Harmonic oscillator

A harmonic oscillator moves around an equilibrium position under a restoring force that is proportional to displacement and directed back toward equilibrium. For a one-dimensional ideal oscillator, Hooke's law is often written F = -kx, where x is displacement from equilibrium and k is a positive force constant. This linear force relationship gives an equation of motion whose solutions are sinusoidal. A mass attached to an ideal spring is a standard mechanical example.

When no damping or external driving force acts, the motion is called simple harmonic motion. The displacement varies periodically in time and can be represented by a sine or cosine function with an amplitude, angular frequency and phase. For an ideal mass-spring oscillator, the angular frequency is the square root of k divided by m, where m is the oscillating mass; the period is 2 pi divided by the angular frequency. In this ideal model, changing the amplitude changes the total energy but not the period.

In a conservative harmonic oscillator, energy alternates between kinetic energy and potential energy while total energy remains constant. At equilibrium, speed and kinetic energy are greatest; at the turning points, displacement and potential energy are greatest and the instantaneous speed is zero. Friction or other energy losses produce damping, so oscillations decay unless energy is supplied. An external periodic force can drive an oscillator, and its response depends on the driving frequency and damping. For small displacements near many stable equilibria, a system's potential energy can often be approximated by a quadratic function, making harmonic motion a useful first approximation.

In quantum mechanics, a harmonic oscillator is a particle in a parabolic potential-energy well. Unlike a classical oscillator, its allowed energies are quantized and its ground-state energy is nonzero. The model is important for describing approximate molecular vibrations and for introducing methods used in quantum theory. At sufficiently high quantum numbers, its probability distribution increasingly resembles the classical distribution.

Oscillator models help explain vibrations in springs and structures, sound-producing systems, electrical resonators and molecular motion. Coupled oscillators can be analyzed in terms of normal modes, in which a system oscillates in characteristic patterns and frequencies. The model is an approximation: real systems can lose energy, experience external forces, have nonlinear restoring forces at larger displacements, or contain interacting components. Its predictions are useful when these effects are small or are included explicitly.