Statistical Physics I (8.044) Spring 2009
Assignment 8
April 8, 2009
Due April 15, 2009
Please remember to put your name
and section number
at the top of what you turn
in.
Your
SECOND TEST
is
Wednesday April 22
at the usual lecture time,
12 pm
,
in an unusual location:
3RD FLOOR OF WALKER MEMORIAL, AKA 50340
.
Readings
The reading for Chapter VI of 8.044 is the Notes on the Canonical Ensemble, the Notes
on Rotational Raman Spectrum of a Diatomic Molecule, which are both available on the
8.044 web page, and Baierlein Chapters 5, 13 and 14.
Problem Set 8
1.
Interpolating Between a TwoState System and a Harmonic Oscillator (10
points)
(a) As a prelude to this problem, show that the heat capacity of a quantum harmonic
oscillator is unchanged if one ignores the zero point energy and writes the energy
eigenvalues as
ε
n
=
n
¯
hω
instead of
ε
n
= (
n
+
1
2
)¯
hω
.
(b) Consider
N
localized noninteracting particles, each of which has a finite number,
n
, of energy levels. The
n
levels are evenly spaced in energy, each separated from
the next by an energy
ε
. Define
θ
≡
ε/k
B
. Show that the partition function is
Z
N
1
where
Z
1
=
Z
h
.
o
.
(
θ/T
)
Z
h
.
o
.
(
nθ/T
)
with
Z
h
.
o
.
(
x/T
) being the partition function for a harmonic oscillator with
¯
hω/k
B
=
x
and with the zero point energy ignored.
(c) Show that the heat capacity of the system in (b) is the difference between the
heat capacities of two harmonic oscillators.
(d) Sketch a plot of the heat capacity of the system in (b) as a function of temper
ature for
n
= 2,
n
= 5 and
n
≫
1.
1
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2.
A ThreeState System (10 points)
A solid contains
N
ions with magnetic moments due to an angular momentum
J
= 1.
That means that each ion can exist in one of three possible states labelled by a
quantum number
m
J
which can take on the values 1, 0 or 1.
The magnetization
of the solid is
M
=
Nμ
(
m
J
)
. In the absence of any applied magnetic field, electric
effects within the solid
1
cause the states with
m
J
=
±
1 to have an energy Δ lower
than the state with
m
J
= 0. Turning on a magnetic field
H
induces a further shift
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 Spring '09
 KrishnaRajagopal
 Physics, Atom, Energy, Fundamental physics concepts

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