Download Most probable occupation number

Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel
Phys 452
Homework
Thursday Jan 12:
assignment #2
5.28, 5.30, 5.31
Phys 452
Quantum statistical mechanics
Most probable occupation number:
• Distinguishable particle
• Identical fermions
• Identical bosons
Nn  dne
Nn 
Nn 
   En 
dn
   En 
e
e
1
dn  1
   En 
1
Phys 452
Quantum statistical mechanics
Most probable occupation number:

N n ( ,  )
N
n 0
n
N

N E
n 0
 ,  
n
n
E
To be determined by E, N
Or by T and characteristic
energy of the system
Phys 452
Quantum statistical mechanics
Calculation of  and :
Case of ideal gas
One spherical shell
kz
kF
“degeneracy”
Fermi
surface
kx
density of states
 d shell
dk
ky
Vk 2

dk
2
2

N
Bravais
k-space


Nk dk
k 0

Volume in k-space
of each individual state
Vunit 

3
V
E

k 0
N k Ek d k
Phys 452
Quantum statistical mechanics
Calculation of  and :
Case of ideal gas : distinguishable particles

N

 m
N  eV 
 2
Nk dk
k 0
E

E


2 

3/2
3
m
 
E
Ve 
2
 2
N k Ek d k
k 0
We identify

1
k BT
and

2 

3/2
 (T )  kBT
3N
2
Phys 452
Quantum statistical mechanics
Meaning of  and :
Case of ideal gas : distinguishable particles
Using
1

k BT
and
 (T )  kBT
3
Etot  Nk BT
2
  N  3  2 2
 (T )   k BT  k BT ln    ln 
  V  2  mk BT



Phys 452
Quantum statistical mechanics
Most probable occupation number
or density of occupation:
  E    / k BT
Maxwell-Boltzmann
n
E

e


• Distinguishable particle
distribution
• Identical fermions
nE 
• Identical bosons
nE 
1
 E    / k BT
e
1
1
e E  / kBT  1
Fermi- Dirac
distribution
Bose-Einstein
distribution
Phys 452
Quiz 3a
What is the maximum possible value
for the density of occupation in case of fermions
at a given temperature T?
A.



B. 1/ e  1
C. 1
D. 0
E. 1/2
Phys 452
Quantum statistical mechanics
Fermi-Dirac distribution:
nE 
1
 E    / k BT
e
1
 (0)  EF
nE 
1
e E  EF / kBT  1
T 0
n 1 if E    0
n  0 if
E    0
Phys 452
Quantum statistical mechanics
Ideal gas of fermions or bosons

N
 n(k )d

V
N 2 
2 0 e 
k
k 0

E
 n(k ) E (k )d

k /2 m    / kBT

2 2

2
V
E 2
2 2m 0 e 
k
k 0
Pb 5.28: Fermions at T=0
Pb 5.29: Bosons
k2
dk
1
k4

k /2 m    / k BT

2 2
dk
1
 (T )
Etot (T )
Relationships between EF, kF and Etot, N
 (T )  0
Predicted in 1924-25
First measured on Rb atoms 1995
First measured on Photons 2010
Bose-Einstein condensation
T  Tc
 0
And all particles condense into ground state
Phys 452
Quiz 3b
What is the maximum possible value
for the density of occupation in case of bosons?
A.



B. 1/ e  1
C. 1
D. 0
E. 1/2
Phys 452
Quantum statistical mechanics
Black-body spectrum
Photons
• Boson: S=1; m=+/-1
• Energy- wavelength:
E 
k  2 / 
• Non-conservation of the
number of photons ( =0)
 ( ) 
 2 c3  e
3
 / k BT
Density of energy

1
Phys 452
Quantum statistical mechanics
Black-body spectrum
Wien displacement law
Bosons
at equilibrium T
Pb 5.30: Wien law
Analogy: lava emits light when hot !
Pb 5.31: Stefan-Boltzmann formula
Phys 452
Quiz 3c
We can find that for the blackbody emission, the relationship
between the optimal wavelength and the temperature is
max
2.9 103

m.K
T
What should be the temperature of the blackbody
to emit principally around 600nm (orange) ?
A. around 2000K
B. around 5000K
C. around 8000K
D. around 10000K
E. around 20000K