Editing Canonical ensemble (section)

=== Quantum mechanical ===

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| header    = Example of canonical ensemble for a quantum system consisting of one particle in a potential well.
| footer    = {{small|The particle's Hamiltonian is [[Schrödinger equation|Schrödinger]]-type, {{math|''Ĥ'' {{=}} ''U''(''x'') + ''p''<sup>2</sup>/2''m''}} (the potential {{math|''U''(''x'')}} is plotted as a red curve). Each panel shows an energy-position plot with the various stationary states, along with a side plot showing the distribution of states in energy.}}
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| image1    = Ensemble quantum 1DOF all states.png
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| caption1  = Plot of all possible states of this system. The available stationary states displayed as horizontal bars of varying darkness according to {{math|{{!}}''ψ''<sub>''i''</sub>(x){{!}}<sup>2</sup>}}.
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| image2    = Ensemble quantum 1DOF canonical.png
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| caption2  = A canonical ensemble for this system, for the temperature shown. The states are weighted exponentially in energy.
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{{details|topic=the representation of ensembles in quantum mechanics|Statistical ensemble (mathematical physics)}}

A statistical ensemble in quantum mechanics is represented by a [[density matrix]], denoted by <math>\hat \rho</math>. In basis-free notation, the canonical ensemble is the density matrix{{citation needed|date=October 2013}}
: <math>\hat \rho = \exp\left(\tfrac{1}{kT}(F - \hat H)\right),</math>
where {{math|''Ĥ''}} is the system's total energy operator ([[Hamiltonian (quantum mechanics)|Hamiltonian]]), and {{math|exp()}} is the [[matrix exponential]] operator. The free energy {{math|''F''}} is determined by the probability normalization condition that the density matrix has a [[trace (linear algebra)|trace]] of one, <math>\operatorname{Tr} \hat \rho=1</math>:
: <math>e^{-\frac{F}{k T}} = \operatorname{Tr} \exp\left(-\tfrac{1}{kT} \hat H\right).</math>

The canonical ensemble can alternatively be written in a simple form using [[bra–ket notation]], if the system's [[stationary state|energy eigenstates]] and energy eigenvalues are known. Given a complete basis of energy eigenstates {{math|{{!}}''ψ''<sub>''i''</sub>⟩}}, indexed by {{math|''i''}}, the canonical ensemble is:
: <math>\hat \rho = \sum_i e^{\frac{F - E_i}{k T}} |\psi_i\rangle \langle \psi_i | </math>
: <math>e^{-\frac{F}{k T}} = \sum_i e^{\frac{- E_i}{k T}}.</math>
where the {{math|''E''<sub>''i''</sub>}} are the energy eigenvalues determined by {{math|''Ĥ''{{!}}''ψ''<sub>''i''</sub>⟩ {{=}} ''E''<sub>''i''</sub>{{!}}''ψ''<sub>''i''</sub>⟩}}. In other words, a set of microstates in quantum mechanics is given by a complete set of stationary states. The density matrix is diagonal in this basis, with the diagonal entries each directly giving a probability.