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Difference between revisions of "Oscillator, harmonic"

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A system with one degree of freedom whose oscillations are described by the equation
 
A system with one degree of freedom whose oscillations are described by the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705301.png" /></td> </tr></table>
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$$\"x+\omega^2x=0.$$
  
The phase trajectories are circles, the period of the oscillations, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705302.png" />, does not depend on the amplitude. The potential energy of a harmonic oscillator depends quadratically on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705303.png" />:
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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 $x$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705304.png" /></td> </tr></table>
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$$U=\frac{\omega^2x^2}{2}.$$
  
 
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.
 
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.
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The oscillations of a quantum-mechanical linear oscillator are described by the [[Schrödinger equation|Schrödinger equation]]
 
The oscillations of a quantum-mechanical linear oscillator are described by the [[Schrödinger equation|Schrödinger equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705305.png" /></td> </tr></table>
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$$\frac{h^2}{2m}\frac{d^2\psi}{dx^2}+\left(E-\frac{m\omega^2x^2}{2}\right)\psi=0.$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705306.png" /> is the mass of a particle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705307.png" /> is its energy, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705308.png" /> is the Planck constant, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o0705309.png" /> is the frequency. A quantum-mechanical linear oscillator has a discrete spectrum of energy levels, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o07053010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070530/o07053011.png" />; the corresponding eigen functions can be expressed in terms of Hermite functions (cf. [[Hermite function|Hermite function]]).
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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" .
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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====
 
====References====

Revision as of 11:40, 3 August 2014

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

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

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)


Comments

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://encyclopediaofmath.org/index.php?title=Oscillator,_harmonic&oldid=32692
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article