# 摄动理论

${\displaystyle A=\epsilon ^{0}A_{0}+\epsilon ^{1}A_{1}+\epsilon ^{2}A_{2}+\cdots \,\!}$

## 一阶无简并摄动理论

### 一阶本征值修正

${\displaystyle Dg(x)=\lambda g(x)\,\!}$ (1)

${\displaystyle D=D^{(0)}+\epsilon D^{(1)}\,\!}$

${\displaystyle D^{(0)}f_{i}^{(0)}(x)=\lambda _{i}^{(0)}f_{i}^{(0)}(x)\,\!}$

${\displaystyle \int f_{i}^{(0)}(x)f_{j}^{(0)}(x)\,dx=\delta _{ij}\,\!}$

${\displaystyle g(x)=f_{n}^{(0)}(x)+{\mathcal {O}}(\epsilon )\,\!}$

${\displaystyle \lambda =\lambda _{n}^{(0)}+{\mathcal {O}}(\epsilon )\,\!}$

${\displaystyle g(x)=\sum _{m}c_{m}f_{m}^{(0)}(x)\,\!}$ (2)

${\displaystyle \lambda _{n}^{(0)}c_{n}+\epsilon \sum _{m}c_{m}\int f_{n}^{(0)}(x)D^{(1)}f_{m}^{(0)}(x)\,dx=\lambda c_{n}\,\!}$

${\displaystyle A_{nm}=\lambda _{n}^{(0)}\delta _{nm}+\epsilon \int f_{n}^{(0)}(x)D^{(1)}f_{m}^{(0)}(x)\,dx\,\!}$

${\displaystyle \lambda =\lambda _{n}^{(0)}+\epsilon \int f_{n}^{(0)}(x)D^{(1)}f_{n}^{(0)}(x)\,dx\,\!}$ (3)

${\displaystyle \lambda _{n}^{(1)}=\int f_{n}^{(0)}(x)D^{(1)}f_{n}^{(0)}(x)\,dx\,\!}$

### 一阶本征函数修正

${\displaystyle g(x)=f_{n}^{(0)}(x)+\epsilon f_{n}^{(1)}(x)\,\!}$ (4)

${\displaystyle \left(D^{(0)}+\epsilon D^{(1)}\right)\left(f_{n}^{(0)}(x)+\epsilon f_{n}^{(1)}(x)\right)=(\lambda _{n}^{(0)}+\epsilon \lambda _{n}^{(1)})\left(f_{n}^{(0)}(x)+\epsilon f_{n}^{(1)}(x)\right)\,\!}$

${\displaystyle D^{(1)}f_{n}^{(0)}(x)+D^{(0)}f_{n}^{(1)}(x)=\lambda _{n}^{(0)}f_{n}^{(1)}(x)+\lambda _{n}^{(1)}f_{n}^{(0)}(x)\,\!}$ (5)

${\displaystyle f_{n}^{(1)}(x)=\sum _{i\neq n}C_{i}f_{i}^{(0)}(x)\,\!}$ (6)

${\displaystyle (D^{(1)}-\lambda _{n}^{(1)})f_{n}^{(0)}(x)=\lambda _{n}^{(0)}\sum _{i\neq n}C_{i}f_{i}^{(0)}(x)-D^{(0)}\sum _{i\neq n}C_{i}f_{i}^{(0)}(x)=\sum _{i\neq n}(\lambda _{n}^{(0)}-\lambda _{i}^{(0)})C_{i}f_{i}^{(0)}(x)\,\!}$

${\displaystyle \int \,f_{j}^{(0)}(x)D^{(1)}f_{n}^{(0)}(x)\,dx=\sum _{i\neq n}(\lambda _{n}^{(0)}-\lambda _{i}^{(0)})C_{i}\int \,f_{j}^{(0)}(x)f_{i}^{(0)}(x)=(\lambda _{n}^{(0)}-\lambda _{j}^{(0)})C_{j}\,\!}$

${\displaystyle f_{n}^{(1)}(x)=\sum _{m\neq n}{\frac {f_{m}^{(0)}(x)}{\lambda _{n}^{(0)}-\lambda _{m}^{(0)}}}\int \,f_{m}^{(0)}(y)D^{(1)}f_{n}^{(0)}(y)\,dy\,\!}$

## 參考資料

1. ^ William E. Wiesel. Modern Astrodynamics. Ohio: Aphelion Press. 2010: 107. ISBN 978-145378-1470.