# Home work

```Homework of Phys 556- Quantum Theory and Matter
(Spring 2010)
HW1 (01/11/10―01/20/10)
Text book:
Problem 4.26 (page 177)
Problem 4.29 (page 177)
Problem 4.31 (page 178)
Challenge problem (extra point):

Calculate Jˆx jm and Jˆ y jm where jm is the common eigenstate of Jˆ 2 and Ĵ z .
HW2 (01/20/10―01/25/10)
Text book:
Problem 4.32 (page 183)
Problem 4.34 (page 189)
Problem 4.37 (page 189)
Challenge problem (extra point):
1
2
(Hint: S  S(1)  S(2) and sm 
If s1  s2  , prove that (a) S 10  2
C
m1  m2
(page 188))
1 1 1 1


 2 1  1 and (b) S 00  0 .
2 2 2 2
s1s2 s
m1m2 m
s1m1 s2 m2 , where Cms11sm2s2m from Table 4.8
HW3 (02/01/10―02/08/10)
Text book:
Problem 5.4 (page 206)
Problem 5.5 (page 206)
Problem 5.7 (page 210)
Challenge problem (extra point):
Find the wave functions of two systems of identical, noninteracting particles: the first consists of
two bosons, and the second of two spin &frac12; fermions.
HW4 (02/08/10―02/15/10)
Text book:
Problem 5.8 (page 211)
Problem 5.12 (page 216) (Hint: for nitrogen, χ is antisymmetric for
S=1/2 and symmetric for S=3/2; ψ is
antisymmetric for L=0, and symmetric for
L=1, 2, and 3)
Problem 5.14 (page 218)
HW5 (02/15/10―02/22/10)
Text book:
Problem 5.17 (page 223) (For copper, P=3.84x1010 N/m2)
Problem 5.18 (page 229)
Problem 5.19 (page 229)
Challenge problem (extra point): Problem 5.16 (page 223)
HW6 (02/22/10―03/01/10)
Text book:
Problem 5.23 (page 233)
Problem 5.24 (page 225)
Problem 5.28 (page 243)
Challenge problem (extra point): Problem 5.31 (page 245)
HW7 (03/01/10―03/10/10)
Text book:
Problem 6.1 (page 254)
Problem 6.2 (page 254)
1
1
2
1
8
1 3
x  ... and
16
(2) refer page 49 for n V n )
(Hint: (1) 1  x  2  1  x  x 2 
Problem 6.4 (page 256)
Challenge problem (extra point): Problem 6.3 (page 255)
HW8 (03/22/10―03/29/10)
Text book:
Problem 6.14 (page 270)
1
( En(0) )2  2 En(0) n V ( x) n  n V 2 ( x) n  and xˆ 
 aˆ  aˆ  )
2 
2mc
2m
Problem 6.20 (page 277) (Hint: L ~ and r ~ a)
(Hint: Er(1),n  
Problem 6.24 (page 280)
Challenge problem (extra point): Calculate energy levels n=2 of a hydrogen atom
corrected to the first order perturbations due to the fine structure and to a weak external magnetic
field Bext (Bext is uniform and along z axis)


 2 En(0)  n
3
(0)
(1)
(1)
0
Enljm j  En  EFS  EZ  En 
   g j B m j Bext , where

n2  j  1 4 

2

1
 2l  2
 2l  1 for j  l  2
.
gj  
2l
1

for j  l 
2
 2l  1
HW9 (03/29/2010―04/05/2010)
Text book:
Problem 7.2 (page 298)

(Hint: the integral

0
x
dx
2
b

2 n

  2n  3!! 1
2  2n  2 !! b 2 n 1
 n.(n  2).(n  4)....5.3.1 for odd n

n !!  n.(n  2).(n  4)...6.4.2 for even n

1
for n  0 or  1

, n  1, 2,3,... where
).
Problem 7.3 (page 298) (Hint: refer example 7.3 on page 296)
Problem 7.8 (page 308)
Challenge problem (extra point): Consider the hydrogen atom in an external
magnetic field. When Bext  Bint (i.e., in the intermediate regime), where neither Hˆ FS
nor Hˆ Z dominates, we must treat the two terms as perturbations to Borh
Hamiltonian Ĥ 0 on an equal footing (i.e. Hˆ p  Hˆ FS  Hˆ Z ). The unperturbed energy

En(0) is 2n2 degenerate and the zeroth-order eigenstate (i.e.,  n(0)  lsjm j
)
are
characterized by l, s, j, and mj. Please use the degenerate perturbation theory to
calculate the first-order correction to the energy En when n=1. (Hint: (1) the
dimension of the subspace is 2n2=2, the matrix element of Hˆ p
1
1
1
is lsjm j Hˆ p lsjm j  lsjmj Hˆ FS  Hˆ Z lsjmj where l  0 , s  , j  , and m j   ; (2)
For given l and s, we have jm j 
c
lsj
ml ms m j
ml , ms
coefficients)
2
2
lsj
ml ms m j
lml sms where c
2
are Clebsch-Gordan
HW10 (04/05/2010―04/012/2010)
Text book:
Problem 9.1 (page 343) (for   nlm  Rnl (r )Ylm ( ,  ) ,
please refer tables 4.3 and 4.7)
Problem 9.4 (page 345)
Problem 9.5 (page 345)
Challenge problem (extra point): Problem 9.7 (page 348) (Hint: the general
solution of a homogeneous linear differential equation ( y ''  ay '  by  0 ) is
a 1 2
a  4b )
y  Ae1x  Be2 x , where 1,2   
2 2
```