Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW1 (01/19/11―01/26/11)
Text book:
Exercise 6.1 ((a)-(d)) (page 386)
Exercise 6.2 (page 386)
Exercise 6.3 (page 386)
Challenge problem (extra point):
Exercise 6.1 (e)
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW2 (01/24/11―01/31/11)
Text book:
Exercise 6.6 (page 386-387)
Exercise 6.18 (page 389)
2
d2
l (l  1)
U
(
r
)

[
V
(
r
)

]U nl (r )  EnlU nl (r )


nl
2
2m dr
2mr 2
with l=0 case, where U nl (r )  rRnl (r ) , and Rnl (r ) is the radial function which
2
[Hint: (1) solve the radial equation 

satisfies the normalization condition  | Rnl (r ) |2 r 2 dr  1 ; (2) solve En 0  0 where n
0
correspodes the quntum nember below which all the eigenenergies En0 are lower
than zero.]
Exercise 6.19 (page 389)
[Hint: (1) express the function  (r ) in terms of Rnl (r )Ylm ( ,  ) and notice that
r
R21 (r ) 
1 r  2 a0
3
and Y11 ( ,  )  
e
sin  ei , (2) normalizing the wave function
3
6a0 2a0
8

in the spherical coordinates:  (r ) (r )dr   Rnl (r ) Rnl (r )r 2 dr  Ylm ( ,  )Ylm ( ,  )d  to

3
0


find A and using  u n eu du  n !, (3) the propability of finding the particle at r is
0
defined as P(r )  Rnl (r ) r 2 ]
2
Challenge problem (extra point):
Exercise 6.21 (page 390)
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW3 (01/31/2011―02/09/2011)
Text book:
Exercise 6.10 (page 387)
Exercise 6.11 (page 388)
Exercise 6.22 (page 390)

[Hint:  u n eu du  n !]
0
Challenge problem (extra point):
1
2
Exercise 6.17 (page 389) [Hint: the potential is V ( )  mgl 2 and
l 2 2  x 2  y 2 ]
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW4 (02/07/2011―02/16/2011)
Text book:
Exercise 6.9: (a) and (b) (page 387) (Hint: the square of the magnitude
of the orbital angular monemtum is the eigen value of Lˆ2 )
Exercise 6.12 (page 388) (Hint: (1) { nlm (r )  Rnl (r ) Ylm ( , ) } are the
eigenfunctions of the hydrogen atom, (2) Lˆ2 nlm (r )  2l (l  1) nlm (r ) , (3)
Lˆz nlm  m nlm , and (4) nlm n l m   nn  ll  mm )
Exercise 6.20 (page 390)
' '
'
'
'
'
Challenge problem (extra point):
The eigenstate of an isotropic harmonic oscillator is given by
 nlm (r , ,  )  Anl r l e

m 2
r
2
n l
2
a
k 0
2k
r 2 k Ylm ( ,  ) . Find (a) the normalized ground state  000 ,

and (b) the first excited states  11m . [Hint:  e ax dx 
2
0
1 
]
2 a
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW5 (02/21/2011―03/02/2011)
Text book:
Exercise 7.8 (page 451) (Hint: you can apply the
1
2
formula j, m j  l  , m j 
1
1
l  mj 
2 l, 1 , m  1 ,  1 
2 l, 1 , m  1 , 1 )
j
j
2l  1
2
2 2
2l  1
2
2 2
l mj 
Exercise 7.27 (page 454) (Hint: use the relation
ˆ
ˆ
ˆ ˆ
ˆ
ˆ ˆ
J 2  ( L  S ) 2  L2  S 2  2 L  S to rewrite the Hamiltonian )
Exercise 7.32 (page 454) (Hint: rewrite the Hamiltonina in terms of Sˆ 2 ,


ˆ
ˆ
ˆ ˆ ˆ ˆ
ˆ
ˆ ˆ
ˆ ˆ ˆ
ˆ
ˆ 1 ˆ
ˆ
ˆ
S132 , S22 , and S z , where S  S1  S2  S3  S13  S2 , S13  S1  S3 , S13  S 2  S 2  S132  S 22 ,
2
ˆ
ˆ
ˆ
ˆ
and Sz  S1z  S2 z  S3z , respectively)
Challenge problem (extra point):
Consider a hydrogen atom which is initially at the energy level of E30 .
Find the shifted energy levels when it is placed in an external uniform magnetic
field. Discuss the corresponding degeneracies.
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW6 (02/28/2011―03/09/2011)
Text book:
Exercise 8.1 (page 484) (Hint: (1) The bosons in this case are spin 0
particles. Therefore, the spin part of the wave function  S ( s1, s2, s3, ) can be ignored in
the total wave function. (2) The symmetric wave function for N noninteracting
identical bosons is given by
 S (1 ,  2 ,  N ) 
N1 ! N 2 ! N N !
P Pˆ n1 (1 ) n2 (2 )  nN ( N ) and N=3 in this cae)
N!
Exercise 8.3 (page 485) (Hint: (1) There is no symmetry restriction on
the states of nonidentical particles system. (2) If there is no orbital angular
momentum, the total wave function is given by the spin part of the wave function
only (i.e., the eigenstates of total angular momentum). (3) You can use the table
provided in the class to find the C.G. coefficients. (4) If the two particles are
identical, only those symmetric states are allowed)
Exercise 8.4 (page 485) (Hint: (1) When the spin part of the wave
function is symmetric, the spatial part of the wave function must be antisymmetric.
(2) Fermions follow the Pauli’s Excusion Principle.)
Challenge problem (extra point):
Use the lower sign part of the recursion relation of the Clebsch-Gordan
coefficients,
 j  m  j
m  1 j1 , j2 , m1 , m2 j , m 

 j1  m1  j1
 j2  m2  j2
m1  1 j1 , j2 , m1 1, m2 j , m 1
m2  1 j1 , j2 , m1 , m2 1 j , m 1
1
2
1
2
1
2
to obtain the Clebsch-Gordan coefficient of l , , m  , m2 l  , m .
,
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW7 (03/28/2011―04/04/2011)
Text book:
Exercise 9.1 (page 562)
Exercise 9.2 (page 562)
Exercise 9.8 (page 563) (Hint: (1) the exact solutions for Ĥ 0 is
1
En(0)   (n  ) and  n(0)  n (2) a n  n n  1 ; a† n  n  1 n  1 )
2
Challenge problem (extra point):
Find the spectroscopic notation 2 S 1 LJ of the ground state configuration of Rh (the
element Rhodium at the 5th period and 9th column of the periodic table). (Hint: use
the simple method provided by Dr. Yu’s to find S, L, and J values)
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW8 (04/04/2011―04/11/2011)
Text book:
Exercise 9.11 (page 564)


2


2
1
1
Exercise 9.21 (page 565) (Hint: Lˆ2x   Lˆ  Lˆ  , Lˆ2y   Lˆ  Lˆ  )
2

 2i

Exercise 9.44 (page 569)
Challenge problem (extra point):
Exercise 9.10 ((a) and (b)) (page 563)
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW9 (04/11/2011―04/18/2011)
Text book:

Exercise 9.13 (page 564) (Hint: (1)  x 2e2 x dx 
4
0

1
3 2 x 4
0 x e dx  8 (1) , (3)

4 2 x
 x e dx 
4
0
2
1/4
5
( ) , (4)
8
4

8 2 x
 x e dx 
4
0
21/4 3
( ) , (2)
8
4
21/4 9
( ) , and (5) (1)  1 ;
16
4
3
1
2
( )  1.225 ; ( z  1)  z( z ) ;  ( ) 
)
3
4
4
( )
4

Exercise 9.16 (page 565) (Hint:


e  ax dx 
2

a
)
Exercise 10.2 (page 613) (Hint: (1) apply the relation
ˆ
ˆ  Aˆ  Bˆ   Aˆ , Bˆ   1  Aˆ ,  Aˆ , Bˆ    1  Aˆ ,  Aˆ ,  Aˆ , Bˆ     ... to Xˆ (t ) , then we have
e A Be
H

 2!  
  3!   
  
i
Xˆ H (t )  e
, (2) Xˆ 
t ˆ
H
ˆ i
Xe
t ˆ
H
 aˆ
2m
1
1
Pˆ
 Xˆ   Hˆ , Xˆ    Hˆ ,  Hˆ , Xˆ     Hˆ ,  Hˆ ,  Hˆ , Xˆ     ...  Xˆ cos(t ) 
sin(t )
 3!  

2! 
m
†
 aˆ  and Pˆ  i
m †
 aˆ  aˆ  )
2
Challenge problem (extra point):
Exercise 10.5 (page 614)
Homework of Phys 556- Quantum Theory and Matter
(Spring 2011)
HW10 (04/18/2011―04/25/2011)
Text book:
 aˆ
2m
Exercise 10.9 (page 614) (Hint: Xˆ 
†
 aˆ  )
1
2
Exercise 10.10 (page 614) (Hint: sin x sin y  (cos( x  y )  cos( x  y )) and
e
 x2 bx
e
b
b2
 ( x  )2 
2
4
)

Exercise 11.3 (page 650) (Hint: (1)  re
0

(2)  e
0

r
2
a2

r
2

r2
a2

2
b
b
 ( x  )2 
2
1 2
cos(qr )dr   e a (eiqr  e  iqr )dr , (3) e x bx  e 2 4 , (4)
20
Challenge problem (extra point):
Exercise 11.8 (page 650)
r2
 2

sin(qr )dr    e a cos(qr )dr ,
q 0

0
x
x
 e dx   e dx )
2

2
0