Hamiltonian Operator

linear-algebra

Definition

Hamiltonian Operator

The Hamiltonian operator $H$ 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, $H$ admits a spectral representation of the form

$$H=\sum _{i\ge 0}{E}_i{P}_i$$

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

Evolution

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

$$U(t)=e^{-\frac{i}{\hslash }tH}$$

and the state at time $t$ is given by

$$|\psi (t)⟩=U(t)|\psi (0)⟩.$$

Since $H$ is hermitian, $U(t)$ is unitary.