Quantum microstate

In quantum statistical mechanics, a quantum microstate is typically identified with a pure state of the system (see pure state ).

Concretely, in finite dimensions a microstate can be represented by either of the equivalent data:

1. A normalized vector $|\psi\rangle \in \mathcal{H}$ with $\langle\psi|\psi\rangle = 1$, where vectors differing by a global phase represent the same physical state: $|\psi\rangle \sim e^{i\theta}|\psi\rangle$.
2. A rank-one projection (density operator) $\rho_\psi = |\psi\rangle\langle\psi|.$

Given an observable $A$ in the observable algebra , the expected outcome in the microstate $|\psi\rangle$ is

This matches the general density-operator formula for expectations.

A particularly important class of microstates in statistical mechanics are energy eigenstates: if $H$ is the Hamiltonian and $H|\phi_n\rangle = E_n|\phi_n\rangle$, then $|\phi_n\rangle$ is a microstate with definite energy $E_n$ (see eigenvector ).

Physical interpretation

• A microstate is the most refined description allowed by quantum theory for an isolated system: it specifies all measurement statistics but generally not sharp values for all observables.
• Superpositions of energy eigenstates are also microstates; “being in a microstate” does not mean having definite classical phase-space coordinates.

Key properties

• Extremality: Microstates are extreme points of the convex set of density-operator states ; they cannot be expressed as nontrivial convex mixtures of other states.
• Zero von Neumann entropy: For $\rho_\psi = |\psi\rangle\langle\psi|$, the von Neumann entropy satisfies $S(\rho_\psi)=0$.
• Projector condition: $\rho_\psi$ is pure iff $\rho_\psi^2=\rho_\psi$ (equivalently, $\operatorname{Tr}(\rho_\psi^2)=1$).
• Basis dependence of “definite values”: A microstate has definite value for an observable only when it lies in an eigenstate of that observable.