# Oscillator, harmonic

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A system with one degree of freedom whose oscillations are described by the equation

$$\ddot x+\omega^2x=0.$$

The phase trajectories are circles, the period of the oscillations, $T=2\pi/\omega$, does not depend on the amplitude. The potential energy of a harmonic oscillator depends quadratically on $x$:

$$U=\frac{\omega^2x^2}{2}.$$

Examples of harmonic oscillators are: small oscillations of a pendulum, oscillations of a material point fastened on a spring with constant rigidity, and the simplest electric oscillatory circuit. The terms "harmonic oscillator" and "linear oscillatorlinear oscillator" are often used as synonyms.

The oscillations of a quantum-mechanical linear oscillator are described by the Schrödinger equation

$$\frac{h^2}{2m}\frac{d^2\psi}{dx^2}+\left(E-\frac{m\omega^2x^2}{2}\right)\psi=0.$$

Here, $m$ is the mass of a particle, $E$ is its energy, $h$ is the Planck constant, and $\omega$ is the frequency. A quantum-mechanical linear oscillator has a discrete spectrum of energy levels, $E_n=(n+1/2)h\omega$, $n=0,1,\ldots$; the corresponding eigen functions can be expressed in terms of Hermite functions (cf. Hermite function).

The term "oscillator" is used in relation to (mechanical or physical) systems with a finite number of degrees of freedom whose motion is oscillatory (e.g. a van der Pol oscillator — a multi-dimensional linear oscillator representing the oscillations of a material point situated in a potential force field with a potential which is a positive-definite quadratic form in the coordinates, see van der Pol equation). There is evidently no unique interpretation of the term "oscillator", or even of "linear oscillator".

How to Cite This Entry:
Oscillator, harmonic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillator,_harmonic&oldid=32693
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article