Hamiltonian Operator

linear-algebra

Definition

Hamiltonian Operator

The Hamiltonian operator is the hermitian operator that represents the total energy of a quantum-mechanical system.

In a closed system, it generates the time evolution of the quantum state via the Schrödinger equation.

Spectral Representation

As a hermitian operator, admits a spectral representation of the form

where are the real-valued energy eigenvalues and are the corresponding orthogonal projections. The eigenvectors of are called energy eigenstates.

Evolution

If is time-independent, then the time evolution operator is

and the state at time is given by

Since is hermitian, is unitary.