Editing Negative temperature (section)

=== Lasers ===
This phenomenon can also be observed in many [[laser|lasing]] systems, wherein a large fraction of the system's [[atom]]s (for chemical and gas lasers) or [[electron]]s (in [[semiconductor]] lasers) are in excited states. This is referred to as a [[population inversion]].

The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for a single mode of a luminescent radiation field at frequency {{mvar|ν}} is
:<math>H = (h\nu - \mu)a^\dagger a.</math>

The density operator in the [[grand canonical ensemble]] is
:<math>\rho = \frac{e^{-\beta H}}{\operatorname{Tr}\left(e^{-\beta H}\right)}.</math>

For the system to have a ground state, the trace to converge, and the density operator to be generally meaningful, {{math|''βH''}} must be positive semidefinite. So if {{math|''hν'' < ''μ''}}, and {{mvar|H}} is negative semidefinite, then {{mvar|β}} must itself be negative, implying a negative temperature.<ref>{{cite journal
 |last1=Hsu |first1=W.
 |last2=Barakat |first2=R.
 |year=1992
 |title=Statistics and thermodynamics of luminescent radiation
 |journal=[[Physical Review B]]
 |volume=46 |issue=11 |pages=6760–6767
 |bibcode=1992PhRvB..46.6760H
 |doi=10.1103/PhysRevB.46.6760
 |pmid=10002377
}}</ref>