Hamiltonian Operator

The Hamiltonian operator is the quantum-mechanical observable corresponding to the total energy of a system, and the generator of unitary time evolution in the Schrödinger equation. In the passage from classical to quantum mechanics, the classical Hamiltonian function on phase space becomes an operator on a Hilbert space — a transformation that replaces classical Poisson brackets with operator commutators and continuous trajectories with unitary evolution. The Hamiltonian operator is the bridge between the variational principles of classical mechanics and the linear algebra of quantum states.

The operator's spectral decomposition determines the allowed energy levels of a system. The fact that the Hamiltonian typically does not commute with other observables is the quantum echo of the non-commutativity that Hamilton first discovered in his quaternions: order matters, in algebra as in physics.