Hamiltonoperatorn

The Hamilton operator, commonly called the Hamiltonian, is a linear, Hermitian operator that represents the total energy of a quantum system. It acts on states in a Hilbert space and governs time evolution via the Schrödinger equation. When energy is measured, possible outcomes are the eigenvalues E_n of H, with eigenvectors |n⟩ satisfying H|n⟩ = E_n|n⟩.

Construction: In non-relativistic single-particle quantum mechanics, H often has the form H = T + V with T = p^2/(2m) and p = -iħ∇. In cases with position-dependent mass or curved spaces, operator ordering must be chosen carefully to preserve Hermiticity and physical behavior.

Time evolution and spectrum: For a state ψ, the Schrödinger equation iħ ∂ψ/∂t = H ψ holds. If H is time-independent, solutions separate as ψ(t) = e^{-iHt/ħ} ψ(0), with energy eigenstates acquiring a phase e^{-iE_n t/ħ}. The spectrum of H can be discrete, continuous, or mixed; eigenstates form a basis for expanding arbitrary states.

Role and extensions: The Hamiltonian generates time translations via U(t) = e^{-iHt/ħ}. In the Heisenberg picture, operators evolve with H. If [H,A] = 0, A is conserved. In many-body physics, H includes sums over particles and interaction terms; in second quantization it is written with creation and annihilation operators.

See also: In classical mechanics, the Hamiltonian is a function H(q,p) on phase space; quantization maps it to an operator. In German-language literature, Hamiltonoperator or Hamiltonoperatoren is used; in English, Hamiltonian is standard.