Physics 742 – Graduate Quantum Mechanics 2
Final Exam, Spring 2014
Due Monday, May 5 by 5 PM
You may use (1) class notes, (2) former homeworks and solutions (available online), or
(3) any math references, such as integral tables, Maple, etc, including any routines I
provided for you. If you cannot do an integral or sum, or if you cannot solve an equation,
try to go on, as if you knew the answer. Some useful integral appear on the last page.
The approximate points (out of 200) are marked for each problem and part. Feel free to
contact me with questions.
Work: 758-4994
Home: 724-2008
Cell:
407-6528
1. [15] A particle of mass m and initial momentum k moving in the +z direction
scatters from a weak potential V  r     z  exp   12   x 2  y 2   . Calculate the
differential cross-section d d  and total cross-section  in the first Born
approximation. Note that the potential is not spherically symmetric, and hence you
cannot choose the momentum change K to be along the z-axis. I recommend working
in Cartesian coordinates. For  , you may leave one integral undone.
2. [20] A particle of mass m lies in the spherical infinite square well, with potential
 0 if r  R ,
V r   
 if r  R .
The radius R of the confining sphere will actually be a function of time, so that it
changes.
(a) [10] The ground state is given in the allowed region r < R by
 r  
N
r 
sin  
r
 R 
Show that this is, in fact, an eigenstate of the Hamiltonian, and determine the
energy and the correct normalization factor N.
(b) [10] The radius R of the allowed region is slowly decreased from R to 12 R . What
is the probability that it will remain in the ground state? The radius is then
suddenly increased from 12 R to R. What is the probability now that it remains in
the ground state?
3. [15] A particle is in a one-dimensional infinite square well, with allowed region
0  x  a . It is initially in the ground, or n  1 state. At t = 0, a perturbation of the
form W  APe t , where A is a small positive number and P is the momentum
operator, is turned on. Calculate to leading order (that’s order A2), the probability that
the particle ends up in the state n at t   for all n > 1.
4. [15] A particle of mass m in one dimension is in the bound state of the Dirac delta
function. This state has wave function and energy given by
 g  x    e  x , Eg   ,
The values of  and  depend on the mass and the strength of the delta function. It
is then affected by a time-dependent perturbation of the form W  t    P cos t  ,
where P is the momentum operator and  is very small. The frequency is high
enough such that    .
(a) [2] Write W  t  in the form W  t   We  it  W †eit . What is W?
(b) [5] For sufficiently high energies, the potential is negligible, and the eigesntates
are approximately momentum eigenstates k . Work out the matrix element
k W  g for these high energy final states. Hint – let any momentum operators
act to the left.
(c) [8] Calculate the rate   g  k

 for a given final state wave number k.
Integrate it over k to get the rate for the transision.
5. [20] A system consists of a superposition of a zero photon state and two one photon
states, so that
 t  0  N

2 0  q,1  i q, 2

where N is a normalization constant, q  qzˆ , and the two polarizations are ε  q,1  xˆ
and ε  q, 2   yˆ .
(a) [3] What is the normalization constant N?
(b) [5] What is the state vector   t  at an arbitrary time? Work in the convention
where the energy of the vacuum is 0.
(c) [12] what is the expectation value of the electric field E  r  at arbitrary time
and arbitrary position for the state you found in part (b)?
6. [20] An electron is trapped in a 3D infinite square well, with allowed region
0  x  a , 0  y  a , 0  z  a . It is in the quantum state nx , ny , nz  1,1, 4 .
(a) [12] Calculate every non-vanishing matrix element of the form
nx , ny , nz R 1,1, 4 where the final state is lower in energy than the initial state
(there will be very few of these).
(b) [8] Calculate the decay rate  1,1, 4  nx , n y , nz

 for this decay in the dipole
approximation for every possible final state. What is the ratio of the two rates?
7. [30] A hydrogen atom is at rest in a strong magnetic field oriented in the z-direction.
This field adds an additional perturbation to the hydrogen atom as given in Eq. (9.23):
W
eB
 S z  12 Lz 
m
The field is sufficiently strong that we can neglect any hyperfine effects. The atom is
initially in the state n, l , m, ms  1, 0, 0, 12 , but it will transition to 1, 0, 0,  12 .
(a) [2] Explain why this transition cannot occur in the electric dipole approximation.
(b) [8] Find all non-vanishing matrix elements coming from the magnetic dipole
transition F  S  12 L  I .
(c) [20] Find a formula for the differential decay rate d  d  for this decay into a
particular polarization ε . You do not need to sum it or integrate it over
polarizations.
8. [15] The Dirac equation in the presence of electromagnetic fields is given by
i

  cα   P  eA  R, t    eU  R, t   mc 2  
t


where P  i . Prove or disprove: The Dirac equation is invariant under gauge
transformations.
Definite Integrals:
For each of the following formulas, m and n and p are assumed to be positive integers.

e
 Ax
x n dx 
0

  n  1
if Re  A   0,
An 1
n
 Ax
 e x dx 
2
0
  n21 
n 1
2A 2
if Re  A   0,
  n  1  n !,   12    ,   32  
a
1
2
 ,   52  
3
4
.
 2a n if n is odd,
  nx 
 dx  0
if n is even,
a 

 sin 
0
a
  nx 
 dx  0
a 
 cos 
0
  nx 
  mx 
  nx    mx 
0 cos  a  cos  a  dx  0 sin  a  sin  a  dx  12 a nm
a
a
2na   n 2  m 2  if n  m is odd
  nx 
  mx 
dx

sin
cos
0  a   a  
0
if n  m is even

a
 4a 2 mn

  nx    mx 
0 sin  a  sin  a  x dx  

a
 2  n 2  m 2 2  if n  m is odd


1 2
if n  m is even
4 a  nm
a 2 n  1
  nx 
  mx 
0 sin  a  cos  a  x dx    m2  n2 
a
m n
  2a 2 m 2  n 2  2 n 2  m 2 2  if n  m is odd

  
 

  nx 
  mx 

x
dx
cos
cos





0  a   a 
1 2

if n  m is even
4 a  nm
a