This field state is the equilibrium state for a field coupled to a reservoir at temperature 
. This state is commonly discussed in statistical and thermal physics courses because it is used to derive the Planck radiation law.
The distribution function which describes a thermal state's statistics is similar to the Maxwell--Boltzmann distribution
is the Kronecker delta; thus, the only non--zero elements of the density matrix are along the diagonal.[14] The properties of the Fock and thermal states are quite similar because neither has off--diagonal terms. The principle difference between these two states is the photon number and its variance. The average photon number of a thermal state is a well known result which is often termed the Planck distribution function
and is derived in most statistical physics texts. The variance of a thermal state is not equal to zero, unlike the Fock state. This quantity is not commonly derived. In order to find the 
, one must know two expectation values: <N> and 
. The former is eq:q15. The latter is easily uncovered with eq:q14 and eq:q16b
It would be helpful if eq:q16 could be rewritten without a sum. Let u be defined as a dummy variable which equals 
. Making this substitution into eq:q16
and observe that
Although eq:q17 may seem to complicate finding 
, it actually simplifies eq:q16 significantly. Since the second derivative is a linear operator and u is not involved in the sum, the second derivative may be pulled outside the sum
Notice that 
and 
since u is greater than zero. This series is geometric and will converge to the value 
; Ergo
The variance in the photon number can now be found
Besides this average and variance of the photon number, a thermal state is very much like a Fock state; however, these photon number properties are quite unique and interesting---especially when contrasted with the remaining two field states. The thermal state's properties are summarized in the following table
