Quantum Hamiltonian

A quantum Hamiltonian is a self-adjoint operator $H$ acting on the system Hilbert space $\mathcal{H}$ (or, more generally, a self-adjoint element of the observable algebra ).

In finite dimension, self-adjointness means $H=H^\dagger$, so $H$ admits a spectral decomposition

where $E_n \in \mathbb{R}$ are eigenvalues (energies) and $P_n$ are the corresponding orthogonal projections. See spectrum of a self-adjoint operator for details in finite dimension.

The Hamiltonian generates unitary time evolution:

for states $\rho$ and observables $A$.

In equilibrium statistical mechanics, for inverse temperature $\beta$ (see inverse temperature beta ), one defines the quantum partition function

and the quantum Gibbs state

Physical interpretation

• $H$ is the energy observable: measuring energy yields an eigenvalue $E_n$ with Born-rule probabilities determined by the state.
• $H$ sets the system’s dynamics (closed system evolution is unitary and energy-conserving when $H$ is time-independent).
• Through $e^{-\beta H}$ it governs thermal weighting: low-energy states are favored when $\beta>0$.

Key properties

• Real spectrum: Self-adjointness implies energies $E_n$ are real and eigenprojections resolve the identity.
• Energy zero-point irrelevance: Replacing $H$ by $H+c\,I$ shifts all energies by $c$ but leaves $\rho_\beta$ unchanged, since $e^{-\beta(H+cI)} = e^{-\beta c} e^{-\beta H}$ and the scalar cancels in normalization.
• Degeneracy and symmetry: If distinct microstates share the same $E_n$, the corresponding eigenspace is degenerate; symmetries often explain degeneracies.
• Conservation laws: If an observable $A$ satisfies $[A,H]=0$, then $A$ is conserved under the Heisenberg evolution $A(t)$.