# 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".

#### References

 [1] L.I. Mandel'shtam, "Lectures on the theory of oscillations" , Moscow (1972) (In Russian) [2] L.D. Landau, E.M. Lifshitz, "Quantum mechanics" , Pergamon (1965) (Translated from Russian)

#### References

 [a1] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) [a2] L.I. Schiff, "Quantum mechanics" , McGraw-Hill (1949)
How to Cite This Entry:
Oscillator, harmonic. Encyclopedia of Mathematics. URL: http://www.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