Quantum partition function

Let $\mathcal H$ be a finite-dimensional Hilbert space and let $H$ be the Hamiltonian (quantum Hamiltonian ), a self-adjoint operator on $\mathcal H$.

For inverse temperature $\beta>0$ (inverse temperature β ), the quantum (canonical) partition function is

where $\operatorname{Tr}$ is the operator trace (trace ).

If $H$ has eigenvalues $\{E_n\}$ (counting multiplicity), then

The partition function normalizes the quantum Gibbs state via $\rho_\beta = e^{-\beta H}/Z(\beta)$.

Physical interpretation

$Z(\beta)$ is the normalization constant that makes thermal probabilities sum to one, and it is the generator of equilibrium thermodynamics for the canonical ensemble. It is the quantum analog of the classical canonical partition function .

Through $Z(\beta)$ one obtains the equilibrium Helmholtz free energy (Helmholtz free energy ), internal energy, entropy, and response coefficients.

Key properties

1. Positivity and finiteness (finite systems).
   In finite dimension, $e^{-\beta H}$ is positive and trace-class automatically, so $Z(\beta)\in(0,\infty)$ for every $\beta>0$.

2. Energy-shift covariance.
   If $H' = H + cI$ for a scalar $c$, then

   $$Z_{H'}(\beta)\;=\;e^{-\beta c} Z_H(\beta).$$

   Consequently, the Gibbs state itself is invariant under adding a constant to energy (since the factor cancels in $\rho_\beta$).

3. Free energy from $\log Z$.
   The equilibrium free energy is

   $$F(\beta)\;=\;-\beta^{-1}\log Z(\beta),$$

   matching statistical free energy .

4. Thermodynamic derivatives (finite-dimensional identities).
   Define the equilibrium mean energy

   $$U(\beta)\;=\;\operatorname{Tr}(\rho_\beta H), \qquad \rho_\beta=\frac{e^{-\beta H}}{Z(\beta)}.$$

   Then

   $$U(\beta)\;=\;-\frac{d}{d\beta}\log Z(\beta).$$

   More generally, if $H(\lambda)=H_0+\lambda V$, then

   $$\frac{\partial}{\partial\lambda}\log Z(\beta,\lambda)\;=\;-\beta\,\langle V\rangle_{\beta,\lambda},$$

   where the expectation is the quantum expectation value in the Gibbs state for $H(\lambda)$.

5. Factorization for independent subsystems.
   If $\mathcal H=\mathcal H_A\otimes\mathcal H_B$ and

   $$H \;=\; H_A\otimes I_B \;+\; I_A\otimes H_B,$$

   then

   $$Z(\beta)\;=\;Z_A(\beta)\,Z_B(\beta).$$

   This is the canonical statement that noninteracting systems have additive free energy.

6. Convexity and stability.
   The map $\beta\mapsto \log Z(\beta)$ is convex. Equivalently, $U(\beta)$ is decreasing in $\beta$ and the heat capacity is nonnegative in the canonical ensemble (in finite dimension).

7. Noncommuting Hamiltonian bounds.
   When splitting a Hamiltonian into noncommuting parts, trace inequalities such as the Golden–Thompson inequality yield useful estimates of $Z(\beta)$ and hence of $F(\beta)$.