Exercises in quantum mechanics; a collection of illustrative problems and their solutions 2nd rev edition

Wilson-Sommerfeld Quantization Condition 1 The Delta Function, Completeness, and Closure 16 Angular Momentum and Coupled States 262 Tensor Operators, and Evaluation of Matrix Elements 29

Exercises in Quantum Mechanics

Kluwer Texts in the Mathematical Sciences

VOLUME 6

A Graduate-Level Book Series

The titles published in this series are listed at the end of this volume

King Fahd University of Petroleum and Minerals,

Dhahran, Saudi Arabia

Library ofCongress Cataloging-in-Publication Data

"CIP-data available from the Publisher on request."

ISBN 978-94-010-5172-9 ISBN 978-94-011-2652-6 (eBook)

DOI 10.1007/978-94-011-2652-6

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© 1992 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1992

Softcover reprint ofthe hardcover lst edition 1992

No part ofthe material protected by this copyright notice may be reproduced or uti1ized in any form or by any means, electronic or mechanical,

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To: Vasso

and

Tn f.p.a un f.uTiv

Wilson-Sommerfeld Quantization Condition 1 The Delta Function, Completeness, and Closure 16

Angular Momentum and Coupled States 262 Tensor Operators, and Evaluation of Matrix Elements 291

326

Preface

This monograph is written within the framework of the quantum mechanical paradigm It is modest in scope in that it is restricted to some observations and solved illustrative problems not readily available in any of the many standard (and several excellent) texts or books with solved problems that have been written on this subject Additionally a few more or less standard problems are included for continuity and purposes of comparison

The hope is that the points made and problems solved will give the student some additional insights and a better grasp of this fascinating but mathematically somewhat involved branch of physics

The hundred and fourteen problems discussed have intentionally been chosen to involve a minimum of technical complexity while still illustrating the consequences

of the quantum-mechanical formalism

Concerning notation, useful expressions are displayed in rectangular boxes while calculational details which one may wish to skip are included in square brackets Beirut

June, 1985

IX

HARRY A MAVROMATIS

Preface to Second Edition

More than five years have passed since I prepared the first edition of this graph

mono-The present revised edition is more attractive in layout than its predecessor, and most, if not all of the errors in the original edition (many of which were kindly pointed out by reviewers, colleagues, and students) have now been corrected

Additionally the material in the original fourteen chapters has been extended with significant additions to Chapters 8, 13, and 14

Three new chapters, Chapters 15-17 were added in order to make this volume more comprehensive and more complete in its coverage of Elementary Quantum Mechanics, principally by including material on angular momentum coupling and tensor algebra

As a result of these additions and the revisions of the original fourteen chapters, the present monograph includes 228, as opposed to the 114 solved exercises in the first edition

The author would like to acknowledge the support of the King Fahd University

of Petroleum and Minerals in this project

Dhahran

October, 1991

Xl

Harry A Mavromatis

Schematic illustration of various approaches to calculating Energy Levels

of Quantum Mechanical Systems

1) Wilson-Sommerfeld Quantization Condition (Chapter 1)

Approaches for Ground State: 1) Uncertainty Principle (Chapter 5)

2) Variational Technique (Chapter 10)

CHAPTER 1

Wilson-Sommerfeld Quantization Condition

The hydrogen atom, when treated using Bohr's admixture of classical and tum concepts involves an electron circulating about a proton (subject to the attractive Coulomb force - (e 2 147rfor2) r) in orbits which satisfy the condition:

quan-27rr = nADe Broglie, n = 1, 2, 3

Since ADe Broglie = hlp this reduces to p27rr = nh, which may be generalized to

the Wilson-Sommerfeld quantization condition:

where § implies a complete cycle, and p and q are conjugate variables

'By construction' Eq (1.1) gives the correct quantized energies for the hydrogen atom But it also gives the correct energy spectrum for a particle in a box with infinite walls:

where a cycle involves x varying from 0 7 a and from a 7 O

Integrating one obtains:

or E = - 8 2' n = 1, 2, 3

Figure 1.1: Potential in Example 1.1

for 0 < x < a with boundary conditions '1/;(0), 'I/;(a) = 0, 'I/;(x) being zero for

'I/;(a) = If sinka = 0 if ka = mr, n = 1, 2, 3 ,

while

Wilson-Sommerfeld Quantization Condition

Wilson-Find the energy levels of a particle in the potential:

On the other hand, solving the Schrodinger equation in the regions 0 < x < a

and a < x < a + b ('IjJ(x) being zero for x:::; 0, x ~ a + b) yields

'IjJ(x) = AsinkIx, 0 < x < a, 'IjJ(x)=Bsink2(x-a-b), a<x<a+b,

since 'IjJ(0) and 'IjJ(a + b) must be zero

The continuity of 'IjJ(x) , d'IjJ(x)jdx at x = a then implies

kl

One notes that the conditions specified in Eq (1.4) and Eq (1.5) are

differ-ent Only if k2 rv kl i.e E > > Vo that is if the total energy is large compared to

the potential energy does Eq (1.5) reduce to Eq (1.4) since then tan kla rv - tan k2b

Wilson-Sommerfeld Quantization Condition

The kinetic energy

l.e

p2

T = - = IVoI - lEI for Ixl < a

2m The Wilson-Sommerfeld quantization condition therefore yields in this case

1/Jo(x) = Asinkx Ixl < a, 1/Jo(x) = ±Bexp {-Klxl} x~ ± a,

(where the subscripts e, 0 stand for even and odd solutions under the interchange

x -+ -x.)

6 Chapter 1

The continuity of 'l/J'f,(x) and d'l/J'f,(x)/dx at x = ± a (independently for the odd

and even solutions) implies:

ka = 2' n = 1, 2, 3 , yielding

I.e

which is identical to Eq (1.6)

Though the Wilson-Sommerfeld quantization condition was superseded by tum Mechanics (with the Schrodinger and Heisenberg formulations in the early twen-ties), as a calculational aid it has the advantage over the Schrodinger equation that it

Quan-is easier to work with since it involves an integral rather than a differential equation However, as indicated in Examples 1.2-3, it generally gives results which are reason-ably accurate (i.e in agreement with Quantum Mechanics) only when the energy is large compared to the potential under consideration

If V(x) = Alxl P one can obtain the form of the energy sequence according to the Wilson-Sommerfeld quantization condition Eq (1.1) as follows:

can be written (for En > 0) as

Wilson-Sommerfeld Quantization Condition

In several cases I(p) can be easily evaluated directly

This problem goes through like Example 1.4, except

Find the energy levels for a particle in the well

Vex) = Alxl, all x (A> 0)

Here p = 1 and Eq (1.8) reduces to:

2/3

n - y'2r;i"1(1) , where

Wilson-Sommerfeld Quantization Condition

EXAMPLE 1.7

Find the energy levels for a particle in the well

{ Alxl, x > 0 (A> 0), V(x) =

00, x < o

This problem goes through as in Example 1.6 except

1(1) = 2101 vT=X dx = 4/3, and hence

Wilson-Sommerfeld Quantization Condition

The results of Example 1.4-14 are summarized in Table 1.1

For two-dimensional systems (or three-dimensional systems where a particle moves

in a plane chosen for convenience to be the x-y plane), where the potential only depends on p,

Wilson-Sommerfeld Quantization Condition

TABLE 1.1

13

Energy levels for various potentials Vex) = Alxl P

Wilson-Sommerfeld

p Range quantization condition Schrodinger result

1 x ;:: 0 E n n = 2/3 ( A2/;2 m )1/3 ( 3" )2/3 2T2 En = ( _ n 1)2/3 ( 4 A2/;2t3( 311: y/3 m 2T2

with the constraint

B2 :2': 4AC I.e E:2': nq,nw

This yields:

or

E = (2np + nq,) nw, where np} = 1, 2, 3

Wilson-Sommerfeld Quantization Condition 15

By comparison, the standard Schrodinger equation result for this system (see Chapters 8 and 12) is:

E = (2n + m - 2)hw : } = 1, 2, 3 (1.31 )

Besides its shortcoming, that it gives quantized energies which are as a rule only approximately correct for energies large compared to the potentials involved, the Wilson-Sommerfeld procedure says nothing about the evaluation of probability dis-tributions, transition rates etc for which there are standard techniques in Quantum Mechanics

On a more positive note the quantization condition Eq (1.1) is also a quence of applying the W.K.B approximation to the Schrodinger equation2 with the modification that nh must be replaced by (n + {1/2}) h in order to get the W K B

conse-approximation result

References

1 H Goldstein, Classical Mechanics, Addison-Wesley (1950), p 300

2 E Merzbacher, Quantum Mechanics, Wiley (1970), p 123

CHAPTER 2

The Delta Function, Completeness, and Closure

The delta function is defined to have the following properties (in one dimension):

of completeness allows one to expand an arbitrary function in terms of any complete orthonormal set Thus if 'IjJ( x) is an arbitrary function,

(2.3)

n

if the complete set 4>n (x) chosen is a discrete set, or

(2.4)

if the complete set 4>k(X) involves continuous functions, where

an = i: 4>~(x)'IjJ(x) dx, a(k) = i: 4>;;(x)'IjJ(x) dx (2.5) Expanding the delta function 6(x - x') in terms of a complete set of discrete functions implies:

The Delta Function, Completeness, and Closure 17 and similarly expanding the delta function in terms of a complete set of continuous functions implies

Considering the first two terms in the expansion of Eq (2.9) as a crude mation one gets the approximate representation:

approxi-2 (7rX 37rX) 4 7rX 27rx h'(x) rv - cos-+cos = -cos-cos ,

Figure 2.1: Plot of c5(x) '" (4/ a) cos (7rX/ a) cos (27rx / a)

Considering the first four terms of the expansion given in Eq (2.9) one gets a somewhat better approximate representation of the delta function:

2 (7rX 37rx 57rx 77rX) c5(x) '" - cos - + cos - -+ cos - -+ cos - - ,

This is plotted in Figure 2.2

One notes that the central maximum (about x = 0) gets progressively sharper and the secondary maxima less important as the number of terms increases Also the normalization of these two expressions (J~oo c5(x) dx) , 8/37r '" 0.85 and 304/105 7r '"

0.92 gets closer to 1 as the number of terms one takes increases

Eq (2.9) resembles a Fourier expansion with the constraint the expansion is zero

at x = ±a/2 and only valid for -a/2 < x < a/2

EXAMPLE 2.3

Consider instead for the complete, discrete set in the expansion of the delta tion, the eigenfunctions of a particle in a harmonic-oscillator potential:

func-The Delta Function, Completeness, and Closure 19

The eigenfunctions of this potential which are non zero at x = 0 are:

1 (1)1/2( 1 )1/4 (X2) (X)

<Pn{x) = 2n/2 n! 7rb2 exp - 2b2 Hn b ' (n even) (2.12)

8 as I x I=SI-a-I

-4-+4-4-~-+~+ -.-L a

Figure 2.2: Plot of 8( x) rv (8/ a) cos (7rX / a) cos (27rx / a) cos (47rx/ a)

Hence an alternative representation of the delta function is:

(2.13) where one need not specify the sum is only over even n since Hn(O) is zero for odd n

The mathematical identity (known as Mehler's formulal )

with the substitution y = 0, X - t x/b becomes:

Figure 2.3: Plot of 8(x/b) = b 8(x) '" (3/ [2 ji ])exp (-x 2 /2b 2 ) (1-2x 2 /3b 2 )

The Eq (2.15) is a standard representation of the delta function in terms of a limit

In Figures 2.3 and 2.4 are plotted two approximate expressions for the delta function using Eq (2.13) The first involves including the first two non-zero terms and the second the first three non-zero terms of this expression,

( 2.16)

The Delta Function, Completeness, and Closure 21

Show that Eq (2.13) is consistent with Eq (2.2) for f(x) = exp (_x 2 /2b 2 )

One notes, interchanging integration and summation, that since Ho(O = 1, and the H's are orthonormal9 ,

1 -00 00 exp -(X2) 2b2 8(x) dx = Ji 1 b 1 -00 00 exp -(X2) b2 E 00 2nn!Hn 1 (X) b Hn(O) dx =

~ b ~ 2:n!Hn(O) f: exp (-~:)Hn(~) Ho(~) dx = ~ ~ 2:n!Hn(O)8noJi = 1, consistent with Eq (2.2)

EXAMPLE 2.5

Show that Eq (2.13) is consistent with Eq (2.1)

With the substitution y2 = e /2, X = y'2, this reduces to

Substitution then yields8 the desired result:

I

EXAMPLE 2.6

Use as a continuous representation of the delta function the set of free-particle

wave functions (V(x) = 0, all x),

1/Jk(X) = Aexp (ikx)

This implies, using Eqs (2.4), (2.5) that:

8(x) = IAI2jOO exp (ikx) dk = IAI2 lim exp (ikx) -: exp (-ikx) = 21AI2 lim sin kx

The Delta Function, Completeness, a.nd Closure

One can obtain, using the property of Eq (2.1) J~ 6(x) dx = 1,

while 6( x) can also be written

with the choice of V (x) similarly determining the detailed form of ¢n (x)

For continuum states Eq (2.21) is replaced by

(2.22) which is analogous to the one dimensional Eq (2.7)

Eqs (2.6), (2.7), (2.21), (2.22) are illustrations of the 'closure' property of mechanical wave functions

The Delta Function, Completeness, and Closure 25 for each permissible value of 1 (1 = 0, 1, 2 ), and where r, r' > O

If I = 0 for example

( 2.24)

Substituting Eq (2.12), the solutions of the one-dimensional harmonic oscillator,

in Eq (2.6), one has analogously:

(2.25)

H 2n + 1(e) = (-nIt (2n + 1)!2e [1F1 ( -nj ~j e)]

Substituting this expression into Eq (2.24) and using Legendre's duplication formula2

The Eq (2.26) involves a sum only over odd n terms, for which Hn(O is zero if e = o

Thus 8(r - r') in Eq (2.26) vanishes if either r or r' is zero Also, an additional factor 2 arises in Eq (2.26) as compared to Eq (2.24) because x, x' extend from

-00 to +00 whereas r, r' extend from 0 -+ 00

Using the Hille-Hardy formula3

one can sum Eq (2.23) Here

Ia(x) = i-a Ja(ix) = J 7r ~:-1 ja-1/2(ix)

26 Chapter 2

Thus Eq (2.23) can be cast into the form:

_ 1 _ {2(rr/)1/2 r(21+1)/4 (_ [(r2 + r/2) ~]) ( 2rr' t 1/2 )} 8(r r ) - ~~ b2 1 _ t exp 2b2 1 _ t 11+1/2 b2 (1 - t) ,

(2.27) where (r, r' > 0), which can be written in terms of spherical Bessel functions:

1 • {4rrl t- I/2 ([(r2 + r/2) 1 + t]) (i2rrltl/2)} 8(r - r ) = ~~ i l b3Vi (1 _ t)3/2 exp - 2b2 1 _ t Jl b2 (1 - t)

The Delta Function, Completeness, and Closure 27 Multiplying Eq (2.26) by Hm(r /b) exp (-r2 /2l?) (modd), and integrating both sides over r one obtains

( ::.) (_r/2)?2exp(-r/2/2b2)",Hn(rl/b) tx) (:) (:) (_r2) (:)

Hm b exp 2b2 - Ji n~d 2nn! 10 Hn b Hm b exp b2 db'

(2.33) where integration and summation have been interchanged

But the integrand in Eq (2.33) is always even Hence the r.h.s of Eq (2.33) can

which is identical with the 1.h.s of this equation

Since completeness enables one to expand any arbitrary function in terms of a complete set, it can also be used to describe what happens if the potential of a system suddenly changes without the wave function undergoing any modification: EXAMPLE 2.9

Suppose a particle is in the ground state of the potential:

Suddenly the potential changes to

One wishes to find the probability the particle will be in any (say the ground) state of this new potential What is involved in this case is a displacement of the equilibrium position (from x to Xl) and a change of frequency (from Wo to wd of the potential,

28 Chapter 2

The potential changes frequency where:

and is displaced so the complete set of states which now describes the system is

¢n (x - Xl, bd To find the probability one merely uses Eq (2.3)

ao = L: ¢o (x - XI ~) ¢o (x, bo) dx = ~ i: exp (-[;:6])

X exp (_ [(X ;b~d2])dX = b6 2b+ o b l bi exp -( [ 2 (b6 xi ] ) + bi) ( 2.34) 1.e laol2 = b~:b~i exp ( - [b6 ~ bi])

an expression which, as expected, is symmetric in bo, bl Thus for instance if Xl =0, laol2 = 0.8 if bl equals either bo/2 or 2 bo etc

This procedure applies equally well if one changes the form of the potential