Homework D
PHY4523
Due: April 6, 2011
1. (30.1)
2. (30.2)
3. (30.3)
4. (30.4) I worked on most of this in class. You may want to use
Li ( z )
d
Lin ( z )  n 1
dz
z
.
d
d
dz
Lin ( z )  Lin ( z )
dT
dz
dT
5. (30.5) Work on bosons with S = 0. Starting from g (k )dk 
N
A
th2
A
(2 )
2
d 2 k , you will find
Li1 ( z ) in 2D. How does Li1(z) behave as z1?
6. (30.6)
7. In class, we discussed a Bose gas (S = 0) contained in a box of volume V. Let’s
consider a Bose gas in a harmonic potential
V (r ) 
1
m(1 x 2  2 y 2  3 z 2 ) .
2
The energy of a particle is
 (n1 , n2 , n3 )  (n1  1/ 2)1  (n2  1/ 2)2  (n3  1/ 2)3 .
Let’s assume we are working at large energies so that we are in a continuum limit and
can ignore zero point energy.
(a) Show that the total number of states with energy less than  = 1 + 2 + 3, G(), can
be written as
 1
 1  2

1
3
G ( )  3
d

d

d


1 
2
0 3 63123 . (5)
 123 0
0
d
G ( ) ?) (5)
d
(c) Again, when the chemical potential goes to zero, this system enters into BEC.
The condition at Tc is then
(b) Find out the density of states, g(), in this case. (Do you see g ( ) 

N   d  g ( )
0
1
e
 / k BTc
1
.
Show that
k BTc 
(123 )1/3 N 1/3
 (3)
1/3
. (10)
(d) You showed in Prob.#5 that a Bose gas contained in a 2D box does not have BEC at
a finite temperature. Can a Bose gas trapped in 2D harmonic potential condense at a
finite temperature? (10)