Problems in Quantum Mechanics

By I. I. Gol’dman and V. D. Krivchenkov

A comprehensive collection of problems of varying degrees of difficulty in nonrelativistic quantum mechanics, with answers and completely worked-out solutions. Among the topics: one-dimensional motion, transmission through a potential barrier, commutation relations, angular momentum and spin, and motion of a particle in a magnetic field. An ideal adjunct to any textbook in quantum mechanics, useful in courses in atomic and nuclear physics, mathematical methods in physics, quantum statistics and applied differential equations. 1961 edition.

• Language: English
• Publisher: Dover Publications
• Release date: May 9, 2012
• ISBN: 9780486173214

Book preview

solutions.

PREFACE TO THE RUSSIAN EDITION

This book contains problems in non-relativistic quantum mechanics which have been solved at seminaria or given as home assignments for 4th year students at the Moscow State University. The selection contains problems of various degrees of difficulty. The problems requiring comparatively lengthy calculations are intended primarily for students of theoretical physics, who as their basic text-book in quantum mechanics have used the book of L. D. Landau and E. M. Lifshitz, Quantum Mechanics.

Didactic experience has shown that mastering of the matrix form of quantum mechanics presents the greatest difficulty. Therefore, in preparation of the present book a great deal of attention was paid to the construction of perturbation matrices and their diagonalization. A relatively large amount of space has been devoted to auxiliary problems on angular momentum and spin, since a serious study of quantum mechanics is not possible without an understanding of these basic notions.

The authors would like to express their gratitude to V. V. Tolmachev, to A. R. Frenkin and V. D. Kukin for their aid in preparing the book and also to E. E. Zhabotinskiĭ for his critical remarks.

I. Gol’dman and V. Krivchenkov

TRANSLATORS’ PREFACE

While the translation was in progress, the authors proposed a number of important revisions to the original Russian edition. These revisions, all of which have been incorporated into the English edition, involve changes in the formulation of some of the problems and solutions, inclusion of a number of new problems, and deletion of a few problems of lesser importance. The authors also drew attention to a number of typographical errors appearing in the formulae of the original Russian edition.

In the English edition the scalar product of two vectors A and B is indicated by the symbol (AB) and the vector product by [AB].

The translators are indebted to G. Bialkowski, who prepared the translation for the Polish edition, for a number of helpful comments and to Ruth Marquit and Olga Lepa for aid in preparing the book for press.

E. Marquit and E. Lepa

December, 1960

PROBLEMS

§1. One-dimensional motion. Energy spectrum and wave functions

1. Determine the energy levels and normalized wave functions of a particle in a potential box. The potential energy of the particle is V = ∞ for x < 0 and x > a, and V = 0 for 0 < x < a.

2. Show that a particle in a potential box (see the preceding problem) satisfies the relation

Show that for large values of n the above result coincides with the corresponding classical solution.

3. Find the probability distribution for different values of the momentum of a particle in a potential box in the nth energy state.

4. Find the energy levels and the wave functions of a particle in a non-symmetric potential well (Fig. 1). Investigate the case V1 = V2.

Fig. 1

5. Find the energy of the bound state of a particle in a potential field of the form

if V0→ ∞ and a → 0 under the condition that V0a = q.

6. To eliminate the quantities ħ, μ,ω defined by

so that the energy E will be expressed in units of ħω (E = ɛħω). The Schrödinger equation for the oscillator in the new variables then takes the form

show that

(b) Find the normalized wave functions and energy levels of the oscillator.

Express the wave function of the nth excited state in terms of the wave function of the ground state with the help of the operator â.

in the energy representation.

Hint

7. On the basis of the results of the previous problem, show by direct multiplication of matrices that for an oscillator in the nth energy state

8. Determine the probability of finding the particle outside of the classical limits for the ground state.

9. Find the energy levels of a particle moving in a potential field of the shape

10. Write the Schrödinger equation for an oscillator in the p representation and determine the probability distribution for different values of momentum.

11. (see Fig. 2) and show that the energy spectrum coincides with the spectrum of an oscillator.

Fig. 2

12. (see Fig. 3).

Fig. 3

13. (Fig. 4). Normalize the wave function for the ground state.

Fig. 4

Consider the limiting cases of small and large values of V0.

14. Find the wave functions of a charged particle in a uniform field V(x) = — Fx.

15. Find the Schrödinger equation in the p representation for a particle moving in a periodic potential field V(x) = V0 cos bx.

16. Find the Schrödinger equation in the p representation for a particle moving in a periodic potential field V(x) = V(x + b).

17. Determine the zones of allowable energy for a particle moving in the periodic potential field shown in Fig. 5. Investigate the limiting case V0 → ∞, b→0 with the condition

V0b = constant.

Fig. 5

18. find the energy levels and the total number of discrete levels in the quasi-classical approximation.

19. Using the quasi-classical approximation, determine the energy spectrum of a particle in the field:

20. Using the quasi-classical approximation, find the mean value of the kinetic energy of a stationary state.

21. Using the result of the preceding problem, find in the quasi-clasical approximation the mean kinetic energy of a particle in the field:

(see Prob. 19).

22. using the quasi-classical approximation and the virial theorem.

23. Derive an expression for the potential energy V(x) in terms of the energy spectrum En in the quasi-classical approximation. Let V(x) be an even function V(x) = V(—x), increasing monotonically for x > 0.

24. Find the energy of the bound state in the well V = — q δ(x).

§2. Transmission through a potential barrier

1. In studying the emission of electrons from metals it is necessary to take into account the fact that electrons with energy sufficient to escape from the metal can, according to quantum mechanics, undergo reflection at the surface of the metal. Consider a one-dimensional model with the potential V = — V0 for x < 0 (inside the metal) and V = 0 for x > 0 (outside the metal) (Fig. 6) and determine the reflection coefficient of an electron of energy E >0 at the surface of the metal.

Fig. 6

2. (see Fig 7),

Fig. 7

find the reflection coefficient of an electron of energy E > 0.

3. Find the transmission coefficient of a particle through a rectangular barrier (see Fig. 8).

Fig. 8

4. Find the reflection coefficient of a particle by a rectangular barrier for E > V0.

5. (Fig. 9) of a stream of particles moving with the energy E < V0.

Fig. 9

6. Calculate in the quasi-classical approximation the transmission coefficient of electrons through the surface of a metal in a strong electric field of intensity F (Fig. 10). Determine the limits of applicability of the approximation.

Fig. 10

7. acts over a large distance from the surface. Taking into account the force due to the electric image, determine the transmission coefficient D through the surface of a metal in an electric field (Fig. 11).

Fig. 11

8. Find approximate expressions for the energy levels and the wave functions of a particle in a symmetric potential field (see

Fig. 12

9. A symmetric field V(x) is represented by two potential wells separated by a barrier (see Fig. 13). Using the quasi-classical approximation, find the energy levels of a particle in the field V(x). Compare the obtained energy spectrum with the energy spectrum of a single well. Calculate the splitting of the energy levels for one of the wells.

Hint. See Appendix I.

Fig. 13

10. Assume that up to the time t = 0 there was an impenetrable partition between two symmetric potential wells (see the preceding problem) and that a particle was in a stationary state in the well on the left. How long after removal of the partition will the particle reach the well on the right?

11. A field V(x) is represented by N identical potential wells separated by equal potential barriers (see Fig. 14). Assuming that the quasi-classical conditions are satisfied, find the energy levels for the field V(x).

Compare the obtained energy spectrum with the energy spectrum for the individual wells.

Fig. 14

12. Find for the quasi-classical case the quasi-stationary levels of a particle in the symmetric field shown in Fig. 15. Find also the transmission coefficient D(E) for particles of energy E < V0.

Fig. 15

13. Find the transmission coefficient through the potential barrier V = q δ(x).

14. Find the quasi-stationary levels of a particle in the potential field V(x) = q {δ(x+a) + δ(x—a

15. Find the transmission coefficient through a potential barrier of the form V(x) = q{δ(x + a) + δ(x — a)}.

16. Consider a one-dimensional model of the scattering of electrons on a fixed particle which can be found in two energy states. Consider the force of interaction to be of the short-range type.

§3. Commutation relations. Uncertainty relation. Spreading of wave Packets

1. are Hermitian,

is valid.

2. are Hermitian, then

where

3.

Hintin the form of a Taylor series.

4. Estimate the energy of the ground state of an oscillator from the uncertainty relation.

5. Estimate the energy of an electron in the Kth shell of an atom of atomic number Z in the non-relativistic and relativistic cases.

6. Estimate the energy of the ground state of a two-electron atom of nuclear charge Z with the help of the uncertainty relation.

7. The magnetic field produced by a free electron depends on the motion of the particle as well as on its intrinsic magnetic moment.

As is known from electrodynamics, the intensity of the magnetic field of the moving charge is of the order of magnitude

and the field intensity of a magnetic dipole of moment µ is of the order

The magnetic moment µ of a free electron can be determined from a measurement of the intensity of the field produced by it if the following two conditions are satisfied:

and

These conditions mean that the region of localization of an electron ∆r should be smaller than the distance from this region to the point of observation of the magnetic field.

Can these two conditions be satisfied simultaneously?

Hint

8. What is the physical sense of the quantity p0 in the expression for the wave function

where the function φ(x) is real?

9.

10. The wave function of a free particle at the time t = The function ψ(x, 0) is real and differs significantly from zero only for values of x lying in