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Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel
Physics 452
Homework
First homework assignment:
Tuesday Jan 10 by 10pm
Assignment # 1:
Problems 5.22, 5.23, 5.24
in the textbook
Second homework assignment:
Thursday Jan 12 by 10pm
Phys 452
Quantum statistical mechanics
N particles ( N
1)
Thermal equilibrium, T
Quantization of the energy
for individual particles
( E1 , E2 , E3 ,...)
Total energy:
E  N1E1  N 2 E2  N3 E3  ...
How many ways we get the configuration
Q  N1 , N2 , N3 ,...
N , N
1
2,
N 3, ... ?
Phys 452
Quantum statistical mechanics
Example: 3 –particle system
• Textbook example
Infinite square well
E
2
2
ma 2

na2  nb2  nc2

• In-class example/ pb 5.23
Harmonic oscillator
3

E     na  nb  nc 
2

For each type of particles:
• List all the possible configurations
• Determine the number of combinations of each configuration
• Determine the probability of each configuration for a given energy
Phys 452
Quiz 1b
Consider 3 distinguishable particles in a harmonic oscillator potential.
If the total energy of the system is E  9  / 2
how many possibilities there are to get the configuration (1,1,1,0,…)?
A. 10
B. 4
C. 1
D. 6
E. 2
Phys 452
Quiz 1c
Consider 3 fermions particles in a harmonic oscillator potential.
If the total energy of the system is E  9  / 2
how many possibilities there are to get the configuration (1,1,1,0,…)?
A. 10
B. 4
C. 1
D. 6
E. 2
Phys 452
Quiz 1d
Consider 3 bosons in a harmonic oscillator potential.
If the total energy of the system is E  9  / 2
how many possibilities there are to get the configuration (1,1,1,0,…)?
A. 10
B. 4
C. 1
D. 6
E. 2
Phys 452
Quantum statistical mechanics
Statistical configuration number:

• Distinguishable particle
• Identical fermions
d nNn
Qdis  N1 , N 2 , N3 ,..  N !
n 1 N n !

Q fermions  N1 , N 2 , N 3 ,..  
n 1
• Identical bosons

dn !
N n ! d n  N n !
Qbosons  N1 , N 2 , N3 ,..  
n 1
N
n
 dn  1!
N n ! d n  1!
Work out example: harmonic oscillator, infinite square well
Phys 452
Quiz 2a
Let’s consider the Carbon atom: with 6 electrons
To be distributed in the energy levels E1 and E2
What is the number of combinations Q(2, 4, 0,....) ?
A. 6!
B. 15
C. 8!
D. 70
E. 45
Phys 452
Quiz 2b
Let’s consider the Carbon atom: with 6 electrons
to be distibuted in the shells (1s)(2s)(2p)
What is the number of combinations Q(2, 2, 2, 0,....) ?
A. 6!
B. 15
C. 8!
D. 70
E. 45
Phys 452
Quantum statistical mechanics
The most probable configuration

Lagrange multipliers, using:
N   Nn
n 1

E   N n En
n 1






G  N1 , N 2 , N3 ,...  ln Q    N   N n     E   N n En 
n 1
n 1




Maximizing Q:
G
0
N n
Expressing Nn in terms of  and 
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
Significance of  and :
   En 
e
E  k BT
dimensionless
 related to the temperature
 related to chemical potential
1

k BT

  (T )
k BT
Phys 452
Quantum statistical mechanics
Calculation of  and :
Case of ideal gas (free electron 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

Nk dk
k 0

N k Ek d k
k 0
Then using

2 

E

E
 m
N  eV 
 2
3/2

1
k BT
3
m
 
E
Ve 
2
 2
We can express

2 

3/2
 (T )  kBT
3N
2
Phys 452
Quantum statistical mechanics
Case of ideal gas : distinguishable particles
  N  3  2 2
 (T )  k BT ln    ln 
  V  2  mk BT
3
E  Nk BT
2



Phys 452
Quantum statistical mechanics
Most probable occupation number:
  E    / k BT
Maxwell-Boltzmann
n
E

e


• Distinguishable particle
statistic
• Identical fermions
nE 
• Identical bosons
nE 
1
 E    / k BT
e
1
1
e E  / kBT  1
Fermi- Dirac
statistic
Bose-Einstein
statistic
Phys 452
Quiz 2c
What is the maximum possible value
for the density of occupation in case of fermions?
A.

B. 1/  e  1
C. 1
D. 0
E. undetermined
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