# 波包

## 歷史背景

$E =h\nu$

## 範例

### 非色散傳播

$\nabla^2 u=\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}$

$u(\mathbf{x},\,t) = e^{i{(\mathbf{k}\cdot \mathbf{x}} - \omega t)}$

$\omega^2 =|\bold{k}|^2 v^2=(k_x^2+k_y^2+k_z^2)v^2$

$u(x,\,t)= A e^{i(kx - \omega t)} + B e^{ - i(kx+\omega t)}$

$u(x,\,t) = \frac{1}{\sqrt{2\pi}} \int^{ \infty}_{ - \infty} A(k) ~ e^{i(kx - \omega(k)t)} \ dk$

$A(k) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{ - \infty} u(x,\,0) ~ e^{ - ikx}\,dx$

$u(x,\,0) = e^{ - x^2 +ik_0 x}$

$A(k) = \frac{1}{\sqrt{2}} e^{-\frac{(k-k_0)^2}{4}}$
$u(x,\,t) = e^{-(x-vt)^2 +ik_0(x-vt)}$

### 色散傳播

$i{ \partial u \over \partial t } = - \frac{1}{2} { \nabla^2 u }$

$\omega = \frac{1}{2}|\bold{k}|^2$

$u(x,\,t) =\frac{e^{ - k_0^2/4}}{\sqrt{1+2it}}\ e^{ - (x - \frac{ik_0}{2})^2/(1+2it)}$

$|u(x,\,t)| = \frac{1}{(1+4t^2)^{1/4}}e^{\frac{ - x^2+2k_0 xt}{1+4t^2}}$

## 參考文獻

1. ^ 1.0 1.1 Joy Manners. Quantum Physics: An Introduction. CRC Press. 2000. ISBN 978-0-7503-0720-8.
2. ^ Hecht, Eugene, Optics. 4th, United States of America: Addison Wesley. 2002, ISBN 0-8053-8566-5 （英文）
3. ^ Toda, Mikito. Geometric structures of phase space in multidimensional chaos.... Hoboken, New Jersey: John Wiley & Sons inc. 2005. ISBN 0-471-70527-6.
• J. D. Jackson (1975). Classical Electrodynamics (2nd Ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-43132-X.
• Leonard I. Schiff (1968). Quantum mechanics (3rd ed.). London : McGraw-Hill.