Editing Microcanonical ensemble (section)

=== Quantum mechanical ===

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| header    = Example of microcanonical 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|1=''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|{{abs|''ψ''<sub>''i''</sub>(x)}}<sup>2</sup>}}.
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| image2    = Ensemble quantum 1DOF microcanonical.png
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| caption2  = An ensemble containing only those states within a narrow interval of energy. As the energy width is taken to zero, a microcanonical ensemble is obtained (provided the interval contains at least one state).
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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>. The microcanonical ensemble can be written using [[bra–ket notation]], in terms of the system's [[stationary state|energy eigenstates]] and energy eigenvalues. Given a complete basis of energy eigenstates {{math|{{ket|''ψ''<sub>''i''</sub>}}}}, indexed by {{math|''i''}}, the microcanonical ensemble is{{citation needed|date=November 2013}}
<math display="block">\hat\rho = \frac{1}{W} \sum_i f{\left(\tfrac{H_i - E}{\omega}\right)} \left|\psi_i\right\rangle \left\langle \psi_i \right|,</math>
where the {{math|''H''<sub>''i''</sub>}} are the energy eigenvalues determined by <math>\hat H |\psi_i\rangle = H_i |\psi_i\rangle</math> (here {{math|''Ĥ''}} is the system's total energy operator, i. e., [[Hamiltonian (quantum mechanics)|Hamiltonian operator]]). The value of {{math|''W''}} is determined by demanding that <math>\hat\rho</math> is a normalized density matrix, and so
<math display="block">W = \sum_i f{\left(\tfrac{H_i - E}{\omega}\right)}.</math>
The state volume function (used to calculate entropy) is given by
<math display="block">v(E) = \sum_{H_i < E} 1.</math>

The microcanonical ensemble is defined by taking the limit of the density matrix as the energy width goes to zero, however a problematic situation occurs once the energy width becomes smaller than the spacing between energy levels. For very small energy width, the ensemble does not exist at all for most values of {{math|''E''}}, since no states fall within the range. When the ensemble does exist, it typically only contains one ([[Kramers theorem|or two]]) states, since in a complex system the energy levels are only ever equal by accident (see [[random matrix theory]] for more discussion on this point). Moreover, the state-volume function also increases only in discrete increments, and so its derivative is only ever infinite or zero, making it difficult to define the density of states. This problem can be solved by not taking the energy range completely to zero and smoothing the state-volume function, however this makes the definition of the ensemble more complicated, since it becomes then necessary to specify the energy range in addition to other variables (together, an {{math|''NVEω''}} ensemble).