130_notes.dvi

33.6 Photon States

It is now obvious that the integernk,αis thenumber of photons in the volumewith wave

number~kand polarization ˆǫ(α). It is called theoccupation numberfor the state designated by
wave number~kand polarization ˆǫ(α). We can represent the state of the entire volume by giving the
number of photons of each type (and some phases). The state vector for the volume is given by the
direct product of the states for each type of photon.

|nk 1 ,α 1 ,nk 2 ,α 2 ,...,nki,αi,...〉=|nk 1 ,α 1 〉|nk 2 ,α 2 〉...,|nki,αi〉...

The ground state for a particular oscillator cannot be lowered. Thestate in whichall the oscillators
are in the ground state is called the vacuum stateand can be written simply as| 0 〉. We can
generate any state we want by applying raising operators to the vacuum state.

|nk 1 ,α 1 ,nk 2 ,α 2 ,...,nki,αi,...〉=

∏

i

(a†ki,αi)nki,αi

√

nki,αi!

| 0 〉

The factorial on the bottom cancels all the

√

n+ 1 we get from the raising operators.

Any multi-photon state we construct isautomatically symmetric under the interchange of
pairs of photons. For example if we want to raise two photons out of the vacuum, we apply two
raising operators. Since [a†k,α,a†k′,α′] = 0, interchanging the photons gives the same state.

a†k,αa†k′,α′| 0 〉=a†k′,α′a†k,α| 0 〉

So the fact that thecreation operators commute dictates that photon states aresymmetric
under interchange.

33.7 Fermion Operators

At this point, we can hypothesize that theoperators that create fermion states do not com-
mute. In fact, if we assume that theoperators creating fermion states anti-commute(as do
the Pauli matrices), then we can show that fermion states are antisymmetric under interchange. As-
sumeb†randbrare the creation and annihilation operators for fermions and that they anti-commute.

{b†r,b†r′}= 0

Thestates are then antisymmetric under interchangeof pairs of fermions.

b†rb†r′| 0 〉=−b†r′b†r| 0 〉

Its not hard to show that theoccupation number for fermion states is either zero or one.