Physics 413: Stat. Mech. - Solutions for Homework 2
Problem 1: Carnot process for a paramagnetic substance
a)
isotherms are straight lines through the origin, M = DH/T .
to nd adiabatic H M curves, start from 0 = Q = CdT HdM
use equation of state to sub
Physics 413: Statistical Mechanics - Homework Solutions 3
Problem 1: Probability of a density uctuation
a)
(10)
4
P (4) =
= 210/1024 = 0.2051
210
(10)
5
= 252/1024 = 0.2461
210
P (4)/P (5) = 5/6 = 0.8333
P (5) =
b)
(
P (4 10 ) =
22
(
P (5 1022 ) =
)
1023
Physics 413: Stat. Mech. - Solutions of Homework 5
Problem 1: Relativistic ideal gas
a) eigenstates are plane waves k = V 1/2 eikx with wavevectors ki = (2/L)ni with ni integer
= c|p| = c
Total energy
2
2
2
kx + ky + kz
N
2
2
2
(ki,x + ki,y + ki,z )
E=c
i
Physics 413: Statistical Mechanics - Homework Solutions 6
Problem 3: Ideal gas in the gravitational eld (15 points)
N
a) ZN =Z1 /N !
H
2
Z1 = d3 pd3 q ep /2mmgz = (2mkB T )3/2 A 0 dz emgz (2mkB T )3/2 A/(mg ),
A1 = kB T ln Z1 = (3/2)kB T ln(2mkB T ) kB T
Physics 413: Statistical Mechanics - Homework Solutions 7
Problem 1: Shifted Gaussian distribution
Since a 1 the summation over n can be replaced by an integration.
a)
a
2
dn n ea(nn0 )
a
2
dn [(n n0 ) + n0 ] ea(nn0 )
=
a
2
= n0
dn ea(nn0 )
= n0
Physics 413: Statistical Mechanics - Homework Solutions 8
Problem 1: Quantum corrections to classical ideal gas (12 points)
The classical (Boltzmann) limit corresponds to
n =
1
e ()
+
1
Expanding the denominator gives
n = e () (1 e () ) + O[(e () )3 ]
Ave
Physics 413: Statistical Mechanics - Homework Solutions 9
Problem 9.1
Problem 9.2
Problem 9.3
Problem 9.4
a)
H=
N
p2
j
i=j
discrete Fourier transformation: xj =
for pj
H=
=
2m
p2
k
k
=
1
N
N
p2
j
j =1
2m
p2
k
k
2m
2m
N
A
(xj xj +1 )2 .
2 j =1
+
k
eikj