MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #3
Problem 1: Clearing Impurities
Since we are asked for an approximate answer we will resort to the central limit theorem.
For
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #1
Due in hand-in box by 12:40 PM, Wednesday, February 13
Problem 1: Doping a Semiconductor
p(x)
0.2
0
l d
x
After diusing impurities into a
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Exam #2
Problem 1 (30 points) Entropy of a Surface Film
The surface tension and heat capacity at constant area CA for a water surface of area A
covered b
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Final Exam, Solutions
Problem 1 (20 points) Binary Alloy
a) Find the number of dierent ways of choosing the n -sites to be vacated and occupied
by atoms.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions, Exam #2
Problem 1 (30 points) Entropy of a Surface Film
and CA are given in terms of T and A so it is reasonable to choose T and A as the vari
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Exam #1
Problem 1 (30 points) Quantum Dots
y
L
0
L
x
A complicated process creates quantum dots (also called articial atoms) on the surface of
a square c
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #2
Due in hand-in box by 12:40 PM, Wednesday, February 20
Problem 1: Two Quantum Particles
A possible wavefunction for two particles (1 and 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Final Exam
Problem 1 (20 points) Binary Alloy
A binary alloy (ZnCu is an example) consists of N atoms and N
atoms. At low
temperatures the system can be
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #7
Due in hand-in box by 12:40 PM, Wednesday, April 3
Problem 1: Free Expansion of a Gas
Before
After
A classical, monatomic, non-ideal gas h
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #10
Due in hand-in box by 4:00 PM, Friday, May 3
Problem 1: Two Identical Particles
A system consists of two identical, non-interacting, spin
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #5
Due in hand-in box by 12:40 PM, Monday, March 11
Problem 1: Correct Boltzmann Counting
The calculation we have done so far to obtain the a
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #6
Due in hand-in box by 12:40 PM, Wednesday, March 20
Problem 1: Sound Waves in a Solid
Sound waves are usually thought of as pressure waves
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #3
Due in hand-in box by 12:40 PM, Wednesday, February 27
Problem 1: Clearing Impurities
p(x)
1/3
0
a
x
In an eort to clear impurities from a
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #4
Due in hand-in box by 12:40 PM, Wednesday, March 6
Problem 1: Heat Capacity at Constant Pressure in a Simple Fluid
For a simple uid show t
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #9
Due in hand-in box by 4;00 PM, Friday, April 19
Problem 1: The Big Bang
Early in the evolution of the universe, when the universe occupied
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Practice Exam #2
Problem 1 (35 points) Weakly Interacting Bose Gas
At low temperatures the entropy and isothermal compressibility of a weakly interacting
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #2
Problem 1: Two Quantum Particles
a)
p(x1 , x2 ) = | (x1 , x2 )|2
1 x1 x2 2
x2 + x2
=
(
) exp[ 1 2 2 ]
x2
x0
x0
0
The gure on
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #4
Problem 1: Heat Capacity at Constant Pressure in a Simple Fluid
Start with the rst law of thermodynamics.
dQ = dU + P dV
/
Th
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #1
Problem 1: Doping a Semiconductor
a) Mentally integrate the function p(x) given in the gure. The result rises from zero at a
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #8
Due in hand-in box by 12:40 PM, Wednesday, April 10
Problem 1: Dust Grains in Space
Astronomers have discovered that there exist in the in
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions, Practice Exam #2
Problem 1 (35 points) Weakly Interacting Bose Gas
@P
@P
a)
dP =
dT +
dV
@T V
@V T
@P
=
2cV 3 from given
@V T
@P
@S
5
=
= aT 3
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Problem Set #11
Due in hand-in box by 4:00 PM, Friday, May 10
Problem 1: Ripplons
(k)
k
We have seen that the bulk motion of a solid or liquid can be des
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Practice Exam #1
Problem 1 (30 points) Collision Products
3
EB/
2
1
1
2
3
4
5
EA/
A certain collision process in high energy physics produces a number of
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #5
Problem 1: Correct Boltzmann Counting
a)
= V
N
4emE
3N
3N/2
= V N [2emkT ]3N/2
using E = (3/2)N kT
S(N, V, T ) = k ln
n
o
= k
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions, Problem Set #11
Problem 1: Ripplons
a)
ky
2/Lx
points/k-volume D(k) =
2/Ly
kx
b)
(2)2
L x Ly
k-volume/point =
L x Ly
(2)2
A
(2)2
=
ky
#() = k
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #10
Problem 1: Two Identical Particles
a)
Fermions:
|1, 1, 0 >
|1, 0, 1 >
|0, 1, 1 >
2
3
2 + 3
T = 0 state
Bosons:
|2, 0, 0 >
|1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #9
Problem 1: The Big Bang
If the expansion is adiabatic,
S = 0.
S =
@F
@T
@
@T
=
V
1 2
(kT )4 V
3 ~3
45 c
4 2 4 3
k T V
45 c3 ~
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #6
Problem 1: Sound Waves in a Solid
We need to nd (@T /@P ) Q=0 . To do this we will use in sequence the rst law, the energy
de
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2013
Solutions to Problem Set #8
Problem 1: Dust Grains in Space
a) H is separable: the 6 variables are statistically independent.
p(1 , 2 , 3 , L1 , L2 , L3