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Physics 219  Problem Set 4
Due Date: March 4, 2008
1. Bose Condensation of Relativistic Particles.
Consider a 3
D
system of massless relativistic particles with energy
ε
(
k
) =
ck
What is the BoseEinstein condensation temperature for such particles in terms of the speed
of light
c
, Planck’s constant
h
, Boltzmann’s constant
k
B
, and the particle density
N
/
V
.
2. Density Matrices.
(a) Consider a system + reservoir in a pure state

ψ
i
, as in class:

ψ
i
=
∑
r
,
s
c
r
,
s

r
i
s
i
where

r
i
describes the state of the reservoir and

s
i
describes the state of the system.
The system by itself will be described by a density matrix
ρ
. Show that
ρ
satisﬁes
tr
(
ρ
) =
1.
(b) Following the logic which we used in the case of the canonical ensemble, show that
the von Neumann entropy of a system described by density matrix
ρ
is
S
=

k
B
tr
(
ρ
ln
ρ
)
Show that this result is basisindependent.
3. Oneparticle Density Matrix.
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 Spring '08
 NAYAK
 mechanics, Energy, Mass, Statistical Mechanics

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