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 | ||
| − | + | $$\ddot x+\omega^2x=0.$$ | |
| − | The phase trajectories are circles, the period of the oscillations, | + | 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. | 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]] | ||
| − | + | $$\frac{h^2}{2m}\frac{d^2\psi}{dx^2}+\left(E-\frac{m\omega^2x^2}{2}\right)\psi=0.$$ | |
| − | Here, | + | 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" . | + | 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==== | ||
Latest revision as of 11:41, 3 August 2014
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) |
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) |
Oscillator, harmonic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillator,_harmonic&oldid=16351