bei48482_FM

For bosons the wave function becomes

B [a(1)a(2)a(1)a(2)] a(1)a(2) 2 a(1)a(2) (9.23)

to give a probability density of

Bosons *BB 2 *a(1)*a(2)a(1)a(2) 2 *MM (9.24)

Thus the probability that both bosons be in the same state is twice what it is for

distinguishable particles!

For fermions the wave function becomes

Fermions F [a(1)a(2)a(1)a(2)] 0 (9.25)

It is impossible for both particles to be in the same state, which is a statement of the

exclusion principle.

These results can be generalized to apply to systems of many particles:

1 In a system of bosons, the presence of a particle in a certain quantum state increases

the probability that other particles are to be found in the same state;

2 In a system of fermions, the presence of a particle in a certain state preventsany

other particles from being in that state.

Bose-Einstein and Fermi-Dirac Distribution Functions

The probability f( ) that a boson occupies a state of energy turns out to be

fBE() (9.26)

and the probability for a fermion turns out to be

fFD() (9.27)

1



eekT 1

Fermi-Dirac

distribution function

1



eekT 1

Bose-Einstein

distribution function

1



 2 

2



 2 

1



 2 

Statistical Mechanics 307

T

he Indian physicist S. N. Bose in 1924 derived Planck’s radiation formula on the basis of

the quantum theory of light with indistinguishable photons whose number is not conserved.

His paper was rejected by a leading British journal. He then sent it to Einstein, who translated

the paper into German and submitted it to a German journal where it was published. Because

Einstein extended Bose’s treatment to material particles whose number is conserved, both names

are attached to Eq. 9.26. Two years later Enrico Fermi and Paul Dirac independently realized

that Pauli’s exclusion principle would lead to different statistics for electrons, and so Eq. 9.27 is

named after them.

Names of the Functions

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