Quantum harmonic oscillator

A quantum system governed by a quadratic potential, with equally spaced energy levels and a nonzero ground-state energy called zero-point energy.

Creation and annihilation operators: Operators that raise or lower the excitation number of a quantum harmonic oscillator. They generate its energy eigenstates and make the equal spacing of levels explicit.
Schrödinger equation: The equation governing how a nonrelativistic quantum state evolves and how its stationary states are determined. The oscillator’s energy levels follow by solving its stationary form.
Molecular vibration: Periodic motion of atoms within a molecule about their equilibrium positions. Small molecular vibrations are approximated by quantum harmonic oscillators.
Classical harmonic oscillator: A classical system whose restoring force is proportional to displacement from equilibrium. Its continuous energies contrast with the quantum oscillator’s discrete levels.
Number state: An eigenstate of a quantum system’s particle-number or excitation-number operator. Each oscillator energy eigenstate is a number state with a definite excitation count.
Hamiltonian (quantum mechanics): The operator representing a quantum system’s total energy and generating its time evolution. The oscillator Hamiltonian combines kinetic energy with a quadratic potential.
Phonon: A quantized normal mode of vibration in a material’s lattice. Each lattice vibration mode behaves as an independent quantum oscillator in the harmonic approximation.
Anharmonic oscillator: An oscillator whose restoring potential is not purely quadratic. Anharmonic terms shift the evenly spaced levels predicted by the ideal model.
Ground state: The lowest-energy state available to a quantum system. The oscillator ground state retains energy even when no excitations are present.
Canonical commutation relation: The noncommuting relation between position and momentum operators, expressed as their commutator equaling iℏ. This relation underlies the ladder-operator algebra and the oscillator’s quantization.

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