energiaoperator

The energiaoperator, commonly referred to as the energy operator or Hamiltonian, is a fundamental object in quantum mechanics that encodes the total energy of a system and generates its time evolution. As a Hermitian (self-adjoint) operator, its eigenvalues are real and correspond to possible energy measurement outcomes. The eigenvectors form a basis for the system’s state space, and the Born rule gives the probabilities of obtaining each energy value when measuring.

In non-relativistic quantum mechanics for a single particle of mass m in a static potential V(r), the energiaoperator takes the form H = -(ħ^2 / (2m)) ∇^2 + V(r). For many-body systems it generalizes to the sum of single-particle kinetic terms plus interaction potentials. In relativistic theories, other forms exist, such as the Dirac Hamiltonian for spin-1/2 particles. The energy spectrum can be discrete (bound states) or continuous (free states) depending on the problem.

Dynamics: The Schrödinger equation iħ ∂ψ/∂t = H ψ governs time evolution; stationary states satisfy H|E⟩ = E|E⟩ and evolve as e^{-iEt/ħ}. The expectation value of energy in a state |ψ⟩ is ⟨H⟩ = ⟨ψ|H|ψ⟩, and the time dependence of observables depends on the commutator with H.

Applications: Determining energy levels of atoms and molecules, predicting spectral lines, and underpinning quantum thermodynamics through partition functions Z = Tr(e^{-βH}). The energiaoperator thus serves as a central link between measurement, dynamics, and thermodynamics in quantum systems.