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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
8.044 Statistical Physics I
Spring Term 2004
Exam #3
Problem 1 (30 points) Weakly Interacting Bose Gas
At low temperatures the entropy and isothermal compressibility of a weakly interacting Bose gas
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 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
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
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
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
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
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, Exam #1
Problem 1 (30 points) Quantum Dots
a) Use Bayes theorem: p(x|y) = p(x, y)/p(y). We are given p(x, y) so we must rst nd
p(y).
p(y) =
=
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
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 #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
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
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
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
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
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
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
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