Hamiltonian operator

Summary

The Hamiltonian of a quantum system is the observable of its total energy. It is a self-adjoint operator $H$ that also drives the time evolution. By Stone's theorem, $U(t) = e^{-itH/\hbar}$ is a group of unitary operators, and state vectors evolve by $\psi(t) = U(t)\psi(0)$. For a particle of mass $m$ in a potential $V$, it is the Schrödinger operator $-\frac{\hbar^2}{2m}\Delta + V$, the sum of kinetic and potential energy. Its spectrum gives the possible energies, and its eigenvectors are the stationary states. It is the quantum counterpart of the Hamiltonian function of Hamiltonian mechanics.