PC3130/3201 QUANTUM MECHANICS
TUTORIAL 4
March 2000
1.
(a)
Prove that the trace of an operator A,
TrA =
 u
i
i
A ui  , is
independent of the particular othonormal basis {ui >} that is chosen for
its evaluation.
2.
(b)
Prove that Tru > < v= < v u >.
(a)
Show that for  = 1, the matrices representing  1 and  3 are respectively
 0 1 0

 
1 
 1 0 1
2

 0 1 0
1 0 0 


 3   0 0 0 


 0 0  1
(b)
Show that if an operator commutes with any two components of the
angular momentum operator  , it must commute with the remaining
~
components.
(c)
3.
Show that Tr  i = 0, i = 1, 2, 3.
Let a and a+ be two Hermitian conjugate operators such that [a, a+] = 1. We
put N = a+ a. Show that
[N, aj ] = -j aj;
[N, a+j ] = j a+j
(j integer > 0)
4.
In question (3), if { n > } are the normalized eigenkets of the operator N, i.e.
N n > = n n >
n = 0, 1, 2, .....
< n n > = nn
Show that
(i)
< i x2 j >


2m

j  1 j  2ij 2 
j

j  1ij 2  ( 2 j  1)ij (no sum over j)
where i, j are non-negative integers.
(ii)
(iii)

1
m ( 2 j  1)ij 
2
(no sum over j)
 i p 2 j 
( x ) 2  x 2    x  2 
( p ) 2 
j  1 j  2ij 2 
j
j  1ij 2


( 2 j  1)
2m
1
m ( 2 j  1)
2
1
( 2 j  1)  E j / 
2
Here, i > and j > are eigenstates of N with eigenvalues i and j
respectively and
Thus, xp 
1/ 2
  
x
 (a   a ) ,
 2m 
(iv)
p  ( m / 2)1/ 2 i( a   a )
Show that the operator N can be expressed as
N=
n n
n
n
5.
Suppose operators a and a  satisfy the following relations
aa   a  a  1 ,
a 2  (a  ) 2  0
These are known as fermionic operators in contrast with the bosonic operators
discussed in the lectures.
(i)
Can the operator a be Hermitian ?
(ii)
Prove that the only possible eigenvalues for the operator N  a  a are zero
and one.
Explain.
6.
Consider the Hamiltonian for the two dimensional motion of a particle of
mass  in a harmonic oscillator potential:
H
(i)
(ii)


1
1
p12  p 22 +  2 ( x12  x 22 )
2
2
1
Show that the energy eigenvalues are given by E n  (n  ) , where
2
n  n1  n2 , with n1 , n2  0,1, 2,......... .
Express the operator  3  x1 p 2  x2 p1 in terms of the annihilation operators
a1 
1
2
 X 1  iP1  ,
a2 
1
2
 X 2  iP2 
and the corresponding creation operates a 1 and a 2 . Here X i2 
Pi 2 
1
p i2 , i = 1, 2.

Show that H,  3   0 .


xi2 ,