# Part I (50 points) ```Part I (50 points)
2006/3/17
1. If U and V are unitary operators and A is an arbitrary operator,
(a) show that UVU † , the unitary transformation of V , remains unitary.
(10 points)
(b) show that A and its unitary transformation UAU † have the same
eigenvalues. (10 points)
2. Consider a particle of mass m in one dimension bound by a δ-function
potential:
V(x) = -V0 (x)
where V0 is positive.
(a) Find the wave function and the binding energy. (20 points)
(b) Find the value of x0 such that the probability of finding the particle
with | x | x0 is equal to 2/3. (10 points)
Part 2
2006/3/17
Problem 1: (20 points)
（a） Two spin- 1 2 particles are held fixed on the z-axis and kept at a
distance d apart. They interact through a dipole-dipole term
H
2  
 1   2  31  r  2  r . Calculate the eigenenergies for
3
d
the singlet and triplet states. (Of course r  (0,0,1) .)
1 0
（b） If at time t  0 the spin function is     , through what interval
0 1
 1   2
0 1
of time will the spin function become     ?
 1 1  0  2
Problem 2: (20 points)
Use the relation for the angular momentum operators and states
J  j, m 
j ( j  1)  m(m  1) j, m  1 to find the absent Clebsch-Gordon
coefficients in the following table ( j1
J 2
2
m1 m2 M 2
1
1
1
0
1
0
0
0
1
0
1
-1
0
0
0
0
0
0
-1 1
0
0
 j2  1)
1
2
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
Problem 3: (10 points)
A plane electromagnetic wave propagating in the z-directon can be described
 
by a vector potential field A( x, t )  ( Ax , Ay ,0)e ikzit . Argue that the photon
has spin s  1 and helicity ms  1 . (Hint: Consider the combination
A  Ax  iA y .)
Quantum Mechanics
Part III
2006/3/17
1. Prove the Thomas-Reiche-Kuhn sum rule
 ( En  En ) n x n
n
where
n
and
Eliminate the
n
2
  ( En  En ) n x n n x n 
n
are eigenstates of
E n  E n
H
2
2m
p2
 V ( x) .
2m
factor in favor of H.)
(Hint:
25 points
2. Show that any matrix that commutes with  is a multiple of
unit matrix.
25 points
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