Editing Ensemble (mathematical physics) (section)

=== Quantum mechanical ===
{{main|Density matrix}}
A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a [[density matrix]], denoted by <math>\hat\rho</math>. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable {{mvar|X}} in quantum mechanics can be written as an operator, <math>\hat X</math>. The expectation value of this operator on the statistical ensemble <math>\rho</math> is given by the following [[trace (linear algebra)|trace]]:
<math display="block">\langle X \rangle = \operatorname{Tr}(\hat X \rho).</math>
This can be used to evaluate averages (operator <math>\hat X</math>), [[variance]]s (using operator <math>\hat X^2</math>), [[covariance]]s (using operator <math>\hat X \hat Y</math>), etc. The density matrix must always have a trace of 1: <math>\operatorname{Tr}{\hat\rho} = 1</math> (this essentially is the condition that the probabilities must add up to one).

In general, the ensemble evolves over time according to the [[von Neumann equation]].

Equilibrium ensembles (those that do not evolve over time, <math>d\hat\rho / dt = 0</math>) can be written solely as a function of conserved variables. For example, the [[microcanonical ensemble]] and [[canonical ensemble]] are strictly functions of the total energy, which is measured by the total energy operator <math>\hat H</math> (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator <math>\hat N</math>. Such equilibrium ensembles are a [[diagonal matrix]] in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In [[bra–ket notation]], the density matrix is
<math display="block">\hat\rho = \sum_i P_i |\psi_i\rangle \langle\psi_i|,</math>
where the {{math|{{ket|''ψ''<sub>''i''</sub>}}}}, indexed by {{mvar|i}}, are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)