16 光的本质¶
文本统计：约 411 个字
光的波动性：光的干涉，光的衍射...
光的粒子性：黑体辐射，光电效应，康普顿效应...
说明光既是粒子又是波，同时光既不是传统的粒子也不是传统的波
16.1 黑体辐射¶
Stefan-Boltzmann law
The total radiated energy per area:
\[
\begin{aligned}
&I(T) = \sigma T^4\\
&\sigma = 5.67 \times 10^{-8} \, \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}
\end{aligned}
\]
Wien displacement law
\[
\begin{aligned}
&T \lambda_m = b\\
&b = 2.898 \times 10^{-3} \, \text{m} \cdot \text{K}
\end{aligned}
\]
其中 \(R(\lambda,T)=\frac{\text{d}l}{\text{d}\lambda}\), 那么 \(I(T) = \int_0^\infty R(\lambda, T) d\lambda\)
Example
With a \(\lambda_m\) of \(500 nm\) for sun radiation, the temperature of sun:
\[
T_{\text{sun}}=\frac{b}{\lambda_m}\approx 6000 K
\]
Planck’s Radiation Law (普朗克辐射定律)
- Wien’s line at short wavelength
- Rayleigh - Jeans line at long wavelengths
- Planck’s Law
\[
R(\lambda, T) = \frac{2\pi hc^2}{\lambda^5 (\mathrm{e}^{\frac{hc}{\lambda kT}} - 1)} = \frac{2\pi hc^2}{\lambda^5} \cdot \frac{1}{(\mathrm{e}^{\frac{hv}{kT}} - 1)}
\]
其中 \(h = 6.63 \times 10^{-34} \, \text{J} \cdot \text{s}\)
普朗克假设光子的能量是量子化的，有
\[
\begin{aligned}
&E=h\nu\\
&h=6.626\times 10^{-34}\text{J} \cdot \text{s}
\end{aligned}
\]
16.2 光电效应¶
与高中的内容基本无异
回忆一下：逸出功，截止频率。。。
16.3 康普顿效应¶
Photon is already particle:
\[
E = h \nu = \hbar \omega, \quad p = \frac{h}{\lambda} = \hbar k
\]
Scattering by electron:
\[
\begin{aligned}
&h \nu + m_0 c^2 = h \nu' + m c^2 \Rightarrow \frac{hc}{\lambda_0} = \frac{hc}{\lambda'} + m_0 c^2 \left[ \frac{1}{\sqrt{1 - (v/c)^2}} - 1 \right]\\
&\left\{
\begin{aligned}
&\frac{h}{\lambda_0} = \frac{h}{\lambda'} \cos \phi + \frac{m_0 v}{\sqrt{1 - (v/c)^2}} \cos \theta, \\
&\frac{h}{\lambda_0} \sin \phi = \frac{m_0 v}{\sqrt{1 - (v/c)^2}} \sin \theta
\end{aligned}
\right.
\end{aligned}
\]
从而得到
\[
\Delta \lambda = \lambda' - \lambda_0 = \frac{h}{m_0 c} (1 - \cos \phi)
\]
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