View source for 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|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|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|van der Pol equation]]). There is evidently no unique interpretation of the term  "oscillator", or even of  "linear oscillator".

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.I. Mandel'shtam,   "Lectures on the theory of oscillations" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "Quantum mechanics" , Pergamon  (1965)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.I. Schiff,   "Quantum mechanics" , McGraw-Hill  (1949)</TD></TR></table>