Harmonic oscillator

Japanese: 調和振動子 - ちょうわしんどうし
Harmonic oscillator

Simple harmonic motion is also called harmonic motion. A mass point or other mechanical system that oscillates harmonically is called a harmonic oscillator. In classical mechanics, if the position coordinate of a mass point that oscillates harmonically is x and the momentum is p, its energy E is the sum of kinetic energy and potential energy, E=p 2 /(2m)+(1/2)kx 2
Here, m is the mass of the mass point, and k is the proportionality constant of the restoring force (-kx). This mass point oscillates with a frequency of ν=(k/m) 1/2 /2π. In quantum mechanics, the energy of a harmonic oscillator cannot take continuous values, but can only take discrete values ​​E=(n+1/2)hν (n=0,1,2,……)
where h is Planck's constant. Therefore, in the lowest energy state (ground state), E=hν/2. This energy is called the zero-point energy. hν is also called the energy quantum. Thermal vibrations of solids and standing waves of electromagnetic waves confined within a finite volume (cavity) are both expressed as a superposition of normal vibrations, and each normal vibration is considered to be a single harmonic oscillator. Therefore, the energy also takes on discrete values. The energy quantum is called a phonon in the case of thermal vibrations of solids, and a photon in the case of standing waves of electromagnetic waves. The positive integer n is interpreted as the number of phonons or photons.

When the amplitude of simple harmonic motion is large, the potential energy of the oscillator contains terms that are equal to or greater than the cube of the displacement x. In this case, the oscillator is called an anharmonic oscillator.

[Yoshiro Kainuma]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

単振動は調和振動ともよばれる。調和振動をする質点や、その他の力学系は調和振動子とよばれる。古典力学では、調和振動をする質点の位置座標をx、運動量をpとすると、そのエネルギーEは、運動エネルギーと位置のエネルギーの和として
  E=p2/(2m)+(1/2)kx2
である。ここにmは質点の質量、kは復元力(-kx)の比例定数である。この質点は振動数ν=(k/m)1/2/2πの振動をする。量子力学では、調和振動子のエネルギーは連続的な値をとることができず、離散的な値
  E=(n+1/2)hν (n=0,1,2,……)
をとる。hはプランク定数である。したがって、エネルギー最低の状態(基底状態)では、E=hν/2である。このエネルギーを零点エネルギーという。またhνをエネルギー量子という。固体の熱振動、有限の体積(空洞)の中に閉じ込められた電磁波の定常波は、いずれも基準振動の重ね合わせとして表され、各基準振動は一つの調和振動子とみなされる。したがって、そのエネルギーも離散的な値をとる。エネルギー量子は、固体の熱振動ではフォノン、電磁波の定常波ではフォトンとよばれる。正整数nは、フォノンまたはフォトンの個数と解される。

 単振動の振幅が大きい場合には、振動子の位置エネルギーには、変位xの三乗以上の項が加わる。この場合の振動子は非調和振動子とよばれる。

[飼沼芳郎]

出典 小学館 日本大百科全書(ニッポニカ)日本大百科全書(ニッポニカ)について 情報 | 凡例

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