In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x:
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called "critically damped."
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.
The Hamiltonian of the particle is:
where m is the particle's mass, ω is the angular frequency of the oscillator, ∧x is the position operator (= x), and ∧p is the momentum operator, given by
The first term in the Hamiltonian represents the possible kinetic energy states of the particle, and the second term represents its respectively corresponding possible potential energy states.
One may write the time-independent Schrödinger equation,
where E denotes a yet-to-be-determined real number that will specify a time-independent energy level, or eigenvalue, and the solution |ψ⟩ denotes that level's energy eigenstate.