## Lec37.pdf-Fermions and Bosons Symmetry is ashowing page 1-5 out of 5

##### Page 1

Fermions and Bosons

Symmetry is a constant theme in physics. As a fundamental

symmetry result of QM, particles divide into two classes

according to their

spin

:

http://

en.wikipedia.org

/wiki/

Identical_particles

fermions

:

half-integer spin

e.g., electrons, protons, neutrons,

3

He

bosons

:

integer spin

0,1,2,3...

e.g., photons,

4

He

The two classes have profoundly different quantum rules for

the states they may be in, for non-interacting particles

1)

any integral number of bosons of the same species

(0,1,2...) may be described by the same quantum state

2)

for fermions of the same species, at most one particle,

i.e., (0, 1), may be described by a given state

1

2

,

3

2

,

5

2

...

http://

en.wikipedia.org

/wiki/

Identical_particles#Symmetrical_and_antisymmetrical_states

##### Page 2

Fermi-Dirac Distribution

Consider a ‘system’ of one orbital or wavefunction

Ψ

It may contain zero fermions, or one fermion.

Take the

energy as 0 for zero fermions, and

ε

for one fermion.

then

so we can find the thermal average state occupation for a

system in diffusive and thermal contact with the reservoir:

or conventionally, for the average occupancy:

the

Fermi-Dirac distribution function

Z

≡

exp (

N

μ

−

ε

s

(

N

)

)/

τ

⎡

⎣

⎤

⎦

s

∑

N

=

0

1

∑

λ

≡

exp(

μ

/

τ

)

'activity'

N

(

ε

)

=

N

(

ε

)

P

(

ε

)

∑

=

λ

exp(

−

ε

/

τ

)

1

+

λ

exp(

−

ε

/

τ

)

=

1

λ

−

1

exp(

ε

/

τ

)

+

1

f

(

ε

)

≡

N

(

ε

)

≡

1

exp(

ε

−

μ

(

)

/

τ

)

+

1

=

1

+

exp (

μ

−

ε

)/

τ

[

]

=

1

+

λ

exp(

−

ε

/

τ

)

##### Page 3

Bose-Einstein Distribution

Now, not just 1 but some arbitrary number

N

of bosons can

have the same wavefunction

Ψ

(or we say we ‘populate’ the

orbital with

N

photons). The energy of each added boson is

ε

. In diffusive and thermal equilibrium with a reservoir

thus

or conventionally, for the average occupancy:

the

Bose-Einstein distribution function

using

x

N

N

=

0

∞

∑

=

1

1

−

x

for

x

<

1, always true here

Z

≡

exp (

N

μ

−

N

ε

)/

τ

[

]

N

=

0

∞

∑

=

exp (

μ

−

ε

)/

τ

[

]

N

N

=

0

∞

∑

=

1

1

−

exp

μ

−

ε

(

)

/

τ

{

}

N

(

ε

)

≡

N

(

ε

)

P

(

ε

)

∑

=

N

exp (

N

(

μ

−

ε

)/

τ

[

]

N

=

0

∞

∑

Z

=

1

exp(

ε

−

μ

(

)

/

τ

)

−

1

f

(

ε

)

≡

N

(

ε

)

=

1

exp(

ε

−

μ

(

)

/

τ

)

−

1

##### Page 4

Fermi-Dirac Distribution

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