Homework of Phys 622-Quantum mechanics II (Spring 2008)
HW1 (01/17/08―01/22/08)
ˆ
1. We know that the square of the angular momentum J 2 commutes with Ĵ  , i.e.,
ˆ
ˆ
̂
[ J 2 , Jˆ  ]  0 . How about the commutator of [ J , Jˆ  ] ? Does J commutes with Ĵ  ?

2. Obtain the matrix elements for the angular momentum operator J with j=1/2 and 3/2 in the
representation of jm , respectively.
3. Verify the relation j ,  m  
( j  m )
  j  m ! 
 2 j !  j  m ! 


1/ 2
J j  m j ,  j .
1
4. Consider the operator Aˆ  ( Jˆ x Jˆ y  Jˆ yJˆ x ) . Calculate the expectation value of  and  2
2
with respect of the state j, m .
ˆ
5. Consider case where j  1 . (a) Find the matrices representing operators J 2 , Jˆ z , Jˆ  , Jˆ x , and
ˆ
Ĵ y . (b) Find the joint eigenstates of J 2 and Ĵ z , and verify that they form an orthonormal
and complete basis. (c) use the matrices of Ĵ , Ĵ , and Ĵ to calculate Jˆ , Jˆ , Jˆ , Jˆ ,


x
y
and Jˆ z , Jˆ x . (d) Verify that Jˆ z3   2 Jˆ z and Jˆ 3  0 .
z

x
y

y
z

Homework of Phys 622-Quantum mechanics II (Spring 2008)
HW2 (01/22/08―02/05/08)
1. Find the eigenvectors  y

of Ŝ y in the matrix representation of spin
1
within
2
ˆ 2
1

1

 , m s  basis , where  , m s  are the joint eigenvectors of S and Ŝ z (hint: solving
2

2

the eigenvalue equation of Ŝ y , which is given by Sˆ y  y  ms   y , and using the


normality of eigenvectors). Verify that these vectors are orthonormal and complete.
2. Use the anticommuation and commutation relations of Pauli matrices to show that
 
 
   


(  A)(  B )  A  B  i  ( A  B ) , where  is defined in terms of the Pauli matrices



0 1
0  i
 
1  
,  2  
, and
such that J  j     with   i 1  j  2  k 3 ,  1  
2 2

 1 0
i 0 


 1 0

 3  
 . Note also that A and B are two vector operators commuting with  , but not
 0 1
necessarily with each other.
i 
3. Show that e j  I cos   i j sin  , where I is the identity matrix,  is an arbitrary real
constant angle, and  j is the Pauli matrix ( j  x , y, z ), respectively.
4. Find the energy levels of a spin
5
particle whose Hamiltonian is given by
2


Hˆ  02 ( Sˆ x2  Sˆ y2 )  0 Sˆ z , where  0 is a constant having the dimensions of energy. Are


the energy levels degenerate?
Homework of Phys 622-Quantum mechanics II (Spring 2008)
HW3 (02/05/08―02/19/08)
1. Show how Ĵ x , Ĵ y , and Ĵ z transform under a rotation of (finite) angle  about the z-axis
ˆ

(i.e. the rotation operator is Rˆ ( z ,  )  e  iJ z /  ). Using these results, determine how the
̂
angular momentum operator J transform under the rotation. (Hint: using the relation
1 ˆ ˆ ˆ
1 ˆ ˆ ˆ ˆ
ˆ
ˆ
e A Bˆ e  A  Bˆ  [ Aˆ , Bˆ ] 
A, A, B 
A, A, A, B   and the commutation relations
2!
3!
of the components of the angular momentum).

2. The operator corresponding to a rotation of angle  about an axis n is given by
 ˆ


Rˆ ( n,  )  e  i nJ /  . Show that the components of the position operator r̂ are rotated
 
 
through an infinitesimal rotation like rˆ '  rˆ  n  rˆ . (i.e. In the case  is infinitesimal,


 

show that r '  Rˆ ( n,   ) r Rˆ  ( n,  )  r   ( n  rˆ ) ).
      
j
j
j
3. Using the Wigner formula
( j  m )! ( j  m )! ( j  m' )! ( j  m' )! 

 cos 
( j  m' )! ( j  m  k )! ( k  m' m )! k! 
2
k
1
to find the rotation matrices d ( j ) and D ( j ) corresponding to j  .
2
d m( j'm) (  )   ( 1) k  m' m
2 j  m  m ' 2 k


 sin 
2

m' m  2 k
Homework of Phys 622-Quantum mechanics II (Spring 2008)
HW4 (02/19/08―03/04/08)
1
1. Obtain the expression for the C.G. coefficient corresponding to j1  1, j2  , and
2
j  j1  j2 .
ˆ ˆ
2. Write in full the matrix elements of J 1  J 2 in the representation of j1 j2 m1 m2 when
j1  j2 
1
.
2
ˆ
1
3. Obtain the eigenkets of J 2 for j1  j2  in terms of j1 j2 m1 m2 .
2
ˆ
ˆ


1
 
 
4. Consider the operator Pˆ  (1   1   2 ) such that J 1   1 and J 2   2 . Show
2
2
2
that the eigenvalues of P̂ are  1 depending on where j  1 or 0.