# 波包

## 歷史背景

${\displaystyle E=h\nu }$

## 範例

### 非色散傳播

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

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

${\displaystyle \omega ^{2}=|\mathbf {k} |^{2}v^{2}=(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})v^{2}}$

${\displaystyle u(x,\,t)=Ae^{i(kx-\omega t)}+Be^{-i(kx+\omega t)}}$

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

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

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

${\displaystyle A(k)={\frac {1}{\sqrt {2}}}e^{-{\frac {(k-k_{0})^{2}}{4}}}}$
${\displaystyle u(x,\,t)=e^{-(x-vt)^{2}+ik_{0}(x-vt)}}$

### 色散傳播

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

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

${\displaystyle u(x,\,t)={\frac {e^{-k_{0}^{2}/4}}{\sqrt {1+2it}}}\ e^{-(x-{\frac {ik_{0}}{2}})^{2}/(1+2it)}}$

${\displaystyle |u(x,\,t)|={\frac {1}{(1+4t^{2})^{1/4}}}e^{\frac {-x^{2}+2k_{0}xt}{1+4t^{2}}}}$

## 參考文獻

1. 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 （英语）
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.