# 双缝实验中光子的动力学

## 双缝实验的经典描述

### 电磁波方程

${\displaystyle \nabla ^{2}\mathbf {E} \ -\ {1 \over c^{2}}{\partial ^{2}\mathbf {E} \over \partial t^{2}}\ \ =\ \ 0}$
${\displaystyle \nabla ^{2}\mathbf {B} \ -\ {1 \over c^{2}}{\partial ^{2}\mathbf {B} \over \partial t^{2}}\ \ =\ \ 0}$

${\displaystyle \nabla \times \mathbf {B} ={1 \over c}{\frac {\partial \mathbf {E} }{\partial t}}}$ .

### 电磁波方程的平面波解

${\displaystyle \mathbf {E} (\mathbf {z} ,t)={\begin{pmatrix}E_{x}^{0}\cos \left(kz-\omega t+\alpha _{x}\right)\\E_{y}^{0}\cos \left(kz-\omega t+\alpha _{y}\right)\\0\end{pmatrix}}=E_{x}^{0}\cos \left(kz-\omega t+\alpha _{x}\right){\hat {\mathbf {x} }}\;+\;E_{y}^{0}\cos \left(kz-\omega t+\alpha _{y}\right){\hat {\mathbf {y} }}}$

${\displaystyle \mathbf {B} (\mathbf {z} ,t)={\hat {\mathbf {z} }}\times \mathbf {E} (\mathbf {z} ,t)={\begin{pmatrix}-E_{y}^{0}\cos \left(kz-\omega t+\alpha _{x}\right)\\E_{x}^{0}\cos \left(kz-\omega t+\alpha _{y}\right)\\0\end{pmatrix}}=-E_{y}^{0}\cos \left(kz-\omega t+\alpha _{y}\right){\hat {\mathbf {x} }}\;+\;E_{x}^{0}\cos \left(kz-\omega t+\alpha _{x}\right){\hat {\mathbf {y} }}}$

${\displaystyle \omega _{}^{}=ck}$

${\displaystyle E_{x}^{0}=\mid \mathbf {E} \mid \cos \theta }$
${\displaystyle E_{y}^{0}=\mid \mathbf {E} \mid \sin \theta }$

${\displaystyle \alpha _{x}^{},\alpha _{y}}$

${\displaystyle \theta \ {\stackrel {\mathrm {def} }{=}}\ \tan ^{-1}\left({E_{y}^{0} \over E_{x}^{0}}\right)}$ .
${\displaystyle \mid \mathbf {E} \mid ^{2}\ {\stackrel {\mathrm {def} }{=}}\ \left(E_{x}^{0}\right)^{2}+\left(E_{y}^{0}\right)^{2}}$ .

${\displaystyle \mathbf {E} (\mathbf {z} ,t)=\mid \mathbf {E} \mid \mathrm {Re} \left\{|\psi \rangle \exp \left[i\left(kz-\omega t\right)\right]\right\}}$

${\displaystyle |\psi \rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}\psi _{x}\\\psi _{y}\end{pmatrix}}={\begin{pmatrix}\cos \theta \exp \left(i\alpha _{x}\right)\\\sin \theta \exp \left(i\alpha _{y}\right)\end{pmatrix}}}$

### 电磁波方程的球面波和柱面波解

#### 球面波

${\displaystyle \mathbf {E} (\mathbf {r} ,t)=\mid \mathbf {E} (\mathbf {r_{0}} ,t)\mid \left({r_{0} \over r}\right)\mathrm {Re} \left\{|\psi \rangle \exp \left[i\left(kr-\omega t\right)\right]\right\}}$

${\displaystyle \mathbf {B} (\mathbf {r} ,t)={\hat {\mathbf {r} }}\times \mathbf {E} (\mathbf {r} ,t)}$

#### 柱面波

${\displaystyle \mathbf {E} (\mathbf {r} ,t)=\mid \mathbf {E} (\mathbf {r_{0}} ,t)\mid \left({r_{0} \over r}\right)^{1/2}\mathrm {Re} \left\{|\psi \rangle \exp \left[i\left(kr-\omega t\right)\right]\right\}}$
${\displaystyle \mathbf {B} (\mathbf {r} ,t)={\hat {\mathbf {r} }}\times \mathbf {E} (\mathbf {r} ,t)}$

### 惠更斯原理

${\displaystyle \mathrm {Re} \left\{|\psi \rangle \exp \left[i\left(kr-\omega t\right)\right]\right\}\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {Re} \left\{|\phi \rangle \right\}}$ .

### 干涉

${\displaystyle r_{1}={\sqrt {L^{2}+x^{2}}}}$

${\displaystyle r_{2}={\sqrt {L^{2}+(x-d)^{2}}}}$ .

${\displaystyle \Delta r\approx {xd \over r_{1}}\approx {xd \over L}}$ .

${\displaystyle |\phi _{1}\rangle +|\phi _{2}\rangle =|\psi \rangle \left\{\exp \left[i\left(kr_{1}-\omega t\right)\right]+\exp \left[i\left(kr_{2}-\omega t\right)\right]\right\}=|\phi _{1}\rangle \exp \left[i\left(k\Delta r-\omega t\right)\right].}$

${\displaystyle \cos ^{2}\left(k\Delta r\right)\approx \cos ^{2}\left(2\pi {xd \over \lambda }\right)}$

${\displaystyle 2\pi {xd \over \lambda }=n\pi \quad n=0,1,2,\cdots }$

${\displaystyle x_{n}={n\lambda \over 2d}\quad n=0,1,2,\cdots }$ .

## 双缝实验的量子描述

### 光子的能量和动量

#### 能量

${\displaystyle \epsilon =\hbar \omega }$

${\displaystyle {N\hbar \omega \over V}={\mathcal {E}}_{c}={\frac {\mid \mathbf {E} \mid ^{2}}{8\pi }}}$ .

${\displaystyle N={\frac {V}{8\pi \hbar \omega }}\mid \mathbf {E} \mid ^{2}}$ .

#### 动量

${\displaystyle {\mathcal {P}}_{c}={N\hbar \omega \over cV}={N\hbar k \over V}}$

### 量子力学中的几率本性

#### 几率幅

1. 两个先后过程的几率叠加得到的几率幅是每一个单独几率幅的乘积。
2. 一个过程可由多种不可区分的方法之一来完成，则它的几率幅是每一种独立方法的几率幅的和。
3. 过程发生的总几率是通过第1条和第2条所给出的几率幅的模平方。

## 註釋

1. ^ 雖然每一點表示一個電子抵達探測屏，這事實並不能表現出電子的粒子性，因為探測器是由離散原子組成的，這可以詮釋為電子波與離散原子彼此之間的相互作用。[3]

## 参考文献

1. ^ Bialynicki-Birula, Iwo. Photon wave function. eprint arXiv:quant-ph/0508202. 08/2005.
2. ^ Jackson, John D. Classical Electrodynamics (3rd ed.). Wiley. 1998. ISBN 0-471-30932-X.
3. ^ Hobson, Art. There are no particles, there are only fields. American Journal of Physics. 2013, 81 (211). doi:10.1119/1.4789885.
4. ^ Dirac, P. A. M. The Principles of Quantum Mechanics, Fourth Edition. Oxford. 1958. ISBN 0-19-851208-2. 中文译文来自陈咸亨中译本《量子力学原理》，科学出版社出版
5. ^ Baym, Gordon. Lectures on Quantum Mechanics. W. A. Benjamin. 1969. ISBN 0-8053-0667-6.