﻿ 宇称算符 - 小时百科

# 宇称算符

对多元函数 $f( \boldsymbol{\mathbf{x}} ) = f(x_1, \dots, x_N)$． 定义宇称算符 $\Pi$ 如下

\begin{equation} \Pi f( \boldsymbol{\mathbf{x}} ) = f(- \boldsymbol{\mathbf{x}} ) \end{equation}

\begin{equation} \left\langle f \middle| g \right\rangle = \int f^*( \boldsymbol{\mathbf{x}} ) g( \boldsymbol{\mathbf{x}} ) \,\mathrm{d}^{N}{x} \end{equation}

\begin{equation} \Pi f( \boldsymbol{\mathbf{x}} ) = \lambda f( \boldsymbol{\mathbf{x}} ) \end{equation}

\begin{equation} f(- \boldsymbol{\mathbf{x}} ) = \pm f( \boldsymbol{\mathbf{x}} ) \end{equation}

1. ^ 分别令 $\boldsymbol{\mathbf{x}}$ 为某对称的两点 $\boldsymbol{\mathbf{x}} _1$ 和 $- \boldsymbol{\mathbf{x}} _1$，使函数值不为零．本征方程要求 $f(- \boldsymbol{\mathbf{x}} _1) = \lambda f( \boldsymbol{\mathbf{x}} _1)$ 且 $f( \boldsymbol{\mathbf{x}} _1) = \lambda f(- \boldsymbol{\mathbf{x}} _1)$，所以必有 $\lambda^2 = 1$，$\lambda = \pm 1$．
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