Quantum Hamiltonian - Knowlpedia

A quantum Hamiltonian is a densely defined self-adjoint operator $H$ on a nonzero complex Hilbert space $\mathcal H$ , representing the system's energy. When $H$ is bounded and belongs to an observable algebra, it is an observable; in finite dimension it has a spectral decomposition

H=\sum_n E_nP_n,

where the real numbers $E_n$ are energy levels and the $P_n$ are their spectral projections.

Dynamics

For a time-independent Hamiltonian, Schrödinger evolution is generated by

U_t=e^{-itH/\hbar}.

States and observables evolve by $\rho(t)=U_t\rho(0)U_t^*$ and $A(t)=U_t^*A(0)U_t$ . An observable commuting with $H$ is conserved.

Equilibrium

When $e^{-\beta H}$ is trace-class for $\beta>0$ , $H$ determines the quantum partition function and quantum Gibbs state:

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

Replacing $H$ by $H+cI$ changes $Z$ by the factor $e^{-\beta c}$ but leaves $\rho_\beta$ unchanged whenever the Gibbs state exists. In infinite dimension, trace-class and domain hypotheses cannot be omitted.