1 L3 . −i¯h∂γ R−1 = R−1 [cos β L3 + sin β(cos α L1 + sin α L2 )] .61 1.434) The first relation is trivial. β.15 Spinning Top to have the principal body axes aligned with the x. (1. z-directions. y. γ ′ )R). omitting the matrix indices as R(α. Let ψ(R) be the wave function of the spinning top describing the probability amplitude of the different orientations which arise from a standard orientation by the rotation matrix R = R(α. R3 (γ) are rotations around the z-axis by angles α. and R2 (β) is a rotation around the y-axis by β. in addition. β.435) which is a consequence of Lie’s expansion formula (1. It can be decomposed. γ. (1. (1. the second follows from the rotation of the generator e−iαL3 /¯h L2 eiαL3 /¯h = cos α L2 − sin α L1 .432) of the 3 × 3 matrices Li . β ′. the rotation e−iβL2 /¯h L3 eiβL2 /¯h = cos βL3 + sin βL1 . (1. respectively. (1. β ′ . γ) = R3 (α)R2 (β)R3 (γ). Under a further rotation by R(α′ . (1. The transformation may be described by a unitary differential operator ′ˆ ′ˆ ′ˆ Uˆ (α′. The angles α. β. β ′ .432) It is easy to check that these generators satisfy the commutation rules (1. The space of these matrices has three degrees of freedom.436) . where L To calculate these we note that the 3 × 3 -matrix R−1 (α. These rotation matrices can be expressed as exponentials Ri (δ) ≡ e−iδLi /¯h . respectively. γ ′ ). γ are referred to as Euler angles. The 3 × 3 rotation matrices make it possible to express the infinitesimal rotations around the three coordinate axes as differential operators of the three Euler angles. The third requires. the wave function goes over into ψ ′ (R) = ψ(R−1 (α′ . −i¯h∂β R−1 = R−1 (cos α L2 − sin α L1 ). β.297) together with the commutation rules (1.414) of angular momentum operators. γ). (1. They are specified by the 3 × 3 orthonormal matrices Rij .431) where δ is the rotation angle and Li are the 3 × 3 matrix generators of the rotations with the elements (Li )jk = −i¯hǫijk .433) ˆ i is the representation of the generators in terms of differential operators. An arbitrary orientation is obtained by applying all finite rotations to each point of the body. γ ′ ) = R−1 (α′ . Inserting (1. after replacing β → θ and γ → ϕ.437). γ ′ ). (1.441) The sign is most simply understood by writing ˆ ξ = ai L ˆ L ξ i.62 1 Fundamentals Inverting the relations (1. L After exponentiating these differential operators we derive ˆ ′ . γ ′ )R−1 Uˆ −1 (α′ . so that U ˆ along the body axes. Lη = RL2 R−1 = − sin γ cos β(cos α L1 + sin α L2 ) + cos γ(cos α L2 − sin α L1 ) + sin γ sin β L3 . β ′ . β. β. as desired. β ′ . β ′ . U(α ˆ −1 (α′ . L (1.437) ˆ 3 = −i¯h∂α . operators agree with those in (1.439) Lζ = RL3 R−1 = cos β L3 + sin β(cos α L1 + sin α L2 ).416). β ′ . we need the components of L They are obtained by rotating the 3 × 3 matrices Li by R(α.427). β.442) 20 When applied to functions not depending on α. γ). β ′ . (1. η. sin β (1.440) Note that these commutation rules have an opposite sign with respect to those in ˆ i :20 Eqs. ˆ i in the final expressions. we find the operand replacing Li → L ators ! ˆ ξ = i¯h − cos γ cot β ∂γ − sin γ ∂β + cos γ ∂α . γ)R(α′. (1. L ˆ η ] = −i¯hL ˆζ . γ ′ )ψ(R) = ψ ′ (R). we find the differential operators generating the rotations [7]: ˆ1 L ˆ2 L ! cos α ∂γ . PATH INTEGRALS . (1. (1. γ ′ )R(α. β ′. the ˆ 1. ˆ η = ai L ˆ L η i. β. ζ = cyclic. L sin β ) sin γ ˆ η = i¯h sin γ cot β ∂γ − cos γ ∂β − L ∂α . γ) into Lξ = RL1 R−1 = cos γ cos β(cos α L1 + sin α L2 ) + sin γ(cos α L2 − sin α L1 ) − cos γ sin β L3 . up to the sign of L H.438) Uˆ (α′. [L ξ. sin β ˆ ζ = −i¯h∂γ . β. γ) = R−1 (α. ˆ ζ = ai L ˆ L ζ i. Kleinert. In the Hamiltonian (1. then. γ ′ )(α. γ)U ˆ (α′. = i¯h cos α cot β ∂α + sin α ∂β − sin β ! sin α = i¯h sin α cot β ∂α − cos α ∂β − ∂γ . β ′ .414) of the operators L ˆξ .434). γ ′ )R(α. 445) with the matrices dLmm′ (β) (L + m′ )!(L − m′ )! = (L + m)!(L − m)! " × β cos 2 !m+m′ #1/2 β − sin 2 !m−m′ (m′ −m. 1. It is easy to check that this like [L ξ ξ property produces the sign reversal in (1. aj ] = i¯hǫijk ak . 2. The can then be obtained by using only the group commutators between L spectrum is " ! # 1 1 1 2 ELΛ = h ¯ (1. In accordance with (1. γ) = DmΛ (−α. aiη . (1. these form the spinor representation of the rotations around the y-axis 1/2 dm′ m (β) = cos β/2 − sin β/2 sin β/2 cos β/2 ! . β. (1.441) with respect to (1. The full spinor representation function D 1/2 (α. and D L (α.444) ˆ 3 . β. −γ). Dmm ′ (α. (1.. σ = 1 0 0 −1 ! . . . γ) = e−iασ3 /2 e−iβσ2 /2 e−iγσ3 /2 .e. β. (1.427) by placing operator hats on the La ’s. the wave functions are L ψLΛm (α. they are vector operators.414). . the magnetic quantum numbers. L ˆη. β. −1/2. i. γ) in (1. are the eigenvalues of L ˆ are the eigenvalues of Lζ . are the components of the body axes.414). γ) Here m′ are the eigenvalues of L mΛ are the representation matrices of angular momentum L. β.449) .447) The indices have the order +1/2.15 Spinning Top where aiξ . γ) = e (1. The correspondence principle is now applied to the Hamiltonian in Eq.63 1.433) the representation matrices of spin 1/2 for the generators L commutation rules (1. σ = 0 −i i 0 ! 3 . The wave functions are the representation functions of the rotation group. −β. L where L(L + 1) with L = 0. . L(L + 1) + − 2Iξ 2Iζ 2Iξ ˆ 2 . β. .443) Λ2 .448) Thus we can write D 1/2 (α. γ are used to specify the orientation of the body axes. If the Euler angles α.446) For j = 1/2. and Λ = −L. (1. the famous Pauli spin matrices: 1 σ = 0 1 1 0 ! 2 . aiζ . L ˆ ζ . The energy spectrum and the wave functions ˆξ . .433). one may decompose ′ −i(mα+m γ) L L dmm′ (β).m′ +m) PL−m′ (cos β). Under rotations these behave ˆ i . .445) is most easily obtained by inserting into the general expresˆ i with the sion (1. (1. (±1) = ∓(ˆ P1 i ′ 1 hi|1mi = ǫ (m). β. The exponential (e−iβ Lˆ 2 )mm′ is equal to (1.433) the representation matrices of spin 1 for the generaˆ i with the commutation rules (1. 2L1 + 1 (1.. PATH INTEGRALS . these are tors L ˆ i )jk = −iǫijk .454) + dβ dβ sin2 β ! The scalar products of two wave functions have to be calculated with a measure of integration that is invariant under rotations: hψ2 |ψ1 i ≡ Z 2π 0 Z 0 π Z 0 2π dαdβ sin βdγ ψ2∗ (α. γ) can also be obtained by inserting into the general exponential (1. β. γ)Dm′ m (α.445) satisfy the orthogonality relation Z 0 2π Z 0 π Z 0 2π L1 ∗ L2 dαdβ sin βdγ Dm ′ m (α. the representation functions (1. 1 + β.β) Pl ≡ (−1)l Γ(l + β + 1) F (−l.64 1 Fundamentals The first and the third factor yield the pure phase factors in (1.451) where the indices have the order +1/2. Hence R(β)(m) = m′ =−1 (m )dm′ m (β).β) The functions Pl (z) are the Jacobi polynomials [8]. l! Γ(β + 1) where F (a.452) (1.456) H. The function 2 1/2 dm′ m (β) is obtained by a simple power series expansion of e−iβσ /2 . γ) 1 2 1 = δm′1 m′2 δm1 m2 δL1 L2 2 8π 2 . (1. −1/2. cos β − √12 sin β 1 1 √ sin β (1 + cos β) 2 2 (1.. In Cartesian coordinates.451). (1.446) form the vector representation 1 (1 + cos β) 2 √1 sin β 2 1 (1 − cos β) 2 d1m′ m (β) = − √12 sin β 12 (1 − cos β) . these become (L ˆ i )ij j (m′ ). The representation functions D 1 (α. γ). γ)ψ1(α. β. (1 + z)/2). l + 1 + α + β. which can be expressed in terms of hypergeometric functions as (α. For j = 1. where ǫijk is the completely antisymmetric tensor with simply (L ˆ i )ij hj|m′ i = ˆ i )mm′ = hm|ii(L ǫ123 = 1. . z) ≡ 1 + a(a + 1) b(b + 1) z 2 ab z+ + .447). c. using the fact that (σ 2 )2n = 1 and (σ 2 )2n+1 = σ 2 : e−iβσ 2 /2 = cos β/2 − i sin β/2 σ 2 .414).455) The above eigenstates (1.453) The rotation functions dLmm′ (β) satisfy the differential equation d2 d m2 + m′ 2 − 2mm′ cos β L − 2 − cot β dmm′ (β) = L(L + 1)dLmm′ (β). Kleinert. β. (1. The vector representation goes over into the ordinary rotation matrices Rij (β) by mapping the states |1mi onto the x ± iˆ y)/2 using the matrix elements spherical unit vectors (0) = ˆz. c c(c + 1) 2! (1. In the spherical basis. β.450) which is equal to (1. ǫ∗i (m)(L (α.445). b. ωζ are the angular velocities measured along the principal axes of the top. ω3 are obtained from the relation ˙ −1 ωk Lk = iRR (1. In terms of Euler angles. The canonical momenta associated with the Lagrangian (1.457) where ωξ . ωη . this is also true for the asymmetric top with Iξ 6= Iη 6= Iζ .458) as ω1 = −β˙ sin α + γ˙ sin β cos α.65 1. ˙ ˙ pβ = ∂L/∂ β˙ = Iξ β. ω3 = γ˙ cos β + α. To find these we note that the components in the rest system ω1 . according to (1. ω2 = β˙ cos α + γ˙ sin β sin α. (1. 3.457) are.461) Considering α. ˙ (1. pγ = ∂L/∂ γ˙ = Iζ (α˙ cos β + γ).455) up to a trivial constant factor. 2 (1.387).464) √ Hence the measure d3 q g in the scalar product (1. R . L = [Iξ (β˙ 2 + α˙ 2 sin2 β) + Iζ (α˙ cos β + γ) 2 (1. ωη = β˙ cos γ + α˙ sin β sin γ.459) After the rotation (1. γ as Lagrange coordinates q µ with µ = 1.387) with the Hessian metric [recall (1. where g = Iξ2 Iζ sin2 β. β. Iζ cos β 0 Iζ (1.463) pα = ∂L/∂ α˙ = Iξ α˙ sin2 β + Iζ cos β(α˙ cos β + γ). ω2 . the Lagrangian is 1 ˙ 2 ]. although the metric gµν is then much more complicated (see Appendix 1C). Incidentally. this can be written in the form (1.12) and (1.462) g = Iξ2 Iζ sin2 β. ωζ = α˙ cos β + γ.460) Explicitly.384) agrees with the rotationinvariant measure (1. these become ωξ = β˙ sin γ − α˙ sin β cos γ. 2. ˙ (1. ˙ (1.439) into the body-fixed system. the Lagrangian reads 1 L = [Iξ (ωξ 2 + ωη 2 ) + Iζ ωζ 2 ].388)]: gµν whose determinant is Iξ sin2 β + Iζ cos2 β 0 Iζ cos β 0 Iξ 0 = .15 Spinning Top Let us also contrast in this example the correct quantization via the commutation rules between group generators with the canonical approach which would start out with the classical Lagrangian. 404).466) This Hamiltonian has no apparent ordering problem. 2 2 4 4 sin β 4 (1. PATH INTEGRALS . treatment of the radial coordinates in H As another similarity with the two-dimensional system in radial coordinates and the particle on the surface of the sphere. we find H. It exists even though there is no apparent ordering problem. inserting the differential operators for the body-fixed angular moˆ The term H ˆ discr is the menta (1.467) Inserting these into (1. Indeed. pˆβ = −i¯h(sin β)−1/2 ∂β (sin β)1/2 = −i¯h(∂β + 1 cot β).469) (1. (1. by analogy with the ˆ of Eq.427). The correct Hamiltonian could be obtained by replacing the classical pβ 2 term in H by the operator g −1/4 pˆβ g 1/2 pˆβ g −1/4 . Kleinert.440) into the Hamiltonian (1.466) gives the canonical Hamiltonian operator ˆ can = H ˆ +H ˆ discr . just as in the radial coordinate expression (1. (1. H (1.462) H. One is therefore tempted to replace the momenta simply by the corresponding Hermitian operators which are.468) with ¯2 Iξ ˆ ≡ −h H ∂β 2 + cot β∂β + + cot2 β ∂γ 2 2Iξ Iζ # 2 cos β 1 2 ∂α − ∂α ∂γ + sin2 β sin2 β " and ! 1 3 ˆ discr ≡ 1 (∂β cot β) + 1 cot2 β = H − .465) we find the classical Hamiltonian cos2 β 1 1 1 2 pβ + + H= 2 2 Iξ Iξ sin β Iζ " ! pγ 2 # 1 2 cos β 2 + pα pγ . the Hamiltonian operator of the symmetric spinning top is correctly given by the Laplace-Beltrami operator (1.381) after inserting the metric (1.66 1 Fundamentals After inverting the metric to g µν 1 = Iξ sin2 β µν 1 0 − cos β 0 sin2 β 0 2 2 − cos β 0 cos β + Iξ sin β/Iζ . 2 pα − Iξ sin β Iξ sin2 β (1.398). pˆα = −i¯h∂α .470) ˆ agrees with the correct quantum-mechanical operator derived The first term H above. according to (1. discrepancy between the canonical and the correct Hamiltonian operator. 2 pˆγ = −i¯h∂γ . (1.391). we observe that while the canonical quantization fails. de/ ~kleinert/199). 315 (1990) (ibid. Phys. and (1C.4)]. just like a particle on the surface of a sphere. The result suggests that a replacement gµν (q)pµ pν → −¯h2 ∆ (1. the correct answer will be given in Chapters 10 and 8. In newer developments of the theory one also allows for the presence of a nonvanishing torsion.405)] to be resolved in this text. Phys. the spinning top corresponds to a particle moving in a space with constant curvature. (1C. γ space? As we shall see in detail in Chapter 10.fu-berlin.15 Spinning Top and the inverse (1. B 236 .42).http/202). where the group quantization rule is inapplicable. Kleinert.472) Thus.21 What is the characteristic non-Euclidean property of the α. 21 If the space has curvature and no torsion. Lett. It is straightforward although tedious to verify that this is also true for the completely asymmetric top [which has quite a complicated metric given in Appendix 1C. The exact definition will be found in Eq. β.physik. 22 H. An important non-Euclidean space of physical interest is encountered in the context of general relativity. the Lagrangian cannot be obtained by reparametrizing a particle in a Euclidean space with curvilinear coordinates. If torsion is present. as we shall see in more detail in Chapter 8. since for a spinning top. the relevant quantity is the curvature scalar R. see Eqs. A 4 . Originally. (10.22 The configuration space may carry curvature and a certain class of torsions (gradient torsion). Mod. Lett.465). Several arguments suggest that our principle is correct. This is an important nontrivial result. .471) produces the correct Hamiltonian operator in any non-Euclidean space. In such a general situation. For the above systems with a Hamiltonian which can be expressed entirely in terms of generators of a group of motion in the underlying space. γ space may be considered as the surface of a sphere in four dimensions. 2Iξ Iη Iζ (1. β. the new quantum equivalence principle will give the same results as the group quantization rule. In this space. The geometry is most easily understood by observing that the α. For the asymmetric spinning top we find (see Appendix 1C) (Iξ + Iη + Iζ )2 − 2(Iξ2 + Iη2 + Iζ2 ) R= . 2329 (1989) (http://www.67 1. In Chapters 10 and 8 we shall present a new quantum equivalence principle which is based on an application of simple geometrical principles to path integrals and which will specify a natural and unique passage from classical to quantum mechanics in any coordinate frame. the correspondence principle has always been a matter of controversy [see the references after (1. gravitating matter was assumed to move in a spacetime with an arbitrary local curvature. the correct correspondence principle can also be deduced from symmetry arguments. this is the correct answer.2). After a long time. hpm |S|p (1.474) to find the true scattering amplitude. This expresses the physical fact that the total probability of an incident particle to re-emerge at some time is unity (in quantum field theory the situation is more complicated due to emission and absorption processes).476) shows the normalization of the states [recall (1. hpb |S|p where (1. this amplitude oscillates with a frequency ω = E/¯h characteristic for free particles of energy E.186)].475) ˆ hpb |pa i = hpb |e−iH(tb −ta )/¯h |pa i = (2π¯h)3 δ (3) (pb − pa ) (1. The energy will be unchanged: E = Eb = p2b /2M. This leading term is commonly subtracted from (1. and it is useful to divide this out.478) H. The probability amplitude for such a process is given by the time evolution amplitude in the momentum representation ˆ (pb tb |pa ta ) ≡ hpb |e−iH(tb −ta )/¯h |pa i. Kleinert.16. Moreover. and define the scattering matrix (S-matrix) by the limit ˆ ai ≡ hpb |S|p lim tb −ta →∞ ˆ ei(Eb tb −Ea ta )/¯h hpb |e−iH(tb −ta )/¯h |pa i. which is separated as follows: ˆ a i = hpb |pa i + hpb |S|p ˆ a i′ .477) ˆ it follows that the scattering From the definition (1. since potential scattering conserves energy. so that this amplitude must contain a leading term. defining the so-called T -matrix by the decomposition ˆ a i ≡ (2π¯h)3 δ (3) (pa − pa ) − 2π¯hiδ(Eb − Ea )hpb |Tˆ|pa i. 1. Long before and after the collision. we remove these oscillations. (1. it will be found far from the potential with some momentum pb . In order to have a time-independent limit.182) and are normalized to unity in a finite volume V . the remaining amplitude contains a δfunction ensuring energy conservation. (1. In the basis states |pm i introduced in Eq. from (1.68 1. PATH INTEGRALS . (1.16 1 Fundamentals Scattering Most observations of quantum phenomena are obtained from scattering processes of fundamental particles.180) which satisfy the completeness relation (1.1 Scattering Matrix Consider a particle impinging with a momentum pa and energy E = Ea = p2a /2M upon a nonzero potential concentrated around the origin. the unitarity is expressed as X m′ ′ ′ ˆ m′′ i = hpm |Sˆ† |pm ihpm |S|p X m′ ˆ m′ ihpm′ |Sˆ† |pm′′ i = 1.474) Most of the impinging particles will not scatter at all.474) and the hermiticity of H matrix is a unitary matrix.473) where the limit tb → ∞ and ta → −∞ has to be taken. hpb |S|p (1.473). the unitarity relation reads Z 1.2 d3 p ˆ ai = hpb |Sˆ† |pihp|S|p (2π¯h)3 Z d3 p ˆ hpb |S|pihp| Sˆ† |pa i = 1.185) between the discrete states |pm i and their continuous limits |pi. and makes pb equal to pa . by integrating this over the phase space of the final momenta [recall (1.69 1.480) Cross Section ˆ a i gives the probability Pp ←p for the scattering The absolute square of hpb |S|p b a from the initial momentum state pa to the final momentum state pb . (2π¯h)3 (1. L By summing this over all discrete final momenta. L6 (1. (1. In the continuous basis |pi.479) m where L3 is the spatial volume. we see that ′ ˆ a i.484) where dΩ = dφb d cos θb is the element of solid angle into which the particle is scattered. or equivalently.483). The differential scattering cross section dσ/dΩ is defined as the probability that a single impinging particle ends up in a solid angle dΩ per unit time and unit current density. and the probability is proportional to the time T: 1 (1.485) .483) we identify dP˙ 1 1 Mp dσ 21 = = 3 2π¯ h |T | . Omitting the unscattered particles. The energy integral removes the δ-function in (1.184)].483) The momentum integral can be split into an integral over the final energy and the final solid angle.481) The factor δ(0) at zero energy is made finite by imagining the scattering process to take place with an incident time-independent plane wave over a finite total time T . For non-relativistic particles. (2π¯h)3 (1.482) Ppb ←pa = 6 T 2π¯hδ(Eb − Ea )|hpb |Tˆ |pa i|2 . we have Ppb ←pa = 1 2π¯hδ(0) 2π¯hδ(Eb − Ea )|hpb |Tˆ |pa i|2 . we find the total probability per unit time for the scattering to take place dP 1 = 6 dt L Z d3 pb L3 2π¯hδ(Eb − Ea )|hpb |Tˆ |pa i|2 . this goes as follows Z d 3 pb 1 M = 3 3 (2π¯h) (2π¯h) (2π¯h)3 Z dΩ Z 0 ∞ dEb pb . R Then 2π¯hδ(0) = dt eiEt/¯h |E=0 = T .16 Scattering Remembering the relation (1.16. ˆ a m i ≈ 1 hpb |S|p hpb m |S|p L3 (1. From (1. and pm b and pa are the discrete momenta closest to pb and pa . p p b a dΩ dΩ j L (2π¯h)3 j (1. 489) In the relativistic case. L3 E (1.484) by E = p2 + M 2 inside the momentum integral. the operator Sˆ in (1. dΩ (2π¯h)2 b a (1. where p = |p|.16. PATH INTEGRALS .491) j= Hence the cross section becomes 1. In a volume L3 . the current density of a single impinging particle is given by the velocity v = p/M as j= 1 p .487) so that the differential cross section becomes dσ M2 = |Tp p |2 . Kleinert.3 Born Approximation To lowest order in the interaction strength. (1.474) is Sˆ ≈ 1 − iVˆ /¯h. (1. this implies Tpb pa ≈ Vpb pa /¯h.492) For a time-independent scattering potential. we have to replace the constant mass M in (1.494) H. (1.488) If the scattered particle√moves relativistically. so that Z ∞ 1 d3 p = dp p2 dΩ (2π¯h)3 (2π¯h)3 0 Z ∞ 1 Z dΩ dEE p.493) where Vpb pa ≡ hpb |Vˆ |pa i = Z d3 x ei(pb −pa )x/¯h V (x) = V˜ (pb − pa ) (1. dΩ (2π¯h)2 b a (1.486) for brevity. the initial current density is not proportional to p/M but to the relativistic velocity v = p/E so that 1 p . = (2π¯h)3 0 Z Z (1.490) E2 dσ = |Tp p |2 . L3 M (1.70 1 Fundamentals where we have set hpb |Tˆ |pa i ≡ Tpb pa . Abramowitz and I. one writes dσ = |fpb pa |2 .4 Partial Wave Expansion and Eikonal Approximation The scattering amplitude is usually expanded in partial waves with the help of Legendre polynomials Pl (z) ≡ Pl0 (z) [see (1.500) where b ≡ l¯h/p is the so called impact parameter of the scattering process.495) we identify fpb pa ≡ − M Rp p . op.498) approximately as an integral fpeib pa = p i¯h Z n h i o db b J0 (qb) exp 2iδpb/¯h (p) − 1 . In terms of θ. 1.16.16 Scattering is a function of the momentum transfer q ≡ pb − pa only.71. 1 .. Stegun.P [v] = − 23 Ze2 M 1 |P| h ¯ Z ∞ −∞ dz √ M.495) The amplitude whose square is equal to the differential cross section is usually denoted by fpb pa .499) to rewrite (1. dΩ (1.e. cit. For small θ. Formula 9.1. lm (1.. This is the eikonal approximation to the scattering amplitude.497) where we have chosen the sign to agree with the convention in the textbook by Landau and Lifshitz [9]. Then (1. i.491) reduces to the so called Born approximation (Born 1926) dσ E2 2 ≈ 2 |Vpb pa | . 2 dΩ (2π¯h) h ¯ (1. As an example. 2π¯h b a (1. the momentum transfer q = pb − pa has the size |q| = 2p sin(θ/2). consider 2 Coulomb scattering where V (r) = Ze /r and (2.71 1. we can use the asymptotic form of the Legendre polynomials23 Pl−m (cos θ) ≈ 1 Jm (lθ).751) yields χei b.498) where p ≡ |p| = |pb | = |pa | and θ is the scattering defined by cos θ ≡ pb pb /|pb ||pa |.501) . b2 + z 2 (1.421)] as fpb pa ∞ h ¯ X = (2l + 1)Pl (cos θ) e2i∂l (p) − 1 2ip l=0 (1. (1.496) By comparison with (1. where ρ is the charge density.P where γ≡ ≈ b 2R !2iγ . H.504) is a dimensionless quantity since e2 = h ¯ cα where α is the dimensionless fine-structure constant 24 e2 = 1/137. is equal to twice the Rydberg energy (see also p.505) α= h ¯c The integral over the impact parameter in (1. Meas. see M. Kleinert. 2M h ¯ 2 2n2 (1. 2.E. This amplitude contains poles at momentum variables p = pn whenever Ze2 M¯h iγn ≡ = −n. 2+2iγ 2ip sin (θ/2) Γ(−iγ) (1. . IEEE Trans. 284 (1989).508) which are the well-known energy values of hydrogen-like atoms with nuclear charge Ze. except for the last phase factor which accounts for a finite screening length. . Instrum.500) can now be performed and yields fpeib pa ≈ h ¯ 1 Γ(1 + iγ) −2iγ log(2pR/¯h) e .507) This corresponds to energies E (n) = − MZ 2 e4 1 p2n =− . . . . . 3. 964).506) Remarkably.. where j is the current density. 24 Throughout this book we use electromagnetic units where the electric field E = −∇φ has the energy density H = E2 /8π + ρφ. PATH INTEGRALS . The magnetic field satisfies Amp`ere’s law ∇ × B = 4πj.P [v] = − =− |P| h ¯ b |P| h ¯ b r − b2 2 Ze M 1 2R ≈ −2 log . The fine-structure constant is measured most precisely via the quantum Hall effect .72 1 Fundamentals The integral diverges logarithmically. (1. Performing the integral up to R yields √ Z 1 R + R2 − b2 Ze2 M 1 R Ze2 M 1 ei dr √ 2 log χb. . pn n = 1. (1. but in a physical sample.210 eV. 38 . Cage et al. this is the exact quantum mechanical amplitude of Coulomb scattering. the potential is screened at some distance R by opposite charges.502) |P| h ¯ b This implies exp χei b. (1. (1.035 997 9 . so that ∇ · E = 4πρ and e2 = h ¯ c α.359 × 10−11 erg = 27. The prefactor EH ≡ e2 /aH = Me4 /¯h2 = 4.503) Ze2 M 1 |P| h ¯ (1. 509) where pb = |pb |.6 Lippmann-Schwinger Equation From the definition (1. hpb |U tb −ta →∞ tb .73 1. (1. ta ) is the time evolution operator in Dirac’s interaction picture (1. (1. ta ) = e−iHt/¯h U ˆI (0.286) it follows that the operator UˆI (tb . exp − h ¯ 2M (1.16 Scattering 1. ta ) satisfies the same composition law (1.510) This treatment of a δ-function is certainly unsatisfactory. ta )|pa i.5) as follows: ˆ ai ≡ hpb |S|p = lim ei(Eb tb −Ea ta )/¯h (pb tb |pa ta ) lim ˆI (tb . ta )e−iH0 t/¯h . UˆI (t.16. ta − t)e−iH0 t/¯h .22.515) . e−iH0 t/¯h U (1. ta ) = lim eiH0 tb /¯h U ta →−∞ ta →−∞ ta →−∞ (1. ta ) = e−iHt/¯h UˆI (0. ta ) = UˆI (0. For this we express the δ-function in the energy as a large-time limit M tb M δ(pb − pa ) = lim δ(Eb − Ea ) = pb pb tb →∞ 2π¯hM/i !1/2 i tb (pb − pa )2 . ta ): ˆI (tb .514) and therefore ˆI (0. e−iH0 t/¯h U − −→ U (1.477) and setting sloppily pb = pa for elastic scattering.474) with the help of the time evolution operator (2.512) Now we observe that ˆI (t. we may proceed with more care with the following operator calculation. We rewrite the limit (1. ta ).286).513) so that in the limit ta → −∞ ˆI (t. ta ) = UˆI (t.511) where UˆI (tb .−ta →∞ (1.16. 1. lim UˆI (tb . A satisfactory treatment will be given in the path integral formulation in Section 2. At the present stage.254) as the ordinary time evolution operator Uˆ (t. the δ-function is removed and we obtain the following expression for the scattering amplitude fpb pa pb = M q 3 2π¯hM/i lim 1 tb →∞ t1/2 b (2π¯h)3 eiEb (tb −ta )/¯h [(pb tb |pa ta )−hpb |pa i] .5 Scattering Amplitude from Time Evolution Amplitude There exists a heuristic formula expressing the scattering amplitude as a limit of the time evolution amplitude. tb )U (1. Inserting this into Eq. ta ) = lim eiH0 tb /¯h e−iHtb /¯h UˆI (0. ta )e−iH0 tb /¯h. ta ) − ˆI (0. 518) This is the famous Lippmann-Schwinger equation. ta )|pb i. we obtain the equation for the scattering matrix ˆ ai = hpb |S|p lim tb . (1.516) Note that in contrast to (1. Eb = Ea . The prefactor has the property ei(Eb −Ea )tb /¯h = lim tb →∞ Eb − Ea − iη ( 0.74 1 Fundamentals which allows us to rewrite the scattering matrix (1.520) It is easy to see that this property defines a δ-function in the energy: ei(Eb −Ea )tb /¯h = 2πiδ(Eb − Ea ). ta )|pb i = hpb |pb i − i hpb |U h ¯ Z 0 −∞ dt ei(Eb −Ea −iη)t/¯h hpb |Vˆ UˆI (0. Inserting this into (1. (1. ta )|pb i .521) Indeed. so that the prefactor can be set equal to one. the time evolution of the initial state goes now only over the negative time axis rather than the full one. For a time-independent potential. The second term contains. Eb − Ea − iη (1.−ta →∞ i(Eb −Ea )tb e " # 1 hpb |pa i − hpb |Vˆ UˆI (0. and using Eqs. ta )|pb i = hpb |pb i − 1 hpb |Vˆ UˆI (0. dξ ξ + iη −∞ Z ∞ (1. and set Eb ≡ Ea + ξ/tb. making any finite function of Eb vanish. the integral can be done and yields hpb |UˆI (0.519) Eb − Ea − iη The first term in brackets is nonzero only if the momenta pa and pb are equal. Eb = Ea .514). (1.516).517) A small damping factor eηt/¯h is inserted to ensure convergence at t = −∞.474).522) Then the Eb -integral is rewritten as eiξ f (Ea + ξ/ta ) .−ta →∞ ei(Eb −Ea )tb /¯h hpb |UˆI (0. (1. however. i/η.291) between free-particle states hpb | and |pb i. In front of the second term. (1. tb →∞ Eb − Ea − iη lim (1. in which case also the energies are equal. Taking the matrix elements of Eq. Eb = 6 Ea . a pole at Eb = Ea for which the limit has to be done more carefully. Kleinert. (1.523) H. we obtain at tb = 0 ˆI (0.291) and (1.511) as ˆ ai ≡ hpb |S|p lim tb . ta )|pb i. (1. ta )|pa i. as a consequence of the Riemann-Lebesgue lemma. PATH INTEGRALS . the prefactor oscillates rapidly as the time tb grows large. let us integrate the left-hand side together with a smooth function f (Eb ). we approximate |x − x′ | ≈ r − x vector in the direction of x. V (x′ )hx′ |U (2π¯h)3 Ea −p2b /2M +iη (1.529) . Multiplying this state by hx| from the left. ta )|pa i = hx|U Z d3 p ˆI (0. we approximate UˆI (0. we obtain Tpb pa = Vpb pa − Z 1 d 3 pc Vpb pc Tp p . ta )|pb i. at large |x|.518). 3 (2π¯h) Ec − Ea − iη c a (1. h ¯ 4π|x − x′ | pa = q 2MEa . where x ˆ is the unit is nonzero only for small x′ . the function f (Ea ) can be taken out of the integral and the contour of integration can then be closed in the upper half of the complex energy plane. Multiplying the original equation (1.519) the formula (1. ta )|pa i = hx|pihp|U (2π¯h)3 d3 p ˆI (0. From Eq. we calculate ˆI (0. (1. ta )|pa i. ta )|pa i ≈ eipa x/¯h − e hx|U 4πr Z d4 x′ e−ipa xˆ x ′ 2M ˆI (0.344) of the free particle. (1.e..359)] ′ ′ (x|x )Ea 2Mi eipa |x−x |/¯h =− . The Lippmann-Schwinger equation can be recast as an integral equation for the T -matrix.477).526) The function (x|x′ )Ea = Z d3 pb ipb (x−x′ )/¯h i¯h e (2π¯h)3 Ea − p2 /2M + iη (1. and inserting a complete set of momentum eigenstates. we consider the wave function (1. ta ) ≈ 1. i. V (x′ )hx′ |U h ¯2 (1. (1. and (1.526) becomes ipa r ˆI (0. with the T -matrix 1 ˆI (0. In three dimensions it reads [see (1.527) is recognized as the fixed-energy amplitude (1. ta )|pa i in x-space.518) by the matrix hpb |Vˆ |pa i = Vpb pc from the left.514). ta )|pa i.16 Scattering In the limit of large ta . hx|pihp|U (2π¯h)3 Z Using Eq.525) To extract physical information from the T -matrix (1. and find the Born approximation (1. this becomes hx|UˆI (0. Thus we obtain from (1.524) it is useful to analyze ˆI (0. yielding 2πi.528) In order to find the scattering amplitude. ta )|pa i. the behavior of the interacting state U ˆ with the initial we see that it is an eigenstate of the full Hamiltonian operator H energy Ea . (1. ta )|pa i = hx|pa i + Z Z 3 ′ dx ′ d 3 pb eipb (x−x )/¯h ˆI (0.493).526) far away from the scattering center.524) hpb |Tˆ |pa i = hpb |Vˆ U h ¯ For a small potential Vˆ .75 1. Under the assumption that V (x′ ) ˆ x′ . the outgoing particles are detected far away ˆ = pˆb . the factor multiplying the spherical wave factor eipa r/¯h /r is the scattering amplitude f (ˆ x)pa .531) . from the scattering center in the direction x ˆ = pb and obtain the formula we may set pa x fpb pa = lim − ta →−∞ M 2π¯h2 Z ˆI (0.510) for the scattering amplitude. For scattering to a final momentum pb . d4 xb e−ipb xb V (xb )hxb |U (1. we have avoided the singular δ-function of energy conservation. whose absolute square gives the cross section. ta )|pa i = hxb |U(0. ta )|pa i.76 1 Fundamentals In the limit ta → −∞. We ˆI (0. We are now prepared to derive formula (1. the amplitude hxb |U from the time evolution amplitude (xb tb |xa ta ) as follows: ˆI (0. ta )|pa i can be obtained observe that in the limit ta → −∞. ta )|pa i in x-space.530) By studying the interacting state UˆI (0. ˆ ta )|pa ie−iEa ta /¯h hxb |U = lim ta →−∞ −2πi¯hta M !3/2 (1. Because of energy conservation. 2 (xb tb |xa ta )ei(pa xa −pa ta /2M )/¯h . . so that it becomes 2 e−i(pa p−pa )ta /2M ¯h . The appropriate limiting formula for the δ-function δ (D) (−ta )D/2 i ta (pb − pa ) = lim √ (pb − pa )2 D exp − ta →−∞ h ¯ 2M 2πi¯hM (1.533) M Now. In this way we obtain a representation of the δ-function by which 2 the Fourier integral (1. As a cross check we insert the free-particle amplitude (1. (1. which is the correct first term in Eq.531) is a reliable starting point for extracting the scattering amplitude fpb pa from the time evolution amplitude in x-space (xb 0|xa ta ) at xa = pa ta /M by extracting the coefficient of the outcoming spherical wave eipa r/¯h /r. ta )|pa ie−iEa ta /¯h = Z 2 d3 xa (xb tb |xa ta )ei(pa xa −pa ta /2M )/¯h . (1. the momentum integration is squeezed to p = pa . (1. (4.580) for later convenience.341) into (1. Kleinert.531). Formula (1. PATH INTEGRALS . (1.531).342) by an obvious substitution of variables. (1. for large −ta . which is unity in the limit performed in that equation.533) goes over into (1. H. and we obtain (1. xa =pa ta /M . Then the right-hand side becomes Z −ta 3 2 d3 p (xb 0|pta ta )ei(pa p−pa )ta /2M ¯h . The exponential on the right-hand side can just as well be multiplied by a 2 2 2 factor ei(pb −pa ) /2M ¯h which is unity when both sides are nonzero. The phase factor ei(pa xa −pa ta /2M )/¯h on the right-hand side of Eq.532) by substituting the dummy integration variable xa by pta /M.534) is easily obtained from Eq. Its complex conjugate for D = 1 was written down before in Eq. is kept in Eq. This follows directly from the Fourier transformation hxb |Uˆ (0.526) associated with unscattered particles.531). (1.531) and obtain the free undisturbed wave function eipa x .509) with ta replaced by −tb . The integral over the Boltzmann factors of all phase space elements.77 1. its thermodynamic properties can be obtained through the following rules: At the level of classical mechanics.536) where kB is the Boltzmann constant. The reason is that h ¯ can really be omitted in calculating any thermodynamic average. If it is brought into contact with a thermal reservoir at a temperature T and has reached equilibrium. 1. Of course. (1. It will be called the inverse temperature.538) is called the classical partition function. In classical statistics it merely supplies us with an irrelevant normalization factor which makes Z dimensionless.q)/kB T e .535) is occupied with a probability proportional to the Boltzmann factor e−H(p. Then we may drop kB in all formulas.q)/kB T . In fact.3806221(59) × 10−16 erg/Kelvin. kB = 1.537) The number in parentheses indicates the experimental uncertainty of the two digits in front of it. 2π¯h (1. The quantity 1/kB T has the dimension of an inverse energy and is commonly denoted by β.539) In the sequel we shall always work at a fixed volume V and therefore suppress the argument V everywhere.ˆx)/kB T . each volume element in phase space dp dq dp dq = h 2π¯h (1. we shall sometimes take T to be measured in energy units kB times Kelvin rather than in Kelvin. the phase space integral is dpn dqn /2π¯h.25 Zcl (T ) ≡ Z dp dq −H(p.17 Classical and Quantum Statistics Consider a physical system with a constant number of particles N whose Hamiltonian has no explicit time dependence.17. for a generalZHamiltonian system with many Y degrees of freedom. forgetting about the factor kB . It contains all classical thermodynamic information of the system. 25 (1. The reader may n wonder why an expression containing Planck’s quantum h ¯ is called classical . (1.17 Classical and Quantum Statistics 1. This leads to the quantumstatistical partition function ˆ Z(T ) ≡ Tr e−H/kB T ≡ Tr e−H(ˆp.1 Canonical Ensemble ˆ and the integral In quantum statistics. the Hamiltonian is replaced by the operator H over phase space by the trace in the Hilbert space. . (1. 1. ˆ If H ˆ is an N-particle Schr¨odinger where Tr O Hamiltonian.539) contains the position operator xˆ in Cartesian coordinates rather than qˆ to ensure that the system can be quantized canonically.78 1 Fundamentals ˆ denotes the trace of the operator O.17.544) Its grand-canonical version at a fixed chemical potential is FG (T. µ) = Tr e−(H−µN )/kB T . To emphasize this relation we shall define the trace of this operator for time-independent Hamiltonians as the quantum-mechanical partition function: ˆ ZQM (tb − ta ) ≡ Tr Uˆ (tb .2 Grand-Canonical Ensemble For systems containing many bodies it is often convenient to study their equilibrium properties in contact with a particle reservoir characterized by a chemical potential µ. Here N The combination of operators in the exponent.542) ZG (T. the quantum-statistical system is referred to as a canonical ensemble. ˆ is the operator counting the number of particles in each state of the ensemble. (1.540) Obviously the quantum-statistical partition function Z(T ) may be obtained from the quantum-mechanical one by continuing the time interval tb − ta to the negative imaginary value i¯h ≡ −i¯hβ.541) tb − ta = − kB T This simple formal relation shows that the trace of the time evolution operator contains all information on the thermodynamic equilibrium properties of a quantum system. At this point we make an important observation: The quantum partition function is related in a very simple way to the quantum-mechanical time evolution operator. PATH INTEGRALS . µ). In cases such as the spinning top. For this one defines what is called the grand-canonical quantum-statistical partition function ˆ ˆ (1. (1. ˆG = H ˆ − µN. Kleinert. the quantization has to be performed differently in the way to be described in Chapters 10 and 8.545) H. (1. ta ) = Tr e−i(tb −ta )H/¯h . the free energy is defined by F (T ) = −kB T log Z(T ). ˆ H (1. In more general Lagrangian systems. the trace formula is also valid but the Hilbert space is spanned by the representation states of the angular momentum operators. The right-hand side of (1.543) is called the grand-canonical Hamiltonian. µ) = −kB T log ZG (T. Given a partition function Z(T ) at a fixed particle number N. For a system of N free particles.550) or. the density is a rapidly growing function of energy. using the grand-canonical free energy. ˆ ˆ ˆ ˆ )/kB T N ˆ e−(H−µ (1. This can be derived from the grand-canonical partition function as N = ZG −1 (T. this becomes ∂ ∂ E=T (−F (T )/T ) = 1 − T ∂T ∂T 2 ! For a grand-canonical ensemble we may introduce an average particle number defined by .544).554) i=1 where each of the particle momenta pi is summed over all discrete momenta pm in (1. (1. µ) ∂T (1.551) The average energy in a grand-canonical system. the sum can be converted into an integral26 N(E) = V N " N Z Y i=1 26 N X d 3 pi Θ(E − p2i /2M). ˆ (1. µ)kB T 2 = ∂ 1−T ∂T ! ∂ ZG (T. however. by analogy with (1. the exception noted in the footnote to Eq.555) Remember. the number of states up to energy E is given by N(E) = X pi Θ(E − N X p2i /2M). ∂µ ∂µ (1.548) Tr e−H/kB T .17 Classical and Quantum Statistics The average energy or internal energy is defined by ˆ BT ˆ −H/k E = Tr He .179) available to a single particle in a finite box of volume V = L3 .546) F (T ).547) and (1. 3 (2π¯h) i=1 # (1. For a large number of particles. For a large V .79 1. µ)kB T ∂ ∂ ZG (T. ∂µ (1.549) N = Tr N Tr e−(H−µN )/kB T . µ). ˆ ˆ )/kB T N ˆ −(H−µ E = Tr He .548).552) can be obtained by forming. µ) = kB T log ZG (T. for example. the derivative E − µN = ZG −1 (T. It may be obtained from the partition function Z(T ) by forming the temperature derivative ∂ ∂ E = Z −1 kB T 2 Z(T ) = kB T 2 log Z(T ). as N =− ∂ FG (T. µ).553) FG (T. (1. (1.184) for systems possessing a condensate. . (1. ˆ ˆ Tr e−(H−µN )/kB T . (1. µ).547) ∂T ∂T In terms of the free energy (1. . (1. Nevertheless. (8. due to the overwhelming exponential falloff of the Boltzmann factor.560) Therefore. (1. As the two functions ρ(e) and e−e/kB T are multiplied with each other. for example.562) This function has a sharp peak at E(T ) = kB T 3N 3N − 1 ≈ kB T .80 1 Fundamentals 3 which is √ simply [V /(2π¯h) ] radius 2ME: N times the volume Ω3N of a 3N-dimensional sphere of N(E) = " V (2π¯h)3 #N ≡ " V (2π¯h)3 #N Ω3N (2πME)3N/2 Γ 3 N 2 +1 (1. Gradshteyn and I. Ryzhik.. M. Formula 3. the result is a function which peaks very sharply at the average energy E of the system. op. the integral for the partition function (1. H. PATH INTEGRALS .7. the density per energy ρ = ∂N /∂E is given by " V ρ(E) = (2π¯h)3 #N 2πM (2πME)3N/2−1 .557) The surface is [see Eqs.118) for a derivation] SD = 2π D/2 /Γ(D/2). (1.117) and (8. cit. Recall the well-known formula for the volume of a unit sphere in D dimensions: ΩD = π D/2 /Γ(D/2 + 1). e−E/kB T . 2 2 kB T 2E (T ) 2 27 (1. .556) . ρ(E)e−E/kB T ∼ e(3N /2−1) log E−E/kB T . .382. S.564) I.563) The width of the peak is found by expanding (1.559) π (1.561) It grows with the very large power E 3N/2 in the energy. 2 2 (1. Kleinert. Γ( 23 N) (1.558) This follows directly from the integral27 SD = Z = Z D d p δ(p − 1) = ∞ −∞ dλ π π −iλ Z D/2 D 2 d p 2δ(p − 1) = e−iλ 2π D/2 = Γ(D/2) Z D d p Z ∞ −∞ dλ iλ(p2 −1) e (1.562) in δE = E − E(T ): ( ) 3N E(T ) 1 3N exp log E(T ) − − (δE)2 + . The position of the peak depends on the temperature T . For the free N particle system.582) is convergent. µ) and can be written approximately as ρ(E.553) we see that the entropy can be calculated directly from the derivative of the grand-canonical free energy S(T.µ)−T S(T. µ)) δ (n − N(T. µ) = E(T.µ)]/kB T . µ) − T S(T.µ)−µN (T. µ) = e−[E(T. this implies the relation FG (T. µ) − µN(T. (1. n = N(T.µ)−µN (T. (1.565) into (1.582).µ)]/kB T . (1. µ) = − ∂ FG (T.573) For the grand-canonical free energy (1. ∂T (1. and we may write ρ(E)e−E/kB T ≈ δ(E − E(T ))N(T )e−E(T )/kB T .567) Using (1. µ). the free energy can thus be expressed in the form F (T ) = E(T ) − T S(T ).570) we find for large N ZG (T.566) Inserting this with (1.569) S(T ) = − F (T ).. (1.e.571) is now strongly peaked at E = E(T. n)e−(E−µn)/kB T ≈ δ (E − E(T. µ). With N being very large.544). µ).17 Classical and Quantum Statistics √ Thus.81 1. the peak at E(T ) of width E(T )/ N can be idealized by a δ-function.575) . the√peak is very sharp. (1.545).565) The quantity N(T ) measures the total number of states over which the system is distributed at the temperature T . The deviation is of a relative order 1/ N. (1.548) we see that the entropy may be obtained from the free energy directly as ∂ (1. µ) = Z dE dn ρ(E.570) where ρ(E. ∂T For grand-canonical ensembles we may similarly consider ZG (T. (1. the exponential is reduced by a factor √ of two with respect to E(T ) ≈ kB T 3N/2. n)e−(E−µn)/kB T (1.568) By comparison with (1. µ)) × eS(T. (1. n)e−(E−µn)/kB T .574) By comparison with (1. as soon as δE gets to be of the order of E(T )/ N. we see that in the limit of a large number of particles N: Z(T ) = e−[E(T )−T S(T )]/kB T . The entropy S(T ) is now defined in terms of N(T ) by N(T ) = eS(T )/kB .µ)/kB e−[E(T.572) Inserting this back into (1. i. we express the canonical partition function ˆ (1. Inserting (1. Kleinert.e. the volume V is the only one which grows with the system. µ. Equivalently. In the arguments of the grand-canonical quantities. V as natural variables.. as follows directly from the definition (1.578) The energy has S. found from the derivative (1. V ) ≡ −p(T. µ.576) renders the last term. a quantity of fundamental interest is the density of states. of course. V ). Z(T ) = X n This can be rewritten as an integral: Z(T ) = Z The quantity ρ(E) = X n δ(E − En ) (1. respectively. we may write F = −µN − pV. When varying the volume. µ.580) e−En /kB T .18 Density of States and Tracelog In many thermodynamic calculations. V ) changes as follows: dFG (T.551) with respect to the chemical potential. the particle number N is replaced by the chemical potential µ. i.574).576) into (1. 1.582) Z(T ) = Tr e−H/kB T as a sum over the Boltzmann factors of all eigenstates |ni of the Hamiltonian:. V ) = −SdT − Ndµ − pdV. we find Euler’s relation: E = T S − Ndµ − pV. The proportionality constant defines the pressure p of the system: FG (T. (1. N.e. µ. N. µ. Thus FG (T. Among the arguments of the grand-canonical free energy FG (T.574) at a fixed volume. µ. (1.576) Under infinitesimal changes of the three variables. To define it.570). N. The canonical free energy and the entropy appearing in the above equations depend on the particle number N and the volume V of the system. V )V. V ). V are the natural variables. i.82 1 Fundamentals The particle number is. PATH INTEGRALS . the definition (1. (1. they are more explicitly written as F (T. FG (T. V ) must be directly proportional to V . (1. V ) and S(T.579) where T.583) H. (1. (1.577) The first two terms on the right-hand side follow from varying Eq. N.581) dE ρ(E)e−E/kB T . (1. An important quantity related to ρ(E) which will appear frequently in this text ˆ − E. (1. Otherwise N(E) jumps at En by half the degeneracy of this level.190) via the relation (4. In Eq. (1.585) (1.588) This formula will serve to determine the energies of bound states from approximations to ω(E) in Section 4. (1.586) is the number of states up to energy E. short tracelog.18 Density of States and Tracelog specifies the density of states of the system in the energy interval (E. if integrated to the energy of a certain level En . Indeed. of the operator H ˆ − E) = Tr log(H X n log(En − E). as we shall see later.589) It may be expressed in terms of the density of states (1. In order to apply this relation one must be sure that all levels have different energies. (1.83 1.584) as ˆ − E) = Tr Tr log(H Z ∞ −∞ ˆ log(E ′ − E) = dE ′ δ(E ′ − H) Z ∞ −∞ dE ′ ρ(E ′ ) log(E ′ − E). It may also be written formally as a trace of the density of states operator ρˆ(E): ˆ ρ(E) = Tr ρˆ(E) ≡ Tr δ(E − H).587) n This equation is correct only with the Heaviside function which is equal to 1/2 at the origin. The function N(E) is a sum of Heaviside step functions (1. (1. (4A.9) we shall exhibit an example for this situation.580): ρ(E) = Z ∞ −i∞ Z ∞ dβ βE dβ βE ˆ e Tr e−β H = e Z(1/kB β).7.590) The tracelog of the Hamiltonian operator itself can be viewed as a limit of an operator ˆ zeta function associated with H: ˆ −ν .584) The density of states is obviously the Fourier transform of the canonical partition function (1.306). is the trace of the logarithm. E + dE). (1. The integration may start anywhere below the ground state energy. 2πi −i∞ 2πi The integral N(E) = E Z dE ′ ρ(E ′ ) (1. not with the one-sided version (1.591) whose trace is the generalized zeta-function ˆ −ν ) = ζHˆ (ν) ≡ Tr ζˆHˆ (ν) = Tr (H h i X n En−ν . for instance from the Bohr-Sommerfeld condition (4.313): X N(E) = Θ(E − En ).592) .210). ζˆHˆ (ν) = Tr H (1. the result is N(En ) = (n + 1/2). 5) so that ˆ · = B ˆ · δB(t) · = B ˆ · δB(t) + i[B ˆ × δB(t)] · . δB(t) · B ˆ · δB(t) − i[B ˆ × δB(t)] · .594) Rn (E) = Tr R(E). . (1A.3) ta Using (1A. Expanding this as in (1.2) e−iH0 (tb −ta ) = cos[B(tb − ta )/2] + iB ˆ ≡ B/|B|. the integrand on the right-hand side becomes n o o n ˆ · sin[B(tb −t)/2] δB(t) · cos[B(t−ta )/2]+iB ˆ · sin[B(t−ta )/2] .589) with respect to E. B.(1A. Suppose now that the magnetic field is not constant but has a small timewhere B dependent variation δB(t). PATH INTEGRALS . (1. δe−iH0 (tb −ta ) = (1A. so that the time evolution operator reads.1) e−iH0 (tb −ta ) = ei(tb −ta )B·/2 . Kleinert.521).4) cos[B(tb −t)/2]+iB We simplify this with the help of the formula [recall (1.595) By integrating this over the energy we obtain the number of states function N(E) of Eq. this reduces to Riemann’s zeta function (2.84 1 Fundamentals For a linearly spaced spectrum En = n with n = 1. The reduced Hamiltonian operator is H in natural units with h ¯ = 1. (1. Then we obtain from (1.448)] σ i σ j = δij + iǫijk σ k (1A.586): X 1 ˆ = Θ(E − En ) = N(E).329) we see that the imaginary part of this quantity slightly above the real E-axis yields the density of states X 1 ˆ − E − iη) = − Im ∂E Tr log(H δ(E − En ) = ρ(E). From the generalized zeta function we can obtain the tracelog by forming the derivative ˆ = −∂ν ζ ˆ (ν)| . 3 .596) − Im Tr log(E − H) π n Appendix 1A Simple Time Evolution Operator Consider the simplest nontrivial time evolution operator of a spin-1/2 particle in a magnetic field ˆ 0 = −B · /2. π n (1.2). 2.293)] Z tb ˆ ˆ ˆ dt e−iH0 (tb −t) δB(t) · e−iH0 (t−ta ) . = ˆ i¯h n i¯h E−H n E − En Recalling Eq.253) [or the lowest expansion term in (1. . Tr log H H ν=0 (1. (1A. ˆ (1A.319): ˆ − E) = Tr ∂E Tr log(H X 1 X 1 1 1 ˆ = (1.593) By differentiating the tracelog (1. we obtain ˆ ˆ · sin[B(tb − ta )/2] .6) B H. we find the trace of the resolvent (1. (1. .293) and using the fact that (B· )2n = B 2n and (B· )2n+1 = B 2n (B· ). z = peiπ/4 . (1A.7) = i[B ˆ 2 = 1. z 2 = ip2 dz = i Rdp.4 Triangular closed contour for Cauchy integral 2 closed contour shown in Fig. the sum of the integrals along the three pieces of the Figure 1.2) √ The first integral converges rapidly to π/2 for R → ∞.9) cos B[(tb +ta )/2−t] δB(t)−sin B[(tb +ta )/2−t] [B Integrating this from ta to tb we obtain the variation (1A. z = R eiϕ . To see this we estimate its absolute value as follows: .8) which can be combined to give n o ˆ ˆ × δB(t)] · +i sin[B(tb −ta )/2] B·δB(t).337) by relating it to the Gauss integral.1) 0 A B Let R be the radius of the arc.Appendix 1B 85 Convergence of the Fresnel Integral and h i ˆ × δB(t)] · B ˆ · δB(t) · B ˆ · δB(t) B ˆ · + i[B ˆ · = B ˆ · B n o ˆ × δB(t)] · B ˆ + [B ˆ · δB(t)]B ˆ − [B ˆ × δB(t)] × B ˆ · . 1. and obtain the equation Z R Z −p2 iπ/4 dp e +e 0 0 R dp e −ip2 dz = dp. z 2 = p2 iπ/4 dz = dp e . The last term goes to zero in this limit.4): cos B(tb − t)/2 cos B(t − ta )/2 δB(t) · ˆ · δB(t) + i[B ˆ × δB(t)] · } +i sin B(tb − t)/2 cos B(t − ta )/2{B ˆ · δB(t) − i[B ˆ × δB(t)] · } +i cos B(tb − t)/2 sin B(t − ta )/2{B + sin B(tb − t)/2 sin B(t − ta )/2 δB · (1A.4 vanishes since the integrand e−z is analytic in the triangular domain: I Z A Z B Z O 2 −z 2 −z 2 −z 2 dze = dze + dze + dze−z = 0. since B we find for the integrand in (1A. z 2 = p2 . (1B. Thus The first term on the right-hand side vanishes. According to Cauchy’s integral theorem. + Z 0 π/4 dϕ iR e−R 2 (cos 2ϕ+i sin 2ϕ)+iϕ = 0. (1B.3). Appendix 1B Convergence of the Fresnel Integral Here we prove the convergence of the Fresnel integral (1. Then we substitute in the three integrals the variable z as follows: 0 A: B 0: AB: z = p.(1A. the second term is equal to δB. Z . Z π/4 . π/4 . 2 . −R2 (cos 2ϕ+i sin 2ϕ)+iϕ . (1B. dϕ iR e dϕ e−R cos 2ϕ .3) . . <R . 0 . 0 . Iζ cos β. or Z ∞ √ 2 dp e−ip = e−iπ/4 π. g 1 {sin β cos β sin γ cos γ(Iη − Iξ )}Iζ .12) and (1. In the dangerous regime α ∈ (π/4 − ǫ.388)] Iξ sin2 β + Iζ cos2 β − (Iξ − Iη ) sin2 β sin2 γ.1) It has the Hessian metric [recall (1.86 1 Fundamentals The right-hand side goes to zero exponentially fast.2) the limiting formula √ −e−iπ/4 π/2. g11 g21 = = g31 = g22 g32 = = Iη + (Iξ − Iη ) sin2 γ. −(Iξ − Iη ) sin β sin γ cos γ.2) rather than (1. The Lagrangian of the asymmetric top with three different moments of inertia reads L= 1 [Iξ ωξ 2 + Iη ωη 2 + Iζ ωζ 2 ]. π/4) with small ǫ > 0. g 1 {cos β[− sin2 γ(Iξ − Iη ) − Iη ]}Iζ .3) and the inverse metric has the components g 11 = g 21 = g 31 = g 22 = g 32 = g 33 = 1 {Iη + (Iξ − Iη ) sin2 γ}Iζ . R sin 2α ϕ=α (1B. g 1 sin β sin γ cos γ(Iξ − Iη )Iζ . Kleinert. g 1 {sin2 β[Iξ − sin2 γ(Iξ − Iη )]}Iζ . (1C. 2 (1C. one certainly has sin 2ϕ > sin 2α. g33 = Iζ . sin 2α (1B. so that R Z π/4 dϕ e −R2 cos 2ϕ <R Z π/4 α α dϕ sin 2ϕ −R2 cos 2ϕ e .337) by substituting p → p Appendix 1C The Asymmetric Top p a/2.462). The determinant is g = Iξ Iη Iζ sin2 β. except for angles ϕ close to π/4 where the cosine in the exponent vanishes.5) which goes to zero like 1/R for large R. (1C.4) The right-hand integral can be performed by parts and yields αR e−R 2 cos 2α + iϕ=π/4 h 2 1 e−R cos 2ϕ . R0 ∞ 2 dp e−ip = (1B. Thus we find from (1B. g (1C. 0. g 1 {sin2 βIξ Iη + cos2 βIη Iζ + cos2 β sin2 γ (Iξ − Iη )Iζ }.6) ∞ which goes into Fresnel’s integral formula (1.4) H. PATH INTEGRALS . = cos γ sin γ(Iη − Iξ )/Iζ . = {cos γ sin γ[Iη2 − Iξ2 + (Iξ − Iη )Iζ ]}/2Iξ Iη . ¯ 11 R = {sin2 β[sin2 γ(Iη3 − Iξ3 − (Iξ Iη − Iζ2 )(Iξ − Iη )) + ((Iξ + Iζ )2 − Iη2 )(Iξ − Iζ )] + Iζ3 − Iζ (Iξ − Iη )2 }/2Iξ Iη Iζ . From this The other components follow from the symmetry in the first two indices Γ νµ Christoffel symbol we calculate the Ricci tensor. (1C. = −[(Iξ − Iη )2 − Iζ2 ]/2Iξ Iη . (1.5) ¯ µν λ = Γ ¯ λ . the Christoffel symbol Γ λ ¯ and R calculated full affine connection Γµν . respectively. = [sin2 γ(Iξ2 − Iη2 − (Iξ − Iη )Iζ ) − Iη (Iξ − Iη + Iζ )]/2 sin βIξ Iη . = 0. = 0. from Γ .6) Contraction with g µν gives the curvature scalar ¯ = [2(Iξ Iη + Iη Iζ + Iζ Iξ ) − I 2 − I 2 − I 2 ]/2Iξ Iη Iζ .70): ¯ 11 1 Γ ¯ 21 1 Γ ¯ 31 1 Γ ¯ 22 1 Γ ¯ 32 1 Γ = [cos β cos γ sin γ(Iη2 − Iη Iζ − Iξ2 + Iξ Iζ )]/Iξ Iη .Appendix 1C The Asymmetric Top 87 From this we find the components of the Riemann connection. = 0. (10. = [cos γ sin γ(Iξ2 − Iη2 − Iζ (Iξ − Iη ))]/2Iξ Iη . = {cos β[sin2 γ(Iη2 − Iξ2 + (Iξ − Iη )Iζ ) + Iη (Iξ − Iη + Iζ )]}/2 sin βIη Iξ . = 0. ¯ 21 2 Γ ¯ 31 2 Γ = {cos β cos γ sin γ[Iξ2 − Iη2 − Iζ (Iξ − Iη )]}/2Iξ Iη . ¯ 22 R ¯ 32 R = {sin2 γ[Iξ3 − Iη3 + (Iξ Iη − Iζ2 )(Iξ − Iη )] + Iη3 − (Iξ − Iζ )2 Iη }/2Iξ Iη Iζ . to be defined in Eq. = 0. ¯ 33 1 Γ ¯ 11 2 Γ = 0. = {sin β[sin2 γ(Iξ2 − Iη2 − Iζ (Iξ − Iη )) − Iξ (Iξ − Iη − Iζ )]}/2Iξ Iη . ¯ 21 R ¯ 31 R = {sin β sin γ cos γ[Iη3 − Iξ3 + (Iξ Iη − Iζ2 )(Iη − Iξ )]}/2Iξ Iη Iζ . the Christoffel symbol defined in Eq.7) ¯ µν λ is equal to the Since the space under consideration is free of torsion. ¯ 33 R = −{cos β[(Iξ − Iη )2 − Iζ2 ]}/2Iξ Iη . ¯ 22 2 Γ ¯ 32 2 Γ ¯ 33 2 Γ ¯ 11 3 Γ ¯ 21 3 Γ ¯ 31 3 Γ ¯ 22 3 Γ ¯ 32 3 Γ ¯ 33 3 Γ = {cos β sin β[sin2 γ(Iξ2 − Iη2 − Iζ (Iξ − Iη )) − Iξ (Iξ − Iζ )]}/Iξ Iη . = {cos γ sin γ[sin2 β(Iξ Iη (Iξ − Iη ) − Iζ (Iξ2 − Iη2 ) + Iζ2 (Iξ − Iη )) + (Iξ2 − Iη2 )Iζ − Iζ2 (Iξ − Iη )]}/Iξ Iη Iζ . = [cos β cos γ sin γ(Iξ2 − Iη2 − Iζ (Iξ − Iη ))]/2Iξ Iη . R ξ η ζ (1C. (1C. = {cos β[sin2 γ(Iξ2 − Iη2 − (Iξ − Iη )Iζ ) + Iη (Iξ + Iη − Iζ )]}/2 sin βIξ Iη . The same thing is true for the curvature scalars R ¯ µν λ and Γµν λ .41). = {sin2 β[sin2 γ(2Iξ Iη (Iη − Iξ ) + Iζ (Iξ2 − Iη2 ) − Iζ2 (Iξ − Iη )) +Iξ Iη (Iξ − Iη ) + Iη Iζ (Iη − Iζ )] − sin2 γ((Iξ2 − Iη2 )Iζ − Iζ2 (Iξ − Iη )) − Iη Iζ (Iξ + Iη − Iζ )}/2 sin βIξ Iη Iζ . Analytische Mechanik . 13. Physica 20. Hermann & Cie. G. North-Holland . Gelfand and G. I and II. 680. I. H. Moser. H. [9] L. 1967. Angular Momentum in Quantum Mechanics. Geometric Quantization. A. Pergamon. Pergamon.M. R. Princeton University Press. 2nd ed. London. Oxford. 5. 1992. Vols. 1949. Lectures on Celestial Mechanics.K. 143 (1975). Phys. L. Springer.88 1 Fundamentals Notes and References For more details see some standard textbooks: I. Gavazzi. 1958. Phys. New York. Phys. Kleinert. Lifshitz and L. a student of K.2. John Wiley and Sons. Lagrange. 1963. H. Mechanik .M.M.M. [3] L. 586 (1980). Springer. 812 (1928). L. Quantum Mechanics. Newton.D. New York. Physica A 103. New York. 2. [5] This was first observed by B. Weierstrass.56) and (4. Mod.I. Prog. Harri Deutsch. see A. 241 (1981). 1971. J. Th´eorie des distributions. Messiah. Schwarz. Mathematische Prinzipien der Naturlehre. (1973). Lifshitz on Statistical Mechanics. Generalized functions. Kawai. Shilov. Springer.R. Academic Press. 1987. McGraw-Hill. L. Springer.M.M. Merzbacher. [8] For detailed properties of the representation matrices of the rotation group. Vols.I-II. Lehrbuch der Theoretischen Physik . Statistical Physics. Berlin. 1965.J.M. Phys. New YorkLondon. London. Quantum Mechanics. Lifshitz. Lifshitz. 1950. 1968. Frankfurt. Vol. Phys. Statistical Physics. Vols. Goldstein. (4. 1965.S. Kawai. Rev. Princeton. T. W. Quantum Mechanics. Hamel. Marsden. 1887. 274 (1954).P. 1977. Landau and E. Foundations of Mechanics.M. C. Amsterdam. [2] The integrability conditions are named after the mathematician of complex analysis H. Podolsky. J. Theoretische Mechanik . DeWitt. 1970. Landau and E. Woodhouse. L. See also the alternative approach by N.E. Schwartz. [6] B.A.57).B. New York. Quantum Mechanics. Darmstadt. 29. Wiley. Kamo and T. Landau and E. Siegel and J. Rev. Landau and E. 50. Berlin. Nuovo Cimento 101A.A. [7] C. Cheng. 1960.D. Mechanics. Classical Thermodynamics. 377 (1957). London. Callen.I-II. Classical Mechanics. Pitaevski. Clarendon. Quantum Mechanics. Lifshitz. Found. Wiss. 1958.S..L. 1950-51. van Winter.D. Lifshitz. Oxford University Press.M. Addison-Wesley. 169 in the above-cited textbook by L. 1960. H. See p. 3rd ed. London. Berlin. Pergamon. PATH INTEGRALS . 1963. Pergamon. Oxford. Theor. Edmonds. Paris. Sommerfeld. Dirac. Benjamin. Abraham and J. 1965. Bose-Einstein condensation will be discussed in Sections 7. who taught at the Humboldt-University of Berlin from 1892–1921.1 and 7. Schiff.L. Pergamon. 1961.Math. The particular citations in this chapter refer to the publications [1] For an elementary introduction see the book H.2. [4] An exception occurs in the theory of Bose-Einstein condensation where the single state p = 0 requires a separate treatment since it collects a large number of particles in what is called a Bose-Einstein condensate. 1964-68.D. Buchgesellschaft.E. Landau and E. P. 32. G. The Principles of Quantum Mechanics. E. Weizel. A. 1723 (1972). E. Berlin. More details are also found later in Eqs. K. London.D.M. Dekker. Reading.4.
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