Knowlpedia

Quantum partition function

For a finite quantum system with Hamiltonian H, the canonical partition function is Tr(exp(-beta H)).

Let HH be a on a finite-dimensional Hilbert space and let β>0\beta>0. The quantum partition function is

Z(β)=Tr⁡(e−βH).Z(\beta)=\operatorname{Tr}(e^{-\beta H}).

If the eigenvalues of HH are EnE_n, counted with multiplicity, then Z(β)=∑ne−βEnZ(\beta)=\sum_ne^{-\beta E_n}. It is finite and strictly positive.

Derived quantities

The partition function normalizes the and determines the canonical free energy and mean energy:

ρβ=e−βHZ(β),F(β)=−β−1log⁡Z(β),⟨H⟩β=−ddβlog⁡Z(β).\rho_\beta=\frac{e^{-\beta H}}{Z(\beta)}, \qquad F(\beta)=-\beta^{-1}\log Z(\beta), \qquad \langle H\rangle_\beta=-\frac{d}{d\beta}\log Z(\beta).

Moreover,

d2dβ2log⁡Z(β)=⟨H2⟩β−⟨H⟩β2≥0.\frac{d^2}{d\beta^2}\log Z(\beta) =\langle H^2\rangle_\beta-\langle H\rangle_\beta^2\ge0.
Independent systems

If H=HA⊗HB\mathcal H=\mathcal H_A\otimes\mathcal H_B and H=HA⊗I+I⊗HBH=H_A\otimes I+I\otimes H_B, then

Z(β)=ZA(β)ZB(β).Z(\beta)=Z_A(\beta)Z_B(\beta).