Statistical Physics 2

1

Topics

 Recap  Quantum Statistics  The Photon Gas  Summary 2

Recap

In classical physics, the number of particles with energy between

E

and

E

+

dE

, at temperature

T

, is given by ( 

f B d E



A

( )

e



dE

where

g(E)

is the

density of states

Boltzmann distribution describes how energy is distributed in an assembly of . The

identical

, but

distinguishable

particles.

3

Quantum Statistics

In quantum physics, particles are described by

wave functions

. But when these overlap, identical particles become

indistinguishable

and we cannot use the Boltzmann distribution.

We therefore need new energy distribution functions.

In fact, we need two: one for particles that behave like photons and one for particles that behave like electrons.

4

Quantum Statistics

In 1924, the Indian physicist Bose derived the energy distribution function for indistinguishable mass-less particles that do not obey the

Pauli exclusion principle

.

The result was extended by Einstein to massive particles and is called the

Bose-Einstein (BE) distribution

f BE

 

e e

1



1

The factor study

e

 depends on the system under 5

Quantum Statistics

The corresponding result for particles that obey the Pauli exclusion principle is called the

Fermi-Dirac (FD) distribution

f FD

 

e e

1



1

Particles, such as photons, that obey the Bose-Einstein distribution are called such as electrons, are called

fermions bosons

.

. Those that obey the Fermi-Dirac distribution, 6

Quantum Statistics

The Boltzmann distribution can be written in the form

f B

 

e e

1

Apart from the

±1

in the denominator, this is identical to the BE and FD distributions.

The Boltzmann distribution is valid when

e



e E/kT >> 1

. This can occur because of low particle densities and energies >> kT 7

Quantum Statistics

Comparison of Distribution Functions

For a system of two identical particles,

1 2

, one in state

n

and the other in state there are two possible configurations, as shown below

m

and ,

1 st configuration 1 2 2 nd configuration



2

n



1

m

8

Quantum Statistics

Comparison of Distribution Functions

The first configuration

1 2

 

n m

is described by the wave function 

nm

(1, 2)

 

n

(1)



m

(2)

9

Quantum Statistics

Comparison of Distribution Functions

The second configuration

2 1

 

n m

is described by the wave function 

nm

(2,1)

 

n

(2)



m

(1)

10

Quantum Statistics

Comparison of Distribution Functions

If the particles were

distinguishable

, then the two wave functions 

nm

(1, 2)

 

n

(1)



m

(2)



nm

(2,1)

 

n

(2)



m

(1)

would be the appropriate ones to describe the system of two (non-interacting) particles 11

Quantum Statistics

Comparison of Distribution Functions

But since in general identical particles are not

distinguishable

, we must describe them using the symmetric or anti-symmetric combinations 

S

 

A



1 2

 

n

1 2

 

n



m



m

 

n

 

n



m



m

  12

Quantum Statistics

Comparison of Distribution Functions

The

symmetric

wave functions describe

bosons

1.

while the

fermions

. Using these wave functions one can deduce the following: 2.

A chance of finding other identical bosons in the same state A

anti-symmetric boson fermion

the same state ones describe in a quantum state

increases

in a quantum state

prevents

the any other identical fermions from occupying 13

Quantum Statistics

Comparison of Distribution Functions

The probability that a particle occupies a given energy state satisfies the inequality

f FD



f B



f BE

All three functions become the same when

E >> kT

14

Quantum Statistics

Density of States

The number of particles with energy in the range

E

to

E+dE

is given by  and the total number of particles N is given by

N

 0   Each function

f(E)

associated with a different density of states

g(E)

is 15

Quantum Statistics

Density of States

The number of

states E

to

E

+

dE

with energy in the range can be shown to be given by 

W d

 ,

h

3

d

 

V

4  2

p dp

where

W V d

 is called the

phase space volume

, is the degeneracy of each energy level, is the volume of the system and momentum of the particle

p

is the 16

The Photon Gas

Density of States for Photons

For photons,

E = pc

, and

W =

2

. (A photon has two polarization states). Therefore,  8 

VE

2 (

hc

) 3

dE

Extra Credit

: Derive this formula due date: Monday after Spring Break 17

The Photon Gas

Distribution Function for Photons

The number of photons with energy between

E

and

E

+

dE

is given by  

BE

8 

VE

2 (

hc

) 3  

e

For photons 

= 0

.

1  1 

dE

18

The Photon Gas

Photon Density of the Universe

The photon density

n(E) dE / V

 is just the integral of over all possible photon energies    0  /   0  8  2

E dE

(

hc

3 ) (

e

 1) This yields approximately   8    3 (2.40) The photon temperature of the universe is

T= 2.7 K

, implying  =

4 x 10 8 photons/m 3

19

The Photon Gas

Black Body Spectrum

If we multiply the photon density

n(E)dE

/

V E

, we get the

energy density u(E)dE

by  8  (

hc

3 ) (

e E

3  1)

dE

This is the distribution first obtained by Max Planck in 1900 in his “act of desperation” 20

Summary

    Particles come in two classes:

bosons fermions

.

and A boson in a state enhances the chance to find other identical bosons in that state. A fermion in a state prevents other identical fermions from occupying the state. When identical particles become distinguishable, typically, when they are well separated and when

E >> kT

Boltzmann distribution , the B-E and F-D distributions can be approximated with the 21