从冰上的水 - Lorentz Model for an Atom

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来自 Wolfgang Kettler 的 AMO 课中提到

原子由一个固定不动正电荷, 和一个质量为 $m$ , 带电为 $q$ 的负电荷组成, 二者由一个劲度系数 $C$ 的弹簧连接. 原子处于外加驱动电场 $E (t) = e^{- i ω t}$ 中, 运动时受到与速度成正比的阻尼 $m γ \dot{x}$ , 可列出牛顿第二定律的运动方程

\begin{array}{r} q E (t) - C x (t) - m γ \dot{x} (t) = m \ddot{x} (t) \end{array}

记

\begin{array}{r} ω_{0} = \sqrt{\frac{C}{m}} \end{array}

上式解得

\begin{array}{r} x (t) = \frac{q E (t)}{m} \frac{1}{ω_{0}^{2} - ω^{2} - i ω γ} \end{array}

是一个 Lorentz shape 的形式.

\begin{array}{r} p = q x = \frac{q^{2} E (t)}{m} \frac{1}{ω_{0}^{2} - ω^{2} - i ω γ} \end{array}

n is the electrons per volume

\begin{array}{r} P = n p = \frac{n q^{2} E (t)}{m} \frac{1}{ω_{0}^{2} - ω^{2} - i ω γ} \end{array}

\begin{array}{r} χ = \frac{P}{ϵ_{0} E} = \frac{n q^{2} E}{ϵ_{0} m} \frac{1}{ω_{0}^{2} - ω^{2} - i ω γ} \end{array}

define plasma frequency

\begin{array}{r} ω_{p} = \sqrt{\frac{n q^{2}}{m ϵ_{0}}} \end{array}

\begin{array}{r} χ = \frac{ω_{p}^{2}}{ω_{0}^{2} - ω^{2} - i ω γ} \end{array}

\begin{array}{r} ϵ_{r} = 1 + χ = 1 + \frac{ω_{p}^{2}}{ω_{0}^{2} - ω^{2} - i ω γ} \end{array}

https://physics.byu.edu/faculty/colton/docs/phy442-winter20/lecture-11-Lorentz-oscillator-model.pdf