1Quantum Mechanics Qualifier ExamFall 2012

For full credit solve 6 out of the following 8 problems. If you solve more than 6, please

specify which are the ones to be counted toward your score.

1. Show that if two observables A and B do not commute with each other but each

commutes with a Hamiltonian H , i.e., [A, B] 6= 0, [H, A] = 0, [H, B] = 0, then thisgenerally implies that the spectrum of the Hamiltonian is degenerate.

2. Suppose that a spin-less particle is in the presence of a periodic potential in one

dimension, V (x a) = V (x) where a is the lattice periodicity. Suppose also that thepotential barrier height between two adjacent lattice sites is finite. Let |n denote theket corresponding to the state of the particle localized at the n-th lattice site, and is

the lattice translation operator: |n = |n + 1. Note that since the barrier height isnot infinite, the states |n are not energy eigenkets. Suppose the Hamiltonian of thesystem satisfies the equation

H|n = E0|n |n+ 1 |n 1.

Show that the state

| =

n=exp(in)|n

is an eigenstate of (5 points). If it is an energy eigenstate as well, find the energy

eigenvalue as a function of (5 points).

3. The wave function of the ground state of a harmonic oscillator with force constant k

and mass m is

0(x) = (/pi)1/4ex

2/2, = m0/~, 20 = k/m. (1)

Obtain an expression for the probability of finding the particle outside the classical

region.

4. (a) Show that in the usual stationary state perturbation theory, if the Hamiltonian

can be written as H = H0 +H with H00 = E00, then the correction E0 is

E0 < 0|H |0 > (2)

2(b) For a spherical nucleus, the nucleons may be assumed to be in a spherical potential

well of radius R given by

Vsp =

0, r < R,, r > R.For a slightly deformed nucleus, it may be correspondingly assumed that the

nucleons are in an elliptical well, again with infinite wall height, that is:

Vel =

0, inside the ellipsoid (x2 + y2)/a2 + z2/b2 = R2,, otherwise,where a = (1 + c/3), b = (1 2c/3), with c 1.

Calculate the approximate change in the ground state energy E0 due to the

ellipticity of the non-spherical nucleus by finding an appropriate H and using

the result obtained in (a).

Hint: Try to find a transformation of variables that will make the well look

spherical.

5. We consider a particle of mass m in a potential V (x) given by

V (x) =

, x < 0 ,0 , 0 < x < L

U , L < x

The particle state function satisfies the time-independent Schrodinger equation

~2

2m

d2

dx2+ V (x) = E

(a) For energies E < U , find solutions (x) inside the well (0 < x < L) that satisfy

the appropriate boundary condition at x = 0, and find solutions (x) outside the

well (x > L) that satisfy the appropriate boundary condition for x.

(b) Impose appropriate matching conditions at x = L and find an equation for the

allowed energies E of the system.

3(c) Show that there are no bound states if UL2 < ~2/2m. Give a simple physical

explanation for this result in terms of the uncertainty principle.

6. A coherent state | of a one-dimensional harmonic oscillator is defined to be aneigenstate of the (non-Hermitian) annihilation operator a,

a| = |

where is generally a complex number.

(a) Show that

| = e||2/2ea|0is a coherent state.

(b) Writing | as a superposition

| =n=0

cn|n

of energy eigenstates |n, show that

|cn|2 = ||2n

n!e||

2

(c) Using the time-evolution operator U(t) = eiHt/~, show that an initial coherent

state | at t = 0 evolves into a coherent state |(t) at time t > 0. What is thevalue of (t)?

7. An electron in an atom has two states of different total angular momentum for each

value of l and m, except for l = 0.

(a) What are the possible eigenvalues j of the total angular momentum J = L + S

for a given value of l?

(b) Show that

+ =l + 1 + 2S L/~2

2l + 1

and

=l 2S L/2~2

2l + 1

are projection operators for states of good total angular momentum j, i.e. that

they act as identity operators for one set of states and yield zero when applied

to the other set of states.

4(c) The most general rotationally invariant Hamiltonian for an electron has the form

H =p2

2m+ V0(r) + V1(r)S L/~2

Any eigenstate of H can be written as a two-component spinor of the form

(r) =s=1/2

Rls(r)Ylm(, )s

where s is the unit spinor defined by zs = 2ss for s = 1/2. Find the differ-

ential equations for Rl,s(r) in terms of V0(r) and V1(r) for any valid combination

of l and j.

(d) Why does the radial differential equation not depend on the quantum number

m?

Hint: The Laplace operator in spherical coordinates is

=1

r2

r

(r2

r

)+

1

r2 sin

(sin

)+

1

r2 sin2

2

2

8. Three spin-half particles with spin operators S1,S2 and S3 are fixed to the corners of a

triangle. The spins interact with each other such that the Hamiltonian of the system

is given by

H = J (S1 S2 + S1 S3 + S2 S3)

Find the energy levels of the system and their degeneracies.