# 变分原理

## 量子力学中的变分原理

${\displaystyle E_{ground}\leq \left\langle \phi |H|\phi \right\rangle }$

### 证明

${\displaystyle \phi =\sum _{n}c_{n}\psi _{n}\,}$

 ${\displaystyle \left\langle \phi |H|\phi \right\rangle \,}$ ${\displaystyle =\left\langle \sum _{n}c_{n}\psi _{n}|H|\sum _{m}c_{m}\psi _{m}\right\rangle \,}$ ${\displaystyle =\sum _{n}\sum _{m}\left\langle c_{n}\psi _{n}|E_{m}|c_{m}\psi _{m}\right\rangle \,}$ ${\displaystyle =\sum _{n}\sum _{m}c_{n}^{*}c_{m}E_{m}\left\langle \psi _{n}|\psi _{m}\right\rangle \,}$ ${\displaystyle =\sum _{n}|c_{n}|^{2}E_{n}\,}$

${\displaystyle \left\langle \phi |H|\phi \right\rangle \geq E_{g}\,}$

### 推广

${\displaystyle \varepsilon \left[\Psi \right]={\frac {\left\langle \Psi |{\hat {H}}|\Psi \right\rangle }{\left\langle \Psi |\Psi \right\rangle }}.}$

• ${\displaystyle \varepsilon \geq E_{0}}$，式中${\displaystyle E_{0}}$是该哈密顿算符的具有最低能量的本征态（基态）。
• ${\displaystyle \varepsilon =E_{0}}$当且仅当${\displaystyle \Psi }$确切地等同于研究体系的基态。

### 变分法应用示例[1]:192-193

#### 一维简谐振子

${\displaystyle H=T+V=-{\frac {\hbar ^{2}}{2\mu }}\left|A\right|^{2}\int _{-\infty }^{+\infty }{e^{-bx^{2}}{\frac {d^{2}}{dx^{2}}}\left(e^{-bx^{2}}\right)dx}+{\frac {1}{2}}\mu \omega ^{2}\left|A\right|^{2}\int _{-\infty }^{+\infty }{e^{-2bx^{2}}x^{2}dx=}{\frac {\hbar ^{2}b}{2\mu }}+{\frac {\mu \omega ^{2}}{8b}}}$

${\displaystyle {\frac {dH}{db}}={\frac {\hbar ^{2}}{2\mu }}-{\frac {\mu \omega ^{2}}{8b^{2}}}=0\Rightarrow b={\frac {\mu \omega }{2\hbar }}}$

## 延伸阅读

• Epstein S T 1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
• Lanczos C, The Variational Principles of Mechanics (Dover Publications)
• Nesbet R K 2003 "Variational Principles and Methods In Theoretical Physics and Chemistry". (New York: Cambridge U.P.)
• Adhikari S K 1998 "Variational Principles for the Numerical Solution of Scattering Problems". (New York: Wiley)
• Gray C G, Karl G and Novikov V A 1996 Ann. Phys. 251 1.

## 外部链接和参考资料

1. ^ , David J., Griffiths; Hu, Xing.Li, Yuxiao. 第7章；变分原理. 量子力学概论. Beijing: 机械工业出版社. 2009: 192–193. ISBN 9787111278771. OCLC 503192483.