Hamiltonoperatorer

Hamiltonoperatorer, or Hamiltonians, are linear operators on a Hilbert space that encode the total energy of a quantum system. In non-relativistic quantum mechanics, the Hamiltonian H is a self-adjoint (Hermitian) operator whose eigenvalues are the possible results of energy measurements and which generates time evolution through the Schrödinger equation iħ ∂ψ/∂t = Hψ.

A common form for a single non-relativistic particle is H = p^2/(2m) + V(x), with p = -iħ∇. In position representation this acts as Hψ(x) = -(ħ^2/2m)∇^2ψ(x) + V(x)ψ(x). For lattice or finite systems, H is represented by a Hermitian matrix. In many-body quantum systems, H typically consists of sums of local energy terms and interaction contributions.

The spectrum of H—its eigenvalues and continuous parts—corresponds to the possible energy measurements of the system. Eigenstates (energy eigenvectors) provide stationary states with simple time evolution up to a phase. The unitary time evolution operator U(t) = exp(-iHt/ħ) governs dynamics in time-independent cases; for time-dependent Hamiltonians H(t) one uses a time-ordered exponential.

Mathematical aspects include self-adjointness and domain considerations, since many Hamiltonians are unbounded operators. These properties ensure real energy values and well-defined, unitary time evolution. The Hamiltonian also serves as the generator of time translations and, via commutators, determines conserved quantities: if [H, A] = 0, the observable A is constants of motion.

Historically, the Hamiltonian formalism connects classical energy functions to quantum operators through quantization, with the Hamiltonian playing a central role in describing both stationary states and dynamical evolution. The term Hamiltonoperatorer is common in several European languages, while English commonly uses Hamiltonian.