Hamiltonian (quantum mechanics)

The Hamiltonian is the operator representing a quantum system’s total energy. Through the Schrödinger equation, it determines how the system’s quantum state changes over time.

Schrödinger equation: A differential equation that describes how a quantum state changes over time. The Hamiltonian appears in its time-dependent form as the generator of state evolution.
Linear operator: A map between vector spaces that preserves vector addition and scalar multiplication. A quantum Hamiltonian acts linearly on state vectors.
Quantum harmonic oscillator: A quantum system with a quadratic potential and equally spaced energy levels. Its Hamiltonian yields an exactly solvable model central to quantum theory.
Lagrangian mechanics: A formulation of classical mechanics based on kinetic and potential energy through the Lagrangian. It offers a distinct route to dynamics, though it can be transformed into a Hamiltonian formulation.
Energy eigenstate: A quantum state that is an eigenvector of the Hamiltonian, with a definite energy eigenvalue. Hamiltonian eigenstates evolve by acquiring a phase without changing their energy distribution.
Hilbert space: A complete inner-product vector space used to represent quantum states and operators. The Hamiltonian acts on the system’s state space, usually a Hilbert space.
Hydrogen atom: The simplest atom, consisting of one proton and one electron bound by electromagnetic attraction. Its Hamiltonian predicts the bound-state energies that produce the hydrogen spectrum.
Time-dependent Hamiltonian: A Hamiltonian whose form explicitly depends on time. External driving or changing parameters make evolution differ from the time-independent case.
Energy eigenvalue: A possible energy measurement value associated with an eigenstate of the Hamiltonian. Its spectrum supplies the system’s allowed energies.
Hermitian operator: An operator equal to its adjoint, with real eigenvalues under standard conditions. A closed system’s Hamiltonian is Hermitian so its energy spectrum is real.

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