Quantum system (statistical mechanics)

A quantum system for statistical mechanics: a Hilbert space, an algebra of observables, and a Hamiltonian, with states described by density operators.

Quantum system (statistical mechanics)

In (finite-dimensional) quantum statistical mechanics, a quantum system can be modeled as a triple $(\mathcal{H}, \mathcal{A}, H)$ where:

$\mathcal{H}$ is a finite-dimensional complex Hilbert space .
$\mathcal{A}$ is an observable algebra , typically a unital $*$ -subalgebra of the algebra of bounded operators on $\mathcal{H}$ .
$H \in \mathcal{A}$ is the Hamiltonian , assumed self-adjoint (so that energies are real).

A state of the system is represented by a density-operator state $\rho$ on $\mathcal{H}$ (equivalently, a positive normalized linear functional on $\mathcal{A}$ ). The expectation value of an observable $A \in \mathcal{A}$ in the state $\rho$ is

\langle A\rangle_\rho = \operatorname{Tr}(\rho A),

using the trace . This is the quantum expectation value .

Time evolution for an isolated system is generated by $H$ via a unitary family $U_t$ ,

U_t = e^{- \frac{i}{\hbar} t H}, \qquad \rho(t) = U_t\,\rho(0)\,U_t^\dagger, \qquad A(t) = U_t^\dagger A(0) U_t.

Physical interpretation

$\mathcal{H}$ encodes the system’s kinematic degrees of freedom (what pure states can exist).
$\mathcal{A}$ encodes which quantities are regarded as measurable observables and how they compose.
$H$ specifies the energy observable and the dynamics, and it determines equilibrium states such as the quantum Gibbs state .

Key properties

Noncommutativity: Typically $\mathcal{A}$ is noncommutative; this expresses incompatibility of observables (order matters for products).
States form a convex set: If $\rho_1,\rho_2$ are states and $0\le \lambda \le 1$ , then $\rho=\lambda\rho_1+(1-\lambda)\rho_2$ is also a state. Extreme points correspond to quantum microstates (pure states).
Subsystems and reduction: If the system is part of a larger one, reduced states are obtained by the partial trace .
Equilibrium and thermodynamics: Given $H$ and an inverse temperature parameter $\beta$ , the canonical equilibrium state and normalization are governed by the quantum partition function .