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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 AddisonWesley  

CONTENTS

Part I

Problems

 FUNDAMENTAL CONCEPTS

Fundamental Concepts

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

mitian op erator A There is no degeneracy

a Prove that Y

a

A  a^ 

is a null op erator b What is the signicance of

Y

a^ ^ a

A  a^ 

a^  a^

c Illustrate a and b using A set equal to Sz of a spin   system

 A spin   system is known to b e in an eigenstate of S  n with

eigenvalue h  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 b Evaluate the disp ersion in Sx  that is

hSx  hSx i^ i

For your own p eace of mind check your answers for the sp ecial cases       and  

 a The simplest way to derive the Schwarz inequality go es as follows First observe

hj  ^ h j  ji  j i 

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 satises

Aji  B ji

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^ ji    d^ ^ exp

ihpix

h

x^  hxi

d

satises the uncertainty relation

q

h x^ i

q

hp^ i 

h 

Prove that the requirement

hx^ jxji  imaginary numb erhx^ jpji

is indeed satised for such a Gaussian wave packet in agreement with b

 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 quantummechanical 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^ i xjxi  x^ jx^ i

 Consider a particle in three dimensions whose Hamiltonian is given by

H 

p m

 V x

By calculating x  p H  obtain

d dt

hx  pi 

p m

 hx  rV i

To identify the preceding relation with the quantummechanical analogue of the virial theorem it is essential that the lefthand side vanish Under what condition would this happ en

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

exp

 ipa

h

ji

You may use

hx^ ji   ^ x exp   

x x

x  h

m

A 

b Obtain a simple expression that the probability that the state is found in the ground state at t   Do es this probability change for t 

 Consider a function known as the correlation function dened by

C t  hxtxi

where xt is the p osition op erator in the Heisenb erg picture Eval uate the correlation function explicitly for the ground state of a onedimensional simple harmonic oscillator

 QUANTUM DYNAMICS 

 Consider again a onedimensional simple harmonic oscillator Do the following algebraically that is without using wave func tions

a Construct a linear combination of ji and ji such that hxi is as

large as p ossible

b Supp ose the oscillator is in the state constructed in a at t   What is the state vector for t  in the Schrodinger picture

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

using i the Schrodinger picture and ii the Heisenb erg picture

c Evaluate hx^ i as a function of time using either picture

 A coherent state of a onedimensional simple harmonic oscil lator is dened to b e an eigenstate of the nonHermitian annihi lation op erator a

aji  ji

where  is in general a complex numb er

a Prove that

ji  ejj

(^)  ea y

ji

is a normalized coherent state

b Prove the minimum uncertainty relation for such a state

c Write ji as

ji 

X^ 

n

f njni

Show that the distribution of jf nj^ 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 nitedisplacement op erator eiplh^ where p is the momentum op erator and l is the displacement distance to the ground state

 QUANTUM DYNAMICS 

a Show that

hxb tb jxa ta i  exp

iScl h

G tb  ta 

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

G tb  ta  

lim N 

Z

dy    dyN

m  ih

 N^ 

exp

i h

X^ N

j 

m 

yj   yj ^ 

m ^ y  j

where   (^) tNb^  ta 

Hint Let y t  xt  xcl t b e the new integration variable

xcl t b eing the solution of the EulerLagrange equation

b Show that G can b e written as

G  lim N 

 m

 ih

 N^  Z

dy    dyN expnT^  n

where n 

y   yN

 and^ n

T (^) is its transp ose Write the symmetric

matrix  

c Show that

Z

dy    dyN expnT^  n 

Z

dN^ y en

T (^)  n 

 N^ 

p

det

Hint Diagonalize  by an orhogonal matrix

d Let

ih m

N

det  det N  pN  Dene j  j matrices  j that con

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

pj     ^ ^ pj  pj  j      N  

e Let t  pj for t  ta  j  and show that  implies that in

the limit   t satises the equation

d dt^

  ^ t

with initial conditions t  ta   d^ t dt ta  

f  Show that

hxb tb jxa ta i 

s

m  ih sin T 

exp

im h sin T 

x b  x a  cos  T   xa xb 

where T  tb  ta 

 Show the comp osition prop erty

Z

dx Kf x t  x t Kf x t  x t   Kf x t  x t 

where Kf x t  x t  is the free propagator Sakurai   by explicitly p erforming the integral ie do not use completeness

 a Verify the relation

i j  

ihe c

ij k Bk

where   m dtx  p  e A c and the relation

m

d^ x dt^

d  dt

 e

E  

c

dx dt

 B  B 

dx dt

b Verify the continuity equation

 t

 r^  j 

a Is the energy sp ectrum continuous or discrete Write down an approximate expression for the energy eigenfunction sp ecied by E 

b Discuss briey what changes are needed if V is replaced b e

V  jxj

 Theory of Angular Momentum

 Consider a sequence of Euler rotations represented by

D ^     exp

i 

exp

i

exp

i 

ei^ ^ cos   ei^ ^ sin  

ei^ ^ sin   ei^ ^ cos  

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 

 An angularmomentum eigenstate jj m  mmax  j i is rotated

by an innitesimal angle  ab out the y axis Without using the explicit form of the d mj (^) 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 ^ 

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

 x  x  y  z f r 

 THEORY OF ANGULAR MOMENTUM 

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^ 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 

 Consider a particle with an intrinsic angular momentum or spin of one unit of h One example of such a particle is the  meson Quantummechanically such a particle is describ ed by a ketvector ji or in x representation a wave function

i^ x  hx iji

where jx ii corresp ond to a particle at x with spin in the ith di rection

a Show explicitly that innitesimal rotations of i^ x are obtained by acting with the op erator

u    i

h

  L  S   

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  S  p  where p is the momentum op er ator

 We are to add angular momenta j   and j   to form j   and states Using the ladder op erator metho d express all

 SYMMETRY IN QUANTUM MECHANICS 

b The exp ectation value

Q  eh j m  j j z ^  r ^ j j m  j i

is known as the quadrupole moment Evaluate

eh j m^ jx^  y ^ j j m  j i

where m^  j j   j     in terms of Q and appropriate Clebsch Gordan co ecients

 Symmetry in Quantum Mechanics

 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 planewave state at t  is given by a complex function ei pxh^  Why do es this not violate timereversal invariance

 Let p^  b e the momentumspace wave function for state ji that is p^   hp^ jiIs the momentumspace wave function for the timereversed state ji given by p^  p^  ^ p^  or ^ p^  Justify your answer

 Read section  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 nk fulllling nk x  a  eik^ a^ nk 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

nk x 

X

Ri

n x^ ^ Ri e

ik 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 tightbinding description of solids Show that the Wan nier functions are corresp onding to dierent sites andor dierent bands are orthogonal ie 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

 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 timereversal invariance prove

h Li 

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 

 The Hamiltonian for a spin  system is given by

H  AS  z  B S  x  S y