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Solutions to Problems

in

Quantum Mechanics

P. Saltsidis, additions by B. Brinne

Most of the problems presented here are taken from the b o ok Sakurai, J. J., Modern Quantum Mechanics, Reading, MA: Addison-Wesley,

2 CONTENTS

Part I

Problems

1. FUNDAMENTAL CONCEPTS 5

1 Fundamental Concepts

1.1 Consider a ket space spanned by the eigenkets fja^0 ig of a Her-

mitian op erator A. There is no degeneracy.

(a) Prove that Y

a^0

(A a^0 )

is a null op erator. (b) What is the signi cance of

Y

a^00 6 =a^0

(A a^00 )

a^0 a^00

(c) Illustrate (a) and (b) using A set equal to Sz of a spin (^12) system.

1.2 A spin 12 system is known to b e in an eigenstate of S~
n^ with

eigenvalue h= 2 , where n^ is a unit vector lying in the xz -plane that makes an angle with the p ositive z -axis. (a) Supp ose Sx is measured. What is the probability of getting +h= 2? (b) Evaluate the disp ersion in Sx , that is,

h(Sx hSx i)^2 i:

(For your own p eace of mind check your answers for the sp ecial cases = 0 ,  = 2 , and  .)

1.3 (a) The simplest way to derive the Schwarz inequality go es as follows. First observe

(h j + ^ h j)
(j i + j i)  0

for any complex numb er ; then cho ose  in such a way that the preceding inequality reduces to the Schwarz inequility.

(b) Show that the equility sign in the generalized uncertainty re- lation holds if the state in question satis es

Aj i = 
B j i

with  purely imaginary.

(c) Explicit calculations using the usual rules of wave mechanics show that the wave function for a Gaussian wave packet given by

hx^0 j i = (2 d^2 )^1 =^4 exp

ihpix^0

h

(x^0 hxi)^2

4 d^2

satis es the uncertainty relation

q

h (
x)^2 i

q

h(
p)^2 i =

h 2

Prove that the requirement

hx^0 j
xj i = (imaginary numb er)hx^0 j
pj i

is indeed satis ed for such a Gaussian wave packet, in agreement with (b).

1.4 (a) Let x and px b e the co ordinate and linear momentum in one dimension. Evaluate the classical Poisson bracket

[x; F (px )]classical :

(b) Let x and px b e the corresp onding quantum-mechanical op era- tors this time. Evaluate the commutator

x; exp

ipx a h

(c) Using the result obtained in (b), prove that

exp

 ip

x a h

jx^0 i; (xjx^0 i = x^0 jx^0 i)

2.3 Consider a particle in three dimensions whose Hamiltonian is given by

H =

p~^2 2 m

• V (~x):

By calculating [~x
~p ; H ] obtain

d dt

h~x
~pi =

p^2 m

h~x
r~V i:

To identify the preceding relation with the quantum-mechanical analogue of the virial theorem it is essential that the left-hand side vanish. Under what condition would this happ en?

2.4 (a) Write down the wave function (in co ordinate space) for the state

exp

 ipa

h

j 0 i:

You may use

hx^0 j 0 i =  ^1 =^4 x 0 1 =^2 exp

2

x^0 x 0

@x 0  h

m!

A :

(b) Obtain a simple expression that the probability that the state is found in the ground state at t = 0. Do es this probability change for t > 0?

2.5 Consider a function, known as the correlation function, de ned by

C (t) = hx(t)x(0)i;

where x(t) is the p osition op erator in the Heisenb erg picture. Eval- uate the correlation function explicitly for the ground state of a one-dimensional simple harmonic oscillator.

2. QUANTUM DYNAMICS 9

2.6 Consider again a one-dimensional simple harmonic oscillator. Do the following algebraically, that is, without using wave func- tions.

(a) Construct a linear combination of j 0 i and j 1 i such that hxi is as

large as p ossible.

(b) Supp ose the oscillator is in the state constructed in (a) at t = 0. What is the state vector for t > 0 in the Schrodinger picture?

Evaluate the exp ectation value hxi as a function of time for t > 0

using (i) the Schrodinger picture and (ii) the Heisenb erg picture.

(c) Evaluate h(
x)^2 i as a function of time using either picture.

2.7 A coherent state of a one-dimensional simple harmonic oscil- lator is de ned to b e an eigenstate of the (non-Hermitian) annihi- lation op erator a:

aji = ji;

where  is, in general, a complex numb er.

(a) Prove that

ji = ejj

(^2) = 2 ea y

j 0 i

is a normalized coherent state.

(b) Prove the minimum uncertainty relation for such a state.

(c) Write ji as

ji =

X^1

n=

f (n)jni:

Show that the distribution of jf (n)j^2 with resp ect to n is of the

Poisson form. Find the most probable value of n, hence of E.

(d) Show that a coherent state can also b e obtained by applying the translation ( nite-displacement) op erator eipl=h^ (where p is the momentum op erator, and l is the displacement distance) to the ground state.

2. QUANTUM DYNAMICS 11

(a) Show that

hxb tb jxa ta i = exp

iScl h

G(0; tb ; 0 ; ta )

where Scl is the action along the classical path xcl from (xa ; ta ) to (xb ; tb ) and G is

G(0; tb ; 0 ; ta ) =

lim N!

Z

dy 1 : : : dyN

m 2  ih"

 (N^2 +1)

exp

i h

X^ N

j =

m 2 "

(yj +1 yj )^2

"m! 2 y (^2) j

where " = (^) (tNb^ +1)ta.

(Hint: Let y (t) = x(t) xcl (t) b e the new integration variable,

xcl (t) b eing the solution of the Euler-Lagrange equation.)

(b) Show that G can b e written as

G = lim N!

 m

2  ih"

 (N^ +1) 2 Z

dy 1 : : : dyN exp(nT^  n)

where n =

y 1 .. . yN

5 and^ n

T (^) is its transp ose. Write the symmetric

matrix .

(c) Show that

Z

dy 1 : : : dyN exp(nT^  n) 

Z

dN^ y en

T (^)  n

 N^ =^2

p

det

[Hint: Diagonalize  by an orhogonal matrix.]

(d) Let

2 ih" m

N

det  det N^0  pN. De ne j  j matrices  j^0 that con-

sist of the rst j rows and j columns of  (^) N^0 and whose determinants are pj. By expanding  (^) j^0 +1 in minors show the following recursion formula for the pj :

pj +1 = (2 "^2! 2 )pj pj 1 j = 1 ; : : : ; N (2.1)

(e) Let (t) = "pj for t = ta + j " and show that (2.1) implies that in

the limit "! 0 ; (t) satis es the equation

d^2  dt^2

= ! 2 (t)

with initial conditions (t = ta ) = 0 ; d(t dt= ta^ )= 1.

(f ) Show that

hxb tb jxa ta i =

s

m! 2  ih sin(! T )

exp

im! 2h sin(! T )

[(x^2 b + x^2 a ) cos (! T ) 2 xa xb ]

where T = tb ta.

2.12 Show the comp osition prop erty

Z

dx 1 Kf (x 2 ; t 2 ; x 1 ; t 1 )Kf (x 1 ; t 1 ; x 0 ; t 0 ) = Kf (x 2 ; t 2 ; x 0 ; t 0 )

where Kf (x 1 ; t 1 ; x 0 ; t 0 ) is the free propagator (Sakurai 2.5.16), by explicitly p erforming the integral (i.e. do not use completeness).

2.13 (a) Verify the relation

[i ; j ] =

ihe c

"ij k Bk

where ~  m ~ dtx = ~p e^

A~ c and^ the^ relation

m

d^2 ~x dt^2

d ~ dt

= e

E~ + 1

2 c

d~x dt

 B~ B~ 

d~x dt

(b) Verify the continuity equation

@  @ t

+ r~^0
~j = 0

(a) Is the energy sp ectrum continuous or discrete? Write down an approximate expression for the energy eigenfunction sp eci ed by E.

(b) Discuss brie y what changes are needed if V is replaced b e

V = jxj:

3 Theory of Angular Momentum

3.1 Consider a sequence of Euler rotations represented by

D (1=2)^ ( ; ; ) = exp

i 3

exp

i 2

exp

i 3

ei(^ +^ )=^2 cos 2 ei(^ ^ )=^2 sin 2

ei(^ ^ )=^2 sin 2 ei(^ +^ )=^2 cos (^2)

Because of the group prop erties of rotations, we exp ect that this sequence of op erations is equivalent to a single rotation ab out some axis by an angle . Find .

3.2 An angular-momentum eigenstate jj; m = mmax = j i is rotated

by an in nitesimal angle " ab out the y -axis. Without using the explicit form of the d( mj^ ) (^0) m function, obtain an expression for the probability for the new rotated state to b e found in the original state up to terms of order "^2.

3.3 The wave function of a patricle sub jected to a spherically symmetrical p otential V (r ) is given by

(~x) = (x + y + 3 z )f (r ):

3. THEORY OF ANGULAR MOMENTUM 15

(a) Is an eigenfunction of ~L? If so, what is the l -value? If not, what are the p ossible values of l we may obtain when ~L^2 is measured?

(b) What are the probabilities for the particle to b e found in various ml states?

(c) Supp ose it is known somehow that (~x) is an energy eigenfunc- tion with eigenvalue E. Indicate how we may nd V (r ).

3.4 Consider a particle with an intrinsic angular momentum (or spin) of one unit of h. (One example of such a particle is the %- meson). Quantum-mechanically, such a particle is describ ed by a ketvector j%i or in ~x representation a wave function

%i^ (~x) = h~x; ij%i

where j~x; ii corresp ond to a particle at ~x with spin in the i:th di- rection.

(a) Show explicitly that in nitesimal rotations of %i^ (~x) are obtained by acting with the op erator

u~" = 1 i

h

( L~ + S~ ) (3.1)

where L~ = h i r^  r~. Determine S~!

(b) Show that ~L and S~ commute.

(c) Show that S~ is a vector op erator.

(d) Show that r~  %~(~x) = (^) h^12 ( S~
~p) ~% where p~ is the momentum op er- ator.

3.5 We are to add angular momenta j 1 = 1 and j 2 = 1 to form j = 2 ; 1 ; and 0 states. Using the ladder op erator metho d express all

4. SYMMETRY IN QUANTUM MECHANICS 17

(b) The exp ectation value

Q  eh ; j; m = j j(3z 2 r 2 )j ; j; m = j i

is known as the quadrupole moment. Evaluate

eh ; j; m^0 j(x^2 y 2 )j ; j; m = j i;

(where m^0 = j; j 1 ; j 2 ; : : : )in terms of Q and appropriate Clebsch- Gordan co ecients.

4 Symmetry in Quantum Mechanics

4.1 (a) Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless nondegenerate system at any given instant of time can always b e chosen to b e real.

(b) The wave function for a plane-wave state at t = 0 is given by a complex function ei~p
~x=h^. Why do es this not violate time-reversal invariance?

4.2 Let (~p^0 ) b e the momentum-space wave function for state j i, that is, (~p^0 ) = h~p^0 j i.Is the momentum-space wave function for the time-reversed state j i given by (~p^0 , (~p^0 ), ^ (~p^0 ), or ^ (~p^0 )? Justify your answer.

4.3 Read section 4.3 in Sakurai to refresh your knowledge of the quantum mechanics of p erio dic p otentials. You know that the en- ergybands in solids are describ ed by the so called Blo ch functions

n;k full^ lling, n;k (x^ +^ a)^ =^ e ik a n;k (x)

where a is the lattice constant, n lab els the band, and the lattice

momentum k is restricted to the Brillouin zone [ =a;  =a].

Prove that any Blo ch function can b e written as,

n;k (x)^ =^

X

Ri

n (x Ri )eik^ Ri

where the sum is over all lattice vectors Ri. (In this simble one di- mensional problem Ri = ia, but the construction generalizes easily to three dimensions.). The functions n are called Wannier functions, and are imp or- tant in the tight-binding description of solids. Show that the Wan- nier functions are corresp onding to di erent sites and/or di erent bands are orthogonal, i:e: prove

Z

dx?m (x Ri )n (x Rj )  ij mn

Hint: Expand the n s in Blo ch functions and use their orthonor- mality prop erties.

4.4 Supp ose a spinless particle is b ound to a xed center by a p otential V (~x) so assymetrical that no energy level is degenerate. Using the time-reversal invariance prove

h L~i = 0

for any energy eigenstate. (This is known as quenching of orbital angular momemtum.) If the wave function of such a nondegenerate eigenstate is expanded as

X

l

X

m

Flm (r )Y (^) lm ( ; );

what kind of phase restrictions do we obtain on Flm (r )?

4.5 The Hamiltonian for a spin 1 system is given by

H = AS 2 z + B (S 2 x S y^2 ):