MasteringPhysics: Assignment Print Viewhttp://session.masteringphysics.com/myct/assignmentPrint?assignmen... [ Print View ] PHYS 1003 Physics 1 (Technological) 2007 Quantum Physics - PHYS1003 Assignment 6 & Tutorial (Chap 39-40) Due at 5:00pm on Friday, October 26, 2007 View Grading Details Quantum Physics - Assignment 6 and Tutorial Questions (Chapter 39-40) There are two sections to this MasteringPhysics exercise. The first consists of 8 compulsory assignment questions, each worth 5 marks and counting towards your final marks. Assignment questions must be completed by the deadline given above. The second section consists of a number of tutorial questions selected by your lecturers as being valuable practice for the course. You are strongly encouraged to complete at least some of these tutorial questions. They are not counted towards your final marks and they can be completed at any time, even after the assignment deadline. Assignments are due on Fridays at 5pm local time. Hints for Using MasteringPhysics Eight attempts are allowed per part and you may request the solution. We recommend that you work through the questions off-line. Don't just use your eight attempts as chances to guess. Pay special attention to the multiple choice questions as the maximum mark declines rapidly with incorrect responses. If necessary, use the Hints to get the correct answer. The cost in marks is small, as discussed in the FAQ page. Make sure MasteringPhysics interprets what you type the way you mean it - e.g. slide the cursor across symbols to check their syntax; explicitly include * for multiplication; use brackets to control the order of operations in an expression; carefully read feedback provided to wrong answers. If using MasteringPhysics to calculate values employing angles, remember that angles are in radians. Remember that, for example, atan means inverse tan when using the math palette. 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Remember the resources available to help you: the textbook - your #1 reference from the unit WebCT page: Unit and Module outlines Lecture notes Assignment solutions Physics resources for each module Multiple Choice Questions Sample and Past Examination papers and more. ask your lecturer, tutor or fellow students on the WebCT discussion forum consult the Duty Tutor (12 to 2, four days per week, Room 201) Physics Student Office, Room 202 University Student Services Assignment Questions The de Broglie Relation 1 of 20 23/10/2007 9:23 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen... Learning Goal: To understand de Broglie waves and the calculation of wave properties. In 1924, Louis de Broglie postulated that particles such as electrons and protons might exhibit wavelike properties. His thinking was guided by the notion that light has both wave and particle characteristics, so he postulated that particles such , where as electrons and protons would obey the same wavelength-momentum relation as that obeyed by light: is the wavelength, Part A Find the de Broglie wavelength for an electron moving at a speed of . . (Note that this speed is low the momentum, and Planck's constant. enough that the classical momentum formula , and Planck's constant is is still valid.) Recall that the mass of an electron is Express your answer in meters to three significant figures. ANSWER: = 7.270×10−10 Part B Find the de Broglie wavelength . Express your answer in meters to three significant figures ANSWER: = 1.05×10−34 of a baseball pitched at a speed of 44.0 . Assume that the mass of the baseball is As a comparison, an atomic nucleus has a diameter of around . Clearly, the wavelength of a moving baseball is too small for you to hope to see diffraction or interference effects during a baseball game. Part C Consider a beam of electrons in a vacuum, passing through a very narrow slit of width . The electrons then head toward an array of detectors a distance 0.9000 away. These detectors indicate a diffraction pattern, with a broad maximum of electron intensity (i.e., the number of electrons received in a certain area over a certain period of time) from the center of the pattern. What is the with minima of electron intensity on either side, spaced 0.529 wavelength of one of the electrons in this beam? Recall that the location of the first intensity minima in a single slit diffraction pattern for light is , where is the distance to the screen (detector) and is the width of the slit. The derivation of this formula was based entirely upon the wave nature of light, so by de Broglie's hypothesis it will also apply to the case of electron waves. Express your answer in meters to three significant figures. ANSWER: = 1.18×10−8 Part D What is the momentum of one of these electrons? Express your answer in kilogram-meters per second to three significant figures. ANSWER: = 5.64×10−26 2 of 20 23/10/2007 9:23 PM The question. This gives a wave with somewhat well-defined position and wavelength. For the wave in the first figure. The principle states that you can never simultaneously know the exact location and momentum of a particle. "Where is the wave?" does not have a well-defined answer. In this problem. the wavelength that you found in Part C is much larger than that of these electrons. the uncertainty principle. and vice versa. This idea is difficult to reconcile with common experience. However. which looks essentially like a single isolated beat cycle.. the less you know about its momentum. any notion of wavelength for such a wave seems strange. A wave like the one shown in the second figure can be built up by adding together waves with different wavelengths. Heisenberg did not call his idea the uncertainty principle. is produced . as we expect for a particle.com/myct/assignmentPrint?assignmen. This is the essence of the indeterminacy principle. Correspondingly.masteringphysics. However. not just unknown. because position and momentum are fundamentally indeterminate. Electron microscopes frequently use accelerating voltages on the through a potential difference of around order of tens of kilovolts. you will consider such a wave packet as 3 of 20 23/10/2007 9:23 PM . The momentum that you found in Part C could be given to an electron by accelerating it .. the position is not well defined at all. you must consider the properties of a wave. you need to work at scales similar to or smaller than the diameter of an atom. yielding wavelengths roughly one thousand times smaller. This could be considered a wave with a very well determined position. Understanding Heisenberg's uncertainty principle is one of the keys to understanding quantum mechanics. then you get a wave the frequencies between packet. According to the de Broglie equation. If you add contributions from all of and . In fact. Heisenberg's Uncertainty Principle Learning Goal: To understand. for the waves described by quantum mechanics. it states that momentum and position are not simultaneously well defined for quantum particles. Rather. and . the wavelength is clearly well defined. To understand it better. he called it the indeterminacy principle. In order to observe the wave nature of the electron.MasteringPhysics: Assignment Print View http://session. The uncertainty principle is more than just a statement about the difficulty of measuring such things experimentally. are added together. the momentum of a wave is directly related to its wavelength. a wave with a beat frequency of Recall that if two waves with similar frequencies. it states that the more you know about the position of the particle. We could just as easily draw a single sharp point at some particular x coordinate. This is much smaller than the usual momentum of electrons used for standard diffraction experiments or electron microscopy. qualitatively and quantitatively. Further. Use your results from Part A. Express your answer in terms of quantities given in Part A and fundamental constants. The product of the uncertainties in the momentum and position of a particle is on the order of Planck's constant. the uncertainty principle is found to state that greater-than-or-equal-to sign indicates that some less than ideal waveforms have greater uncertainty that the minimum value of .. Part D In an atom. If we take this to be the uncertainty in the in its momentum? 4 of 20 23/10/2007 9:23 PM . Let the distance between the two nodes of the wave be the uncertainty in position by .com/myct/assignmentPrint?assignmen. ANSWER: = This gives you the general idea of what the uncertainty principle states mathematically. ANSWER: = Part C What is the value of the product ? Use to find the uncertainty in the momentum of the particle.. simply being one beat cycle of this wave. can be rewritten in terms of the wave number . . what is the minimum uncertainty meters.masteringphysics. find the wave numbers as and .MasteringPhysics: Assignment Print View http://session. Recall that wave number corresponding to frequencies . this will give a useful approxmation. By looking more . Part A The de Broglie relation is defined by and . . Since the beat frequency is given Express your answer as two expressions separated by a comma. = Part B Find an expression for the uncertainty in the wave number. Express your answer in terms of quantities given in Part A. and the wave travels at speed . While not exactly correct. The rigorously at the definition of the uncertainty. . Using the fact that . an electron is confined to a space of roughly electron's position. and . Use ANSWER: . the uncertainty in position is given by . ANSWER: = 1. What is the minimum uncertainty in the momentum of the pebble? Express your answer in kilogram meters per second to one significant figure. ANSWER: = 1. Uncertainty in the Atomic Nucleus Rutherford's scattering experiments gave the first indications that an atom consists of a small. Consider a nucleus with a diameter of roughly Part A Consider a particle inside the nucleus. this corresponds to an uncertainty in the speed of about meters per second. you will use the uncertainty principle to get a rough idea of the kinetic energy of a particle inside the nucleus. meters. In this problem. it would be impossible to measure the position of a pebble to such accuracy. The justification for this sort of estimation lies in the rigorous statistical definition of the uncertainty. which typically ranges from a bit less than an electron volt up to a few dozen electron volts. The uncertainty is the uncertainty Hint A. His experiments also allowed for a rough determination of the size of the nucleus.MasteringPhysics: Assignment Print View http://session. ANSWER: = 3.masteringphysics. What . In practice.1 of its momentum? To find this.. much less its speed. use in its position is equal to the diameter of the nucleus. A good estimate for the energy scale of a particle can often be found by calculating the energy the particle would have if you set the momentum equal to the minimum uncertainty in momentum.8 Notice that this energy is similar to the energy scale for electrons in an atom.. Such tiny values are the reason that you are unaware of the uncertainty principle in everyday situations. 5 of 20 23/10/2007 9:23 PM . Express your answer in kilogram meters per second to two significant figures. dense.0×10−19 For a 100-gram pebble.05×10−24 Part E What is the kinetic energy of an electron with momentum kilogram meters per second? Express your answer in electron volts to two significant figures. positively charged nucleus surrounded by negatively charged electrons.com/myct/assignmentPrint?assignmen. it is sufficient now for you to know that this will give a reasonably good order-of-magnitude estimate of the energy for a variety of quantum systems. The uncertainty relation Hint not displayed Express your answer in kilogram-meters per second to two significant figures. Part F Suppose that you know the position of a 100-gram pebble to within the width of an atomic nucleus ( meters). Part B Determine the kinetic energy Part B. Use kilograms as the mass of the particle. ANSWER: = 2. this cannot happen. Hint B. The Wavelength of an Electron An electron has de Broglie wavelength 2. Find the kinetic energy as a function of momentum Part not displayed Express your answer in joules to three significant figures.MasteringPhysics: Assignment Print View http://session. . ANSWER: = 0.. If a particle's average momentum were to fall below that point.81 Compare this to the normal energy scale for electrons in an atom. then the uncertainty principle would be violated. Using the minimum momentum of a particle in the nucleus. which is on the order of single electron volts. find the minimum kinetic energy of the particle. Since the uncertainty kilogram-meters per second as principle is a fundamental law of physics.97×10−18 6 of 20 23/10/2007 9:23 PM . Note that since our calculations are so rough.com/myct/assignmentPrint?assignmen.1 Choosing the kinetic energy formula Hint not displayed Express your answer in millions of electron volts to two significant figures. this serves as the mass of a neutron or a proton. This difference can be seen in calculations of the energy output per unit mass from coal-burning power plants. The characteristic energy scale for the nucleus seems to be roughly one million times that for electrons in an atom.85×10−10 Part A Determine the magnitude of the electron's momentum Hint A. ANSWER: = 2.33×10−24 .1 of the electron.1 The de Broglie wavelength Hint not displayed Express your answer in kilogram meters per second to three significant figures.10×10−20 Part B The uncertainty sets a lower bound on the average momentum of a particle in the nucleus. which utilize chemical energy (energy associated with the electrons of an atom) compared to the energy output from nuclear reactors (power plants that harness the much higher energies of nuclei).masteringphysics.. ANSWER: = 2. where and are real.6 Width of a Wave Function A particle is described by a wave function Part A If the value of Hint A.MasteringPhysics: Assignment Print View http://session. ANSWER: = 18. what effect does this have on the particle's uncertainty in position? .3 Find how the FWHM varies with respect to Part not displayed ANSWER: The particle's uncertainty in position will decrease.1 The relation between electron volts and joules Hint not displayed Express your answer in electron volts to three significant figures.. The particle's uncertainty in position will increase. There is no effect on the particle's uncertainty in momentum. Part C Determine the electron's kinetic energy in electron volts. Hint C..masteringphysics. Schrödinger Equation and the Particle in a Box Learning Goal: To become familiar with the Schrödinger equation and its solution for the simple case of the particle in a box. positive constants.1 is increased. Part B If the value of Hint B.com/myct/assignmentPrint?assignmen.1 is increased. How to approach the problem Hint not displayed Part A.2 Find the FWHM of Part not displayed Part A. 7 of 20 23/10/2007 9:23 PM . The particle's uncertainty in momentum will increase. what effect does this have on the particle's uncertainty in momentum? How to approach the problem Hint not displayed ANSWER: The particle's uncertainty in momentum will decrease. There is no effect on the particle's uncertainty in position. Consider a particle in a potential well with infinitely high walls. ). is beyond the scope of introductory physics. . for proper value for is a solution to the Schrödinger equation for the particle in a box. you are able to check a solution.. is a mathematical solution to any Schrödinger equation. where is Planck's constant divided by (i. of the wave function in this .e. However. the particle in a box is also important as an illustration of many key concepts from quantum mechanics. . What is the second derivative. solving the Schrödinger equation allows you to determine the particle wave functions. Finding solutions to the Schrödinger equation. for most potentials..MasteringPhysics: Assignment Print View http://session. Although it does have some real-world applications. . Given a potential energy function . . once it is presented to you. and ? . The potential energy function is formally written as where is the width of the box. You will do this for the simple case of the particle in a box. By inspection. You will prove this and calculate the . you should be able to see that functions Part A Consider the interval interval? Express your answer in terms of ANSWER: . = Part B What is in the interval . and are clearly valid solutions outside the interval . The most important equation in quantum mechanics is the Schrödinger equation. with respect to . The quantum mechanical particle in a box has a particularly simple potential energy function.masteringphysics.com/myct/assignmentPrint?assignmen. Express your answer in terms of ANSWER: = 0 Part C 8 of 20 23/10/2007 9:23 PM . so the . It is claimed that each of the functions .. The points where the domains meet are exactly the points where the potential energy becomes infinite. First.. Both derivatives are continuous and 0 are continuous and To check the second criterion. the energy of a particle with wave function . . it must satisfy several criteria. one of the main ideas of quantum theory--the quantization of energy--may be seen in the discrete allowed energy levels: . . except for points where the potential energy becomes infinite (as it does at the walls of the box). . you have proven that. where their domains meet. Find the expression for the left side of the Schrödinger equation valid on the interval .masteringphysics. You will soon determine if it is also a physical solution. . . = This entire expression is just side of the equation to be constants are equal. is an as yet undetermined constant: the energy of the particle. it must be normalizable. so you don't have to check for continuity there. and . you can check the first criterion by noting that the two functions that they have the same value at and . Part F For a solution to be a physical solution. In this context. Since you have already found the right multiplied by a positive constant ( ). In this case.. if the two is a mathematical solution to the Schrödinger equation for the particle in a box. and . Express your answer in terms of ANSWER: . it must be continuous everywhere. and 0. What is Express your answer in terms of ANSWER: = . . . . Finally. in the interval ? Part D Combine your answers from Parts A and B. . you can see how the quantization of energy is a natural consequence of applying the boundary conditions to solutions of the Schrödinger equation.com/myct/assignmentPrint?assignmen. multipled by a positive constant. 9 of 20 23/10/2007 9:23 PM . simply take the first derivative of functions in their domains. and . Even in the simple case of the particle in a box. it must have a continuous derivative everywhere. Second.MasteringPhysics: Assignment Print View http://session. Part E Combine your answers from Parts C and D to find the value of Express your answer in terms of ANSWER: = . . and Hint not displayed Hint A.masteringphysics. this equation reduces to Use this equation to find the unique positive value of Part F. An electron in this long chain molecule behaves very consists of a chain of carbon atoms. Recall that photons are described by the .. The third criterion requires that there exist some value of such that . and Find the wavelength in terms of energy of a photon in terms of the photon's energy .com/myct/assignmentPrint?assignmen. Part A. and .2 state. Part not displayed Express your answer in terms of ANSWER: = . Part A Find the wavelength of the photon that must be absorbed by an electron to move it from the and that the electron has mass . roughly much like a particle in a box.2. Since is zero outside of the interval . is the and is Planck's constant divided by Find the wavelength is the speed of light. The important light-absorbing molecule in human eyes is called retinal.1. Retinal long. is the angular frequency.1 Evaluate the integral . where .1 equations frequency.MasteringPhysics: Assignment Print View http://session. Find the energy of the photon of the state for the electron and the energy of the Find the difference between the energy Hint A. state of a box to the state. Assume that the box has length Part A.a Energy of the state of a particle in a box 10 of 20 23/10/2007 9:23 PM . Vision and the Particle in a Box Though the particle in a box (infinite potential well) seems like a very unrealistic potential.a Relation between Express your answer in terms of ANSWER: = . . it can actually be used to explain a bit about how humans see. . All three contain retinal bonded to a large protein.masteringphysics. Part C In a human eye.. The molecule would have to be shorter to be more sensitive to red light and longer to be more sensitive to blue light. = Express your answer in terms of ANSWER: .1 Comparing wavelengths Hint not displayed ANSWER: The molecule would have to be shorter to be more sensitive to both red and blue light. The way that retinal bonds to the protein can change the length of the potential well within which the electrons are confined. and . The three different types are most sensitive to red. = Part B The retinal molecule has 12 electrons that are free to move about the chain. respectively. . Express your answer in nanometers to two significant figures. Use the equation that you found in Part A to determine the wavelength of this photon.MasteringPhysics: Assignment Print View http://session. ANSWER: = 570 A photon with this wavelength lies in the green part of the spectrum. . and blue light. the speed of light . Planck's constant (divided by ) . green. and . The molecule would have to be longer to be more sensitive to red light and shorter to be more sensitive to blue light. Thus. Hint not displayed Express your answer in terms of . Use the length of the retinal molecule given in the introduction as the length of the box and use for the mass of the electron. the lowest energy photon that can be absorbed by this molecule would be the one that moves an electron from the 6th state to the 7th. there are three types of cones that allow us to see colors. How would the length have to change from that given in the introduction to make the molecule more sensitive to blue or red light? Hint C. these 12 electons fill the first 6 states of the box (with 2 electrons in each state).com/myct/assignmentPrint?assignmen. 11 of 20 23/10/2007 9:23 PM . ANSWER: . The molecule would have to be longer to be more sensitive to both red and blue light. For reasons that you may learn later. masteringphysics. allowing you to see. They can be done at any time. ANSWER: Part B What is the probability of finding the particle in the region to ? ANSWER: Part C Add the probabilities calculated in parts (a) and (b). The following questions are from the textbook and have no hints or other feedback. This impulse gets fed to the brain. Chapter 39 De Broglie Waves in the Bohr Model The hypothesis that was put forward by Louis de Broglie in 1924 was astonishing for a number of reasons. even after the assignment deadline. it changes shape.34 A particle is in the ground level of a box that extends from Part A What is the probability of finding the particle in the region between 0 and . where it is processed along with the signals from all of the other light-sensing cells in the eye.MasteringPhysics: Assignment Print View http://session. where is normalized.nn' are from the textbook and have no hints and other feedback.. but another astonishing aspect was how well the hypothesis fit in with certain parts of existing physics. ? Calculate this by integrating to . ANSWER: 0. In this problem. we explore the correspondence between the de Broglie picture of the wave nature of electrons and the Bohr model of the hydrogen atom. Problem 40.500 Tutorial Questions The following questions have been selected to allow you to use MasteringPhysics to build your Physics skills. Part A What is the de Broglie wavelength Hint A. Once retinal has absorbed a photon. Questions marked as 'Problem n.1 of the electron in the first Bohr energy level of the hydrogen atom? Speed in the Bohr model Hint not displayed 12 of 20 23/10/2007 9:23 PM .. They are for no credit but you are strongly encouraged to complete as many as possible.com/myct/assignmentPrint?assignmen. initiating a cascade of effects that eventually creates a nerve impulse. from to . Named questions have full MasteringPhysics hints and feedback. An obvious reason is that associating a wavelike nature with particles is far from intuitive. Hint A. ANSWER: .. What is the and the de Broglie wavelength ? relationship between the circumference of the orbit of the th energy level Part E.2 De Broglie wavelength Hint not displayed Express your answer in terms of electron . = Answer not displayed Part D Part not displayed Part E In the previous parts.2 De Broglie wavelength Hint not displayed Express your answer in terms of ANSWER: .2 Find the orbital circumference at the th energy level in hydrogen Part not displayed Express your answer in terms of ANSWER: and . there does exist a definite relationship. and . However.1 Find the Broglie wavelength for the th energy level. .masteringphysics.1 of the electron in the third ( ) Bohr energy level of the hydrogen atom? Speed in the Bohr model Hint not displayed Hint C. = Answer not displayed Simultaneous Measurements of Position and Velocity 13 of 20 23/10/2007 9:23 PM . you saw that there is not equality between the de Broglie wavelength of an electron in the hydrogen atom and the circumference of its orbit.com/myct/assignmentPrint?assignmen.MasteringPhysics: Assignment Print View http://session. . Part not displayed Part E.. . and the magnitude of the charge on the = Answer not displayed Part B Part not displayed Part C What is the de Broglie wavelength Hint C. the mass of the electron . In quantum mechanics. the x component of the electron's is ? Use the following velocity. is the particle's uncertainty in the x component of . where . ANSWER: = Answer not displayed Part B Part not displayed The Uncertainty Principle: Virtual Particles The uncertainty principle can be expressed as a relation between the uncertainty the time interval during which the system remains in that state. Part A The x coordinate of an electron is measured with an uncertainty of expression for the uncertainty principle: . This idea is at the base of the theory of virtual particles. and Hint A. their interaction is interpreted in terms of emission and absorption of . which is then absorbed by the other electron after a photons: One of the two electrons emits a photon with energy short period of time.MasteringPhysics: Assignment Print View http://session. In symbols. What is .2 Find the electron's minimum uncertainty in velocity Part not displayed Express your answer in meters per second to three significant figures.1 How to approach the problem Hint not displayed 14 of 20 23/10/2007 9:23 PM . How long can the photon survive before it is absorbed without violating the uncertainty principle? Hint A. their interaction would be described in terms of the electrostatic force. . in the energy state of a system and The energy-time uncertainty principle says that the longer a system remains in the same energy state.. Another implication is that physical processes can violate the law of energy conservation as long as the violation occurs for only a short time.1 How to approach the problem Hint not displayed Part A.. Part A Consider two electrons that interact with each other.masteringphysics. . where is Planck's constant. determined by the uncertainty principle. where is the uncertainty in the x coordinate of a particle.com/myct/assignmentPrint?assignmen. is Planck's constant. Classically. if the minimum percentage uncertainty in a simultaneous measurement of momentum. the higher the accuracy (or the smaller the uncertainty) a measurement of that energy can be. and or .11×10−31 Answer not displayed m Part B A proton moves with the same speed.67×10−27 ANSWER: for the mass of a proton.63×10−34 ANSWER: .17 By extremely careful measurement.170 component along this axis with a standard deviation of 3. Part A What is the minimum uncertainty in the x-component of the velocity of the car's center of mass as prescribed by the Heisenberg uncertainty principle? Use for Planck's constant... What is its de Broglie wavelength? for the mass of an electron. He claims that this method enables him to detect and its momentum simultaneously the position of a particle along an axis with a standard deviation of 0. = Answer not displayed Part B Part not displayed Problem 39.19 A scientist has devised a new method of isolating individual particles.masteringphysics. Express your answer in terms of ANSWER: . Use 1. Answer not displayed m Problem 39. 15 of 20 23/10/2007 9:23 PM . Answer not displayed ANSWER: Problem 39. The car has a mass of .com/myct/assignmentPrint?assignmen.20×10−25 Part A . you determine the x-coordinate of a car's center of mass with an uncertainty of only . Determine its de Broglie wavelength.3 Part A An electron moves with a speed of 5. for Planck's constant and 9.MasteringPhysics: Assignment Print View http://session.60×106 Use 6. 27 Part A What is the de Broglie wavelength of an electron that has been accelerated from rest through a potential increase of 900 ? Use 6. ANSWER: Answer not displayed m Part B What is the de Broglie wavelength of a proton accelerated from rest through a potential decrease of 900 Use 1. Outside the well. is one of the simplest in quantum mechanics. ANSWER: Answer not displayed Problem 39. The figure is a graph of potential energy versus position.60×10−19 for the charge on an electron. but it also shows more of the qualities that are characteristic of quantum systems.. you will .e. because classically the particle would be trapped in the potential well. The case of a particle in an infinite potential well. Such states consider a particle in a state with energy are called bound states.com/myct/assignmentPrint?assignmen. Answer not displayed m ? Chapter 40 The Finite Square-Well Potential: Bound States Learning Goal: To understand the qualities of the finite square-well potential and how to connect solutions to the Schrödinger equation from different regions.11×10−31 for the mass of an electron. also known as the particle in a box. for ) the solutions take the form . The closely related finite potential well is substantially more complicated to solve. Use the Heisenberg uncertainty principle to evaluate the validity of this claim. The potential energy function for a finite square-well potential is where is a positive number that measures the depth of the potential well and is the width of the well. In this problem. which shows why this is called the square-well potential.63×10−34 for Planck's constant. where and form and and are constants . 16 of 20 23/10/2007 9:23 PM .masteringphysics. 9...MasteringPhysics: Assignment Print View http://session.67×10−27 ANSWER: for the mass of a proton. Inside the well (i. the solutions take the . and 1. where and are constants . You may have worked similar problems before. as a mass on a spring using classical mechanics.. this must become Notice that the wave function is nonzero in the entire domain by classical mechanics since between classical and quantum mechanics. What does this imply about the constants ANSWER: in the region and . and in practice you would never expect to use quantum mechanics to describe a mass on a spring. this must ? (Be careful of signs..masteringphysics.MasteringPhysics: Assignment Print View http://session. Part A For a one-dimensional wave function to be normalizable. Part B Now.) Answer not displayed Part C Part not displayed Part D Part not displayed Part E Part not displayed Part F Part not displayed Part G Part not displayed Classical and Quantum Harmonic Oscillators Consider a harmonic oscillator with mass and . since . goes to infinity. but nonzero. probability of finding the particle hundreds of kilometers away from the potential well. As goes to infinity or negative infinity. This "tunneling" into the classically forbidden region is a key difference is some small. there . What does this imply about the constants ANSWER: and in the region ? . Also notice that. As goes to negative infinity. even though this region would be forbidden anywhere in the region.com/myct/assignmentPrint?assignmen. consider the wave function become zero. it must go to zero as Consider the wave function zero. Keep in mind that this system would be enormous by quantum standards. it is interesting to see what 17 of 20 23/10/2007 9:23 PM . but this time you will use the solution to the Schrödinger equation for the harmonic oscillator. Nonetheless. released from rest at a of the state of the harmonic oscillator? ..2 Find Part not displayed Part A.. quantum mechanics predicts here.1 Finding from equilibrium. Throughout this problem. what is the kinetic energy of the particle if it has a speed of 1.MasteringPhysics: Assignment Print View http://session. use Part A Let this oscillator have the same energy as a mass on a spring.50 Part A Classically. of energy to excite an electron in a box from the ground level to the first excited level. with the same displacement of Hint A.com/myct/assignmentPrint?assignmen. ANSWER: = Answer not displayed and .70 ? is in a box with a width L. Part B Part not displayed Part C Part not displayed Problem 40. ANSWER: Problem 40.3 It takes Part A What is the width L of the box? Use for Planck's constant and Answer not displayed for the mass of an electron. What is the quantum number from and Hint not displayed Part A.5 A particle with mass of 5.3 Find the energy of the oscillator Part not displayed Express the quantum number to three significant figures.masteringphysics. ANSWER: Answer not displayed J 18 of 20 23/10/2007 9:23 PM . . ANSWER: Part B The electron makes a transition from the n = 1 to n = 4 level by absorbing a photon. and 9. organic molecule used in a dye laser behaves approximately like a particle . in a box with width 4. Odd numbered problems have brief 19 of 20 23/10/2007 9:23 PM . ANSWER: Answer not displayed m Part B What is the wavelength of the photon emitted when the electron undergoes a transition from the second excited level to the first excited level? ANSWER: Answer not displayed m Further Questions There are many more questions available at the end of every chapter of the textbook. Part B What is L if the ground state energy of the particle equals the kinetic energy calculated in part (A)? Use 6.masteringphysics.39 Photon in a Dye Laser.63×10−34 ANSWER: for Planck's constant.MasteringPhysics: Assignment Print View http://session. 6.com/myct/assignmentPrint?assignmen. Calculate the wavelength of this photon..11×10−31 for the mass of an electron.8 Part A Find the excitation energy from the ground level to the third excited level for an electron confined to a box that has a width of .20 Part A What is the wavelength of the photon emitted when the electron undergoes a transition from the first excited level to the ground level? Use 3.63×10−34 for Planck's constant. Use for Planck's constant and Answer not displayed for the mass of an electron.00×108 for the speed of light in a vacuum. Answer not displayed ANSWER: Problem 40. Answer not displayed m Problem 40. Use for the speed of light in a vacuum. An electron in a long. com/myct/assignmentPrint?assignmen.. Answers to all 'Exercises' and 'Problems' can be requested during lunch time Duty Tutor sessions.86% avg.. answers at the back of the book. Summary 8 of 21 items complete (37. score) 39.masteringphysics.76 of 40 points 20 of 20 23/10/2007 9:23 PM .MasteringPhysics: Assignment Print View http://session.