Solutions to problems in sakurai's quantum mechanics p saltsidis, b brinne

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in Quantum Mechanics

P Saltsidis, additions by B Brinne

1995,1999

Most of the problems presented here are taken from the book

Sakurai, J J.,Modern Quantum Mechanics, Reading, MA: Addison-Wesley,1985

I Problems 3

1 Fundamental Concepts 5

2 Quantum Dynamics 7

3 Theory of Angular Momentum 14

4 Symmetry in Quantum Mechanics 17

5 Approximation Methods 19

II Solutions 23 1 Fundamental Concepts 25

2 Quantum Dynamics 36

3 Theory of Angular Momentum 75

4 Symmetry in Quantum Mechanics 94

5 Approximation Methods 103

1

3

1 Fundamental Concepts

1.1 Consider a ket space spanned by the eigenkets fja0

ig of a mitian operator A There is no degeneracy.

(a) Suppose Sx is measured What is the probability of getting

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

re-
Aj i=
Bj 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

is an eigenstate of the coordinate operator x What is the sponding eigenvalue?

corre-1.5 (a) Prove the following:

(i) hp0

jxj i = ih @@p0

hp0

j i;(ii) h jxj i = Z

2.1 Consider the spin-procession problem discussed in section 2.1

in Jackson It can also be solved in the Heisenberg picture Using the Hamiltonian

H =;



eBmc



Sz =!Sz;

write the Heisenberg equations of motion for the time-dependent operatorsSx(t),Sy(t), andSz(t) Solve them to obtainSx;y;z as func- tions of time.

2.2 Let x(t) be the coordinate operator for a free particle in one dimension in the Heisenberg picture Evaluate

[x(t);x(0)]:

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

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

exp

;ipa

h

! 1=2 1

A:

(b) Obtain a simple expression that the probability that the state

is found in the ground state at t = 0 Does 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;

wherex(t) is the position operator in the Heisenberg picture uate the correlation function explicitly for the ground state of a one-dimensional simple harmonic oscillator.

Eval-2.6 Consider again a one-dimensional simple harmonic oscillator.

Do the following algebraically, that is, without using wave tions.

func-(a) Construct a linear combination of j0i and j1i such thathxi is as large as possible.

(b) Suppose the oscillator is in the state constructed in (a) att = 0 What is the state vector for t > 0 in the Schrodinger picture? Evaluate the expectation value hxi as a function of time for t > 0

using (i) the Schrodinger picture and (ii) the Heisenberg 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 lator is de ned to be an eigenstate of the (non-Hermitian) annihi- lation operator a:

oscil-aji=ji;

where  is, in general, a complex number.

(a) Prove that

ji=e;j  j

2 =2ea y

j0i

is a normalized coherent state.

(b) Prove the minimum uncertainty relation for such a state (c) Writeji as

ji= 1 X

n=0f(n)jni:

Show that the distribution of jf(n)j 2 with respect 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 be obtained by applying the translation ( nite-displacement) operatore; ipl=h (wherep is the

ground state.

(e) Show that the coherent state ji remains coherent under evolution and calculate the time-evolved state j(t)i (Hint: di- rectly apply the time-evolution operator.)

time-2.8 The quntum mechanical propagator, for a particle with mass

m, moving in a potential is given by:

(a) What is the potential?

(b) Determinethe constantAin terms of the parameters describing the system (such as m, r etc ).

(a) Show that

hxbtb jxata i=exp

iScl

h

 ( N +1) 2

;

12"m!2y2j 

(Hint: Let y(t) = x(t);xcl(t) be the new integration variable,

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

(b) Show that G can be written as

G = limN

!1



m2ih"

 ( N +1) 2 Z

dy1:::dyNexp(;nTn)

2 6 4

y1

yN

3 7

5 and nT is its transpose Write the symmetric matrix.

are pj By expanding 0

j+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

d2

dt2 =;!2(t)

with initial conditions (t = ta) = 0;d(t=t a )

dt = 1 (f) Show that

2.16 A particle in one dimension (;1< x <1) is subjected to a constant force derivable from

V = x; ( > 0):

(a) Is the energy spectrum continuous or discrete? Write down an approximate expression for the energy eigenfunction speci ed by

E.

V is replaced be

V = jxj:

3 Theory of Angular Momentum

3.1 Consider a sequence of Euler rotations represented by

D

(1=2)( 

;i32

3.2 An angular-momentum eigenstatejj;m = mmax=jiis rotated

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

3.3 The wave function of a patricle subjected to a spherically symmetrical potentialV (r) is given by

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

(a) Is an eigenfunction of ~L? If so, what is the l-value? If not, what are the possible values of l we may obtain when ~L2 is measured?

(b) What are the probabilities for the particle to be found in various

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

where j~x;ii correspond to a particle at ~x with spin in the i:th rection.

di-(a) Show explicitly that in nitesimal rotations of%i(~x)are obtained

by acting with the operator

u~"= 1;i~"h
(~L + ~S) (3.1)

where ~L =  h

ir^ r~ Determine S~ !

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

(c) Show that S~ is a vector operator.

(d) Show that r~ ~%(~x) = 1

 h 2(~S
~p)~% where ~p is the momentum ator.

oper-3.5 We are to add angular momenta j1 = 1 and j2 = 1 to form

j = 2;1; and 0states Using the ladder operator method express all

(nine) j;m eigenkets in terms of jj1j2;m1m2 i Write your answer as

where + and 0 stand for m1;2 = 1;0; respectively.

3.6 (a) Construct a spherical tensor of rank 1 out of two dierent vectors ~U = (Ux;Uy;Uz) and ~V = (Vx;Vy;Vz) Explicitly write T(1)

 1;0 in terms of Ux;y;z and Vx;y;z.

(b) Construct a spherical tensor of rank 2 out of two dierent vectors ~U and ~V Write down explicitly T(2)

3.8 (a) Write xy, xz, and (x2

;y2) as components of a spherical (irreducible) tensor of rank 2.

(b) The expectation value

Clebsch-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 be chosen to be real.

(b) The wave function for a plane-wave state at t = 0 is given by

a complex function ei~p
~x=h Why does this not violate time-reversal invariance?

4.2 Let (~p0) be the momentum-space wave function for state j i, that is, (~p0) =h~p0

j i.Is the momentum-space wave function for the time-reversed state j i given by (~p0, (;~p0), (~p0), or (;~p0)? Justify your answer.

4.3 Read section 4.3 in Sakurai to refresh your knowledge of the quantum mechanics of periodic potentials You know that the en- ergybands in solids are described by the so called Bloch functions

n;k full lling,

n;k(x + a) = eika n;k(x)

where a is the lattice constant, n labels the band, and the lattice momentumk is restricted to the Brillouin zone [;=a;=a].

Prove that any Bloch function can be written as,

h~Li= 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)Ylm(;);

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

4.5 The Hamiltonian for a spin 1 system is given by

H = AS2z+B(S2x ;S2y):

Solve this problem exactly to ... < /p>

= 1 X < /p>

n=0 < /p>

1n! < /p>

 < /p>

i < /p>

h < /p>

 < /p>

i < /p>

h < /p>

 < /p>

i < /p>

h... translation in momentum space This can < /p>

be justied by calculating the following operator < /p>

n=0 < /p>

1n! < /p>

 < /p>

ix < /p>

h < /p>

... < /p>

n=1 < /p>

nn!(ia)n < /p>

 < /p>

: (1.42)(c) We have now < /p>

x < /p>

exp < /p>

ipxa < /p>

h < /p>