Physics 742 – Graduate Quantum Mechanics 2
Midterm Exam, Spring 2015
Please note that some possibly helpful formulas and integrals appear on the second page.
Each question is worth 20 points, with points for each part marked separately.
1. A quantum system has two dimensions. In a certain choice of basis, a single particle is in
one of two states:
1
1    ,  2 
0
 
1 1
 
5  2
However, it isn’t known which of these two states it is in. There is a probability P1  83 that it
is in the first state and a probability P2  85 that it is in the second one
(a) [6] What is the state operator  for this system in this basis? Check that Tr     1 .
0 1
(b) [6] In this basis, the Hamiltonian is H  E0 
 . What is H for this state operator?
1 0
(c) [8] Show that the time derivative of  is zero for this system.
2. Work this problem in the Heisenberg formalism. A particle of mass m in one dimension
has potential V  AX .
(a) [8] Deduce the time derivative of X and P.
(b) [5] Write P(t) at all times in terms of P(0).
(c) [7] Write X(t) at all times in terms of X(0) and P(0).
3. A particle of mass m lies in a potential V  x   B x . Using the WKB method, estimate the
energy of the n’th quantum state.
4. Estimate the ground-state energy of a particle of mass m in one dimension in the potential
2
V  x   B x using the variational principle with trial wave function   x   e Ax 2 .
5. A particle of mass m lies in the infinite square well with allowed region 0 < x < a. To this is
added a small perturbation W  x    cos  x a  .
(a) [3] In the limit   0 , what are the exact eigenstates and energies?
(b) [7] To first order in  , what is the ground state wave function?
(c) [10] To second order in  , what is the ground state energy?
Possibly
Helpful
Formulas:
Heisenberg Picture
d
i
A  t    H  t  , A  t  
dt

State Operators:
d
1
  t    H ,   t  
dt
i
Spin-Orbit Coupling
g 1 dVc  r 
WSO 
L S
4m 2 c 2 r dr
Definite integrals: In each of the formulas below, n, m and p are positive integers.


0
x n e  x dx  12 
2

a
0
  n 1 2
  n21  ,   m    m  1 !,   12    ,   32  
1
2
 ,   52  
 2a  n  n odd,
  nx 
sin 
dx  

0
n even
 a 
 0
  nx 
cos 
 dx  0,
 a 

a
a
a
  nx    mx 
  nx 
  mx 
sin 
sin 
cos 
dx   cos 
dx   nm ,




0
0
2
 a   a 
 a 
 a 
2an   n 2  m 2   n  m odd,
a
  nx 
  mx 


sin
cos
dx
0  a   a    0
n  m even

a
a
  nx    mx 
  px 
0 sin  a  sin  a  cos  a  dx  4  n,m p   m,n p   p,nm 

a

a
0
a
  nx 
  mx 
  px 
cos 
 cos 
 cos 
 dx   n ,m  p   m ,n  p   p ,n  m 
4
 a 
 a 
 a 
Indefinite Integrals:
  a  b x  dx  ax 
2
3
bx3/2 ,

a  b x dx 
1
b2

4
 5 a  b x

5/2

4
 a a b x
3

3/ 2


3
4
