# 波包

## 历史背景

${\displaystyle E=h\nu }$

## 范例

### 非色散传播

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

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

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

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

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

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

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

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

### 色散传播

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

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

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

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

## 参考文献

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.