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Statistical mechanics, structure of matter Definition

Quantum statistical mechanics

Builds on Statistical ensembleHilbert spaceSelf-adjoint operatorTrace class operator

Definition

Quantum statistical mechanics studies quantum systems through ensembles of quantum states. A system is given by a complex separable Hilbert space ℋ\mathcal{H} and a self-adjoint Hamiltonian HH on ℋ\mathcal{H}.

  1. A density operator is a positive trace-class operator ρ\rho on ℋ\mathcal{H} with tr⁡ρ=1\tr \rho = 1; it is the quantum statistical ensemble.
  2. The expected value of a bounded self-adjoint observable AA in the state ρ\rho is tr⁡(ρA)\tr(\rho A).
  3. If e−βHe^{-\beta H} is trace class for some β>0\beta > 0, the canonical (Gibbs) state is ρβ=e−βH/Z(β)\rho_\beta = e^{-\beta H} / Z(\beta), with partition function Z(β)=tr⁡e−βHZ(\beta) = \tr e^{-\beta H}.

The time evolution of a density operator is ρ(t)=e−itH/ℏρ eitH/ℏ\rho(t) = e^{-itH/\hbar} \rho \, e^{itH/\hbar}.

Remarks

Example: a spin 12\tfrac{1}{2} in a magnetic field has ℋ=ℂ2\mathcal{H} = \C^2 and H=diag⁡(−ε,ε)H = \operatorname{diag}(-\varepsilon, \varepsilon) with ε>0\varepsilon > 0. Then Z(β)=2cosh⁡(βε)Z(\beta) = 2 \cosh(\beta \varepsilon) and ρβ=diag⁡(eβε,e−βε)/Z(β)\rho_\beta = \operatorname{diag}(e^{\beta \varepsilon}, e^{-\beta \varepsilon}) / Z(\beta), a positive matrix of trace 11. Non-example: the operator diag⁡(1,−12,12)\operatorname{diag}(1, -\tfrac{1}{2}, \tfrac{1}{2}) on ℂ3\C^3 has trace 11 but is not positive, so it is not a density operator. For the free particle in L2(ℝ3)L^2(\R^3), e−βHe^{-\beta H} is not trace class; one first confines the particle to a box. For infinite systems, Gibbs states are replaced by KMS states on a C*-algebra of observables.

Reading path

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Hilbert space, concept, no evidence levelHilbert spaceconceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptStatistical ensemble, concept, no evidence levelStatistical ensembleconceptTrace class operator, concept, no evidence levelTrace class operatorconceptQuantum statistical mechanics, definition, no evidence levelQuantum statistical m…definition
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  1. Hilbert space concept
  2. Self-adjoint operator concept
  3. Statistical ensemble concept
  4. Trace class operator concept
  5. Quantum statistical mechanics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_statistical_mechanics
About this page
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MSC82B10, 82C10, 46L60
DefinesQuantum statistical mechanics
UsesStatistical ensemble, Hilbert space, Self-adjoint operator, Trace class operator
Changed3 November 2026

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