Hamiltonian operator

The quantum-mechanical operator corresponding to a system’s total energy. Its eigenvalues give the energies of stationary states, and it generates time evolution through the Schrödinger equation.

Schrödinger equation: A differential equation governing how a quantum state changes in time. The Hamiltonian determines the generator of time evolution in its time-dependent form.
Observable: A measurable physical quantity represented in quantum mechanics by a self-adjoint operator. Energy is an observable, so its operator must satisfy the corresponding mathematical requirements.
Quantum harmonic oscillator: A quantum system with a quadratic potential and equally spaced energy levels. Its Hamiltonian yields a solvable spectrum used throughout quantum theory.
Lagrangian: A function of generalized coordinates, velocities, and time whose action yields equations of motion. It is an alternative classical description from which a Hamiltonian can often be derived.
Unbounded operator: An operator that maps vectors of arbitrarily small norm to outputs of arbitrarily large norm. Many physical Hamiltonians are unbounded, making their domains essential to rigorous definitions.
Energy eigenstate: A quantum state that is an eigenvector of the Hamiltonian, with a definite energy. Applying the Hamiltonian to this state returns its energy times the state.
Hilbert space: A complete inner-product space used to represent quantum states and operators. The Hamiltonian acts on the system’s state space, subject to domain conditions.
Hydrogen atom: The simplest atom, consisting of one proton bound to one electron. Its Coulomb Hamiltonian explains the characteristic bound-state energy spectrum.
Hamiltonian mechanics: A formulation of classical mechanics based on coordinates, conjugate momenta, and a Hamiltonian function. The quantum operator develops the energy-generating role of its classical counterpart.
Self-adjoint extension: A self-adjoint operator extending a symmetric operator defined on a restricted domain. Different extensions can represent distinct physical boundary conditions and energy spectra.

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