Hamiltonoperátor

The Hamilton operator, also known as the Hamiltonian, is a fundamental concept in quantum mechanics representing the total energy of a system. It is an operator that, when acting on a quantum state, yields the energy of that state. In classical mechanics, the Hamiltonian is a function describing the total energy of a system as a sum of its kinetic and potential energies. In quantum mechanics, this classical function is promoted to an operator.

The time-evolution of a quantum system is governed by the Schrödinger equation, which explicitly involves the Hamilton operator. Specifically, the time-dependent Schrödinger equation is given by iħ ∂/∂t |ψ(t)⟩ = H |ψ(t)⟩, where H is the Hamilton operator, |ψ(t)⟩ is the quantum state vector at time t, and ħ is the reduced Planck constant. The eigenvalues of the Hamilton operator correspond to the possible energy levels of the system, and the corresponding eigenvectors are the stationary states, which do not change their probability distribution over time.

The form of the Hamilton operator depends on the specific system being studied. For a single particle in a one-dimensional potential V(x), the Hamilton operator is typically expressed as H = -ħ²/2m ∂²/∂x² + V(x), where the first term represents the kinetic energy operator and the second term represents the potential energy operator. More complex systems, such as multi-particle systems or systems in the presence of electromagnetic fields, will have more intricate Hamilton operators. The Hamilton operator is a Hermitian operator, ensuring that its eigenvalues (energies) are real numbers.