Representations of the Electromagnetic Field

Abstract

A quantum mechanical state is completely described by its density matrix ρ. The density matrix ρ can be expanded in different basises {|ψ ρ = i,j D ij |ψ i ψ j for a discrete set of basis states (158) The c-number matrix D ij = i | ρ ψ j is a representation of the density operator. One example is the number state or Fock state representation of the density ma-trix. ρ nm = n,m P nm |n (159) The diagonal elements in the Fock representation give the probability to find exactly n photons in the field! A trivial example is the density matrix of a Fock State |k ρ = |k and ρ nm = δ nk δ km (160) Another example is the density matrix of a thermal state of a free single mode field: ρ = exp[− + a/k b T ] T r{exp[− + a/k b T ]} (161) In the Fock representation the matrix is: ρ = n exp[− b T ][1 − exp(− b T)] |n (162) = n n (1 + n+1 |n (163) with = T r(a + aρ) = [exp(b T) − 1] −1 (164) 37