# 变分原理

## 量子力学中的变分原理

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

### 证明

$\phi = \sum_{n} c_{n}\psi_{n} \,$

 $\left\langle\phi|H|\phi\right\rangle \,$ $= \left\langle\sum_{n}c_{n}\psi_{n}|H|\sum_{m}c_{m}\psi_{m}\right\rangle \,$ $= \sum_{n}\sum_{m}\left\langle c_{n}\psi_{n}|E_{m}|c_{m}\psi_{m}\right\rangle \,$ $= \sum_{n}\sum_{m}c_{n}^*c_{m}E_{m}\left\langle\psi_{n}|\psi_{m}\right\rangle \,$ $= \sum_{n} |c_{n}|^2 E_{n} \,$

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

### 推广

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

• $\varepsilon \geq E_0$，式中$E_0$是该哈密顿算符的具有最低能量的本征态（基态）。
• $\varepsilon = E_0$当且仅当$\Psi$确切地等同于研究体系的基态。

## 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.