Quantum Statistics

1
Chapter 7

• Quantum Statistics

2
Boltzmann w/ Chemical Potential
Grand Canonical Ensemble
Reservoir, Heat Bath UR, NR, TR, mR
System Es, Ns, Ts, ms
Thermal and diffusive equilibrium TR Ts mR ms
3
Boltzmann w/ Chemical Potential
Grand Canonical Ensemble
Reservoir, Heat Bath UR, NR, TR, mR
System Es, Ns, Ts, ms
Thermal and diffusive equilibrium TR Ts mR ms
4
Carbon Monoxide Poisoning
DE 0.00 eV
O
O
Fe2
DE -0.70 eV
C
O
DE -0.85 eV
5
Carbon Monoxide Poisoning
DE 0.00 eV
O
O
Fe2
DE -0.70 eV
C
O
DE -0.85 eV
6
Carbon Monoxide Poisoning
DE 0.00 eV
O
O
Fe2
DE -0.70 eV
C
O
DE -0.85 eV
7
Carbon Monoxide Poisoning
DE 0.00 eV
O
O
Fe2
DE -0.70 eV
C
O
DE -0.85 eV
8
Carbon Monoxide Poisoning
DE 0.00 eV
O
O
Fe2
DE -0.70 eV
C
O
DE -0.85 eV
9
Binding Probabilities
10
Binding Behaviors
11
Langmuir Isotherms
12
Langmuir Isotherms
13
Langmuir Isotherms
k T
Zint
vQ
vQ
Po
V/N
Zint
m
14
Fermions and Bosons

• Fermions
• ½ integer spin
• Cannot have more than one in the same single
  particle state
• e, m, ms, x, y, z
• Bosons
• Integer spin
• Can have more than one in the same single
  particle state
• e, m, ms , x, y, z

15
Volume Ratios
Quantum mechanical system
Classical mechanical system

• Wavefunction volume
• Particle volume

16
Fermions and Bosons
1
11
2
12
3
13
4
14
5
15
6
7
8
9
10
17
Fermions
Suppose we have a system that is a metal with an
s orbital valence band. We can put one electron
in each of the ms states (spin up and spin down)
of the s orbital. The metal is in contact with
the environment at some temperature, T.
reservoir
sys
18
Fermions
19
The Fermi-Dirac Distribution
Low T
High T
kBT
20
Bosons
First, the partition function.
21
Bosons
Next, the occupation number
22
The Bose-Einstein Distribution
23
The Bose-Einstein Distribution
24
Fermions and Bosons
25
The Bose-Einstein Condensate
26
The Bose-Einstein Condensate
27
The Bose-Einstein Condensate
28
The Bose-Einstein Condensate
29
Blackbody Radiation
Spectral Radiance
Wilhelm Wien 1864-1928
30
Blackbody Radiation
Spectral Radiance
John William Strutt 3rd Baron Rayleigh 1842-1919
Sir James Hopwood Jeans 1877-1946
31
Blackbody Radiation
Imagine light bouncing around in a very hot oven.
Standing waves are created.
32
Blackbody Radiation
Classically, occupying high frequencies would
obey second law but contradict what we observe
when we open a hot oven.
33
Blackbody Radiation
Max Planck 1858-1947
Two descriptions that describe either end of the
spectrum but not the middle. Enter Max Planck.
34
Blackbody Radiation
Lets start with a partition function. We can
assume a collection of photons or harmonic
oscillators since the energy is the same form.
This is relative to the ground state energy,
which we can always set to zero.
35
Blackbody Radiation
36
Blackbody Radiation
Electron in an atom absorbs a photon.
The chemical potentials must balance to maintain
diffusive equilibrium.
37
Blackbody Radiation
l 2L
l 2L/2
l 2L/3
l 2L/4
l 2L/5
l 2L/6
l 2L/7
L3 V
L
L
L
38
Polarization
We need a factor of 2 to account for x and y
components of polarization.
39
Blackbody Radiation
The total energy contained in a box of photons at
a particular temperature.
40
Blackbody Radiation
nz
dn
n dq
n sin q df
q
ny
f
nx
41
Blackbody Radiation
42
Blackbody Radiation
43
Change of Variables
44
Energy Density Weins Law
45
Blackbody Radiation
Change of variables to l
46
Blackbody Radiation
Change of variables to l
47
Planck Spectrum for various T
48
Various Evaluations
Wave Type Frequency (Hz) Wavelength T(K)
gamma 1021 0.3 pm 1.76 x 1010
X-Ray 1018 0.3 nm 1.76 x 107
UV 1015 300 nm 17,600
VIsible 6 x 1014 500 nm 11,000
IR 1014 3 µm 1760
microwave 1010 3cm 0.176
49
Cosmic Background
50
Microwave Background

• 5 waves/cm ? ? 0.19 cm, f 160 GHz
• T 2.72 K

51
Blackbody Radiation
52
Planck Spectrum for various T
53
An Oven _at_ 375o F
L3 0.125 m3
0.5 m
0.5 m
0.5 m
54
An Oven _at_ 375o F
Fraction of spectrum in the visible
55
An Oven _at_ 375o F
56
Solar Spectrum
Incorrect Solar Spectrum from only changing
x-axis (ehc/l)   
Correct Solar Spectrum from full change of
variables   
l (m)
57
Solar Spectrum
Fraction of spectrum in the visible
58
Heat Capacity and Entropy
59
Heat Capacity and Entropy
60
Stefan Law
Side view
Front view
R
f
q
c dt
61
Stefan Law
62
Stefan Law
63
Stefan Law
Side view
R
q
64
Sun Temperature from Power
P/A 1370 W/m2
sun
1.50 x 1011 m
Rsun 6.96 x 108 m
65
Sun Temperature from Power
66
Thermal equilibrium meansthe oven and blackbody
areat the same temperature, thus their spectra
are the same
blackbody
oven
67
Bose-Einstein Condensate
L
l 2L/3
l L
l 2L/n
l 2L
68
Bose-Einstein Condensate
l 2L/3
l L
l 2L/n
l 2L
L
69
Bose-Einstein Condensate
70
Bose-Einstein Condensate
71
Bose-Einstein Condensate
72
Bose-Einstein Condensate
73
Bose-Einstein Condensate
74
Heme Occupation