Problems solutions in nonrelativistic quantum mechanics

Nonrelativistic quantum mechanics - Problems, exercises, etc.. Title: Problems and solutions in nonrelativistic quantum mechanics.. Title: Nonrelativistic quantum mechanics.. To remedy

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Library of Congress Cataloging-in-Publication Data

Capri, Anton Z

Problems & solutions in nonrelativistic quantum mechanics / Anton Z Capri

p cm

Includes bibliographical references

ISBN 9810246331 (alk paper) — ISBN 9810246501 (pbk.: alk paper)

1 Nonrelativistic quantum mechanics - Problems, exercises, etc I Title: Problems and solutions in nonrelativistic quantum mechanics II Title: Nonrelativistic quantum

mechanics III Title

QC174.24.N64 C374 2002

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Copyright © 2002 by World Scientific Publishing Co Pte Ltd

All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher

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This book is printed on acid-free paper

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To bkaidrite, who knows that physics is simple because "everything equals zero"

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P r e f a c e

Soon after the first edition of Nonrelativistic Quantum Mechanics appeared,

I received numerous requests for solutions to the problems in that book To remedy this situation I started by writing out solutions to the more difficult

problems, but as I proceeded with the third edition of Nonrelativistic Quantum Mechanics I also revised some of the problems and added quite a few others

Since in constructing these new problems I had to solve them, in the first place,

to be sure that they were indeed problems that students could solve, I finally went on to write out solutions to all the problems However, I did not simply want a compendium of solutions of the Schrodinger equation since with pro-grams such as Maple or Mathematica these solutions are accessible to every student Instead I wanted to concentrate on problems that teach quantum me-chanics It is with this in mind that I began to collect and solve problems My idea was to provide a means for students to learn quantum mechanics by "doing it" This is why the book begins with extremely simple problems and progresses

to more difficult ones

Some of the problems extend results that are usually taught in a course on quantum mechanics But, by having the students obtain the results themselves they are more likely to retain the ideas and at the same time gain confidence in their own abilities

As usual, I tested most of these problems on my students Sometimes they came up with very original ways of looking at old problems I have learned a lot from my students It is this learning process that led me to occasionally intro-duce more than one way of solving a problem since the solutions are intended

to help students to obtain a better understanding of the techniques involved in tackling problems in quantum mechanics

The notation and methods used are those explained in Nonrelativistic tum Mechanics and I frequently refer to chapters from that book The chapter headings are also the same as in Nonrelativistic Quantum Mechanics Never-

Quan-theless, the present book is independent and should serve as a companion to any of the numerous excellent books on quantum mechanics Throughout the book I have used Gaussian units since these are the units most commonly used

in atomic physics I also tried to arrange the problems according to increasing degree of difficulty This, was not always possible since it would have meant losing the possibility of arranging them according to topic

It is a pleasure to thank Professor M Razavy for his generous help in, not

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only providing me with some wonderful problems and supplying me with merous references, but also for his constant moral support

nu-Of course the students who suffered through the courses in which I subjected them to all sorts of quantum problems also deserve my heartfelt thanks To their credit, the undergraduates seldom complained On the other hand, there was many an evening, after I had assigned some more than usually difficult problems

in the graduate course on quantum mechanics, that walking down the hall of the fourth floor of the physics building I heard my name muttered with less than flattering epithets Nevertheless, the graduate students survived and many, after they completed their degree, even thanked me for what they had learned

It is my hope that these problems and solutions will be of use to future generations of physics students At any rate they should provide more enter-tainment than solving cross word puzzles

A.Z.Capri

Edmonton, Alberta

July, 2002

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C o n t e n t s

4 One-Dimensional Problems 29

4.11 Wronskian and Non-degeneracy in 1 Dimension 40

5.5 Reflection and Transmission Amplitudes: Phase Shifts 53

7 P h y s i c a l I n t e r p r e t a t i o n 79

9.9 Commutators of x and p with L 132

10.16 Angular Momentum of a Two-Particle System 182

10.19 Runge-Lenz Vector: Constant of the Motion 186

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11 Transformation Theory

11.6 Heisenberg Picture: SHO and Constant Force 203

12.10 Electron in Box Perturbed by Electric Field 248

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12.20 van der Waals' Interaction 260

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- -1 • JO Jí ' 11NI I

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19 Scattering - T i m e Independent 389

19.2 Rodrigues Formula: Spherical Bessel Functions 392

19.13 Effective Range, Scattering Length: Yukawa Potential 405

19.18 Born Approximation: Spherically Symmetric Potential 413

19.22 Scattering from an Inverse Square Potential 418

19.25 n-p Scattering: Singlet and Triplet States 422

19.28 WKB s-Wave Phase Shift: Attractive Potential 426 19.29 WKB s-Wave Phase Shift: Hulthén Potential 427

20.2 Identical s = 1/2 Particles in I = 0,1 States 439

20.4 Identical s = 1/2 Particles in Centre of Mass 441

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20.12 Diagonalization of Quadratic Hamiltonian 452

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m is the mass of the earth = 6 x 1027 g ,

M is the mass of the sun = 2 x 1033 g ,

G is the gravitational constant = 6.67 x 1 0- 8 dyn cm2/g2 and

Proceeding as for the hydrogen atom we find that we need only replace e2 by

Also we see that since

1.3 Photons in a B e a m

For a monochromatic beam of electromagnetic radiation of wavelength (A «

5000 Á) , intensity 7 = 1 watt/m", calculate the number of photons passing an

i 4 H r vnulTÜl\ Aiurn ó

1.4 Hydrogen Atom and de Broglie

Show that if one assumes that the circumference of a stationary state orbit of an electron in a hydrogen atom is an integral multiple of the de Broglie wavelength, one also obtains the correct energy levels

1.5 Vibrations in NaCl

The shortest possible wavelength of sound in sodium chloride is twice the lattice spacing, about 5.8 x 1 0- 8 cm The sound velocity is approximately 1.5 x 105 cm/sec

a) Compute a rough value for the highest sound frequency in the solid

b) Compute the energy of the corresponding phonons, or quanta of vibrational energy

c) Roughly what temperature is required to excite these oscillations appreciably?

Solution

The effect of grinding up the crystal into a powder is to limit the maximum wavelength of a standing wave in the crystal to roughly the size of the crystal particles This changes the integral in the Debye expression [1.1] for the internal energy from

where Ám a x = 1 0- 5 cm « size of the powder particles If we now estimate xmin

at room temperature by using that vo ps 500 m / s , we get xm¡„ « 8 x 1 0- 4 < < 1

Therefore, we can approximate the exponential in the last integral by e x ss 1 + a:

For a collection of atoms with energies E n , n — 1, 2, 3, submerged in

a background of radiation at a temperature T, the following transitions may

occur:

1) spontaneous from n —>• m E n > E m

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are known respectively as the "Einstein Coefficients" of spontaneous and

Coeffi-cient" of induced absorption [1.2] Use these equations together with Planck's

radiation law for the radiation density p(v) at equilibrium to show that

1) the Einstein coefficients of induced absorption and emission are the same, that is that Byim — Bum and that

2) the Einstein coefficients of spontaneous and induced emission are related by

c ó

S o l u t i o n

In equilibrium, at a temperature T, if the number of atoms in the state n and

m is given by N n and N rn respectively, we have that

where N is the total number of atoms Therefore,

N m = N n e (E n -E m )/k B T _ N n e hv/k B T (1.7.35)

Also at equilibrium the number of transitions from n —»• m equals the number

of transitions from m —> n Thus, we have

Therefore, comparing these two equations we see that we have

Sci-entific Publishing Co Pte Ltd., section 1.12, (2002)

[1.2] F.K Richtmyer, E.H Kennard, and J.N Cooper, Introduction to Modern

Physics, 6th edition, sec 13.12, McGraw-Hill, New York, (1969)

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C h a p t e r 2

Review of Classical

M e c h a n i c s

2.1 Lagrangian and Hamiltonian for SHO

Find the Lagrangian for a harmonic oscillator Use the definition of conjugate momentum p to find it and also the Hamiltonian H

2.2 Lagrangian and Hamiltonian: Simple

Furthermore, the expression for the conjugate momentum (2.2.9) shows that it

is just the angular momentum

2.3 Bohr-Sommerfeld Quantization: SHO

Use Bohr-Sommerfeld quantization to calculate the energy levels of a dimensional simple harmonic oscillator

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The motion is bounded by ± x0 = ±\/2E/k The two signs in front of p indicate

that we have momentum to the right + and then momentum to the left — Therefore,

2.4 Bohr-Sommerfeld: Particle in a Box

Use Bohr-Sommerfeld quantization to calculate the energy levels of a particle

confined to a box of length L For simplicity assume this is a "one-dimensional

so that we have the particle moving to the right with constant momentum V¿mE

Assume that B is constant and derive the equation of motion for L Show that

the gyroscope precesses with the angular Larmor frequency

u)L — MB

S o l u t i o n

For a gyroscope with angular momentum L we have that if the angular velocity

is Q and the moment of inertia is 7 then

The equation of motion is

This yields immediately a first integral

But, for no magnetic field (B = 0) we have the gyroscope at rest and we

immediately get 0 = 0 for 5 = 0 So we see that the integration constant vanishes Hence,

S o l u t i o n

The system of quantization proposed by Bohr and Sommerfeld is applicable to systems with repetitive (periodic) motion It is not applicable to systems with unbounded motion

Consider the time-dependent Schrodinger equation and put

Solution

If we put

V = Ae iS ' h

in the Schrodinger equation we get

Now let H -» 0 then we get

(2.7.30) This is just the Hamilton-Jacobi equation

2.8 W K B Approximation

the case of a one-dimensional Schrodinger equation Find the equations for Wo

and Wi and solve the equation for Wq This is the so-called

(2.8.34)

So we have an equation for W\

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2.9 Dumbbell Molecule: Bohr-Sommerfeld

A dumbbell molecule of moment of inertia I is rotating about its centre of mass

a) Write the langrangian for this system and find the canonical momentum b) Use Bohr-Sommerfeld quantization to find the energy levels of this molecule

[2.1] A.Z Capri, Nonrelativistic Quantum Mechanics 3rd edition, World

Sci-entific Publishing Co Pte Ltd., section 2.4, (2002)

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C h a p t e r 3

E l e m e n t a r y S y s t e m s

3.1 Commutator Identities

a) Verify the identity

[AB,C] = A[B,C] + [A,C]B

b) Using the result above and [x, p] = ih prove that

b) By applying this identity we immediately get

[a;2,p] = x[x,p] + [x,p]a; = xih + ihx = 2ihx (3.1.2)

If we replace x 2 above by x n we can provide a proof by induction since

[x n , p ] = x n - 1 [x,p]+[x n - 1 ,p}x (3.1.3)

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and assuming the result holds true for n — 1 we immediately get the result for

n Also, we have already shown the result to be true for n = 2

The second part may be proved in two different ways

i) Since g(x) is analytic at x — 0 we can write a Maclaurin series

Assume that the potential V is complex of the form V = U + iW Show that

W corresponds to a sink or source of probability

Hint: Show that

Next, we multiply the first equation by and the second by $ and subtract the two equations to get

(*¥*) = (**V2tf - l-V2**) + 2 i W W (3.2.11) After rearranging and using the definitions

3.3 Group and Phase Velocity

a) In deep water the phase velocity of water waves of wavelength A is

v = I9±

2 t t

What is the group velocity?

b) The phase velocity of a typical electromagnetic wave in a wave guide has the form

So we see that w = \fgk Therefore,

du) 1 i—— 1 u> 1 I gX

°' = ü ' 2 ^ = ? ' = ü = 2 ] l k ' ( 3'3 1 5 ) b) This time we have

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So, c 2 k 2 = ui 2 — uIq Differentiating this equation with respect to k we get

c) AB if A and B are linear

-f) Ail> = exp(AV>) •

S o l u t i o n

In all cases, to see if the operator is linear, we have to check whether

where Aj and A2 are constants The results are:

a) Linear since

J K(x,y) [XiAifii(y) + \ 2 Aifi 2 (y)] dy

BB ^"(Ai^i -(- A 2 ^2) — AiBB ^ifii X 2 BB 'ifi 2

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where we have used the fact that B is linear We now operate on this equation

But by an argument the same as for part b) above and induction we see that

exp A is linear In fact, by the same sort of argument, if A is linear then any function of A say f(A) is also linear

f) Not linear because

3.5 Probability Density

(3.4.26)

a) Compute, in closed form, the probability density p(t, x) for the wave function

^(í,a;) = J dk A(k) exp—i - kx

To see the spreading of the wave packet we find the values of x such that p has

dropped to roughly 1/e of its maximum value These points are obtained by setting the argument of the exponential equal to —1 Then we find

Now, use the results of problem 3.1 Then,

[Lx, Ly] = [yPz - zpy,zp x - xp z ]

A beam of free particles is moving along the x-axis with velocity v such that

there is one particle in a volume V

a) What is the corresponding normalized, time-dependent wave function for

(3.7.40)

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22 CHAPTER 3 ELEMENTARY SYSTEMS

The particle is confined to the volume V This means that the normalization integral is over the volume V So,

3.8 Time Evolution of Wave Function

A free particle has the wave packet at time t = 0 given by

„ikx

¥ ( 0 , * ) = [

J —( dk

k 2 + a2

Determine an expression for the wavefunction for a later time t Do not attempt

to evaluate the resulting integral

Hint: Use the equation

3.9 OPERATOR HAMILTONIAN 2 3

where

So we have to find an appropriate f(k) and carry out the integral To do this

we use the initial condition that

To make an operator out of this we must do two things:

a) We must replace p by the operator

i ax

and

b) rearrange the order of the operators P o p and x op so that the resultant

Hamil-tonian is hermitian The easiest, and by no means unique, way of doing this is

to symmetrize the operators Therefore we propose

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