S3-单体格林函数 | Fuming's Blog

单体格林函数简引

格林函数的定义

由量子力学我们可以知道，当我们知道一个系统的哈密顿量 $H$ ,那么我们可以求出它的本征值与本征函数，以及其它的物理量。但是求解哈密顿量 $H$ 的本征函数是困难的，那我们试图找到一个包含 $H$ 所包含的信息等价的算符，即格林算符

\begin{equation} G\left( z\right) =\left( z-H\right) ^{-1}=\frac{1}{z-H} ,z\text{为复参量} \end{equation}

这里的 $z$ 完全可以是 $E_{n}$ .

对于定态问题，有哈密顿量的本征方程(已知)

\begin{equation} H\left\vert \varphi_{n}\right\rangle =E_{n}\left\vert \varphi_{n} \right\rangle \end{equation}

且要求本征集正交归一

\begin{equation} \langle\varphi_{n}\left\vert \varphi_{m}\right\rangle =\delta_{mn} \end{equation}

完备关系

\begin{equation} \sum_{n}\left\vert \varphi_{n}\right\rangle \left\langle \varphi _{n}\right\vert =1 \end{equation}

将式(1)插入完备恒等式

\begin{align} G\left( z\right) & =\frac{1}{z-H}\sum_{n}\left\vert \varphi_{n} \right\rangle \left\langle \varphi_{n}\right\vert \nonumber\\ & =\sum_{n}\frac{\left\vert \varphi_{n}\right\rangle \left\langle \varphi _{n}\right\vert }{z-E_{n}} \end{align}

可以看出 $G\left( z\right)$ 中包含 $H$ 的本征值与本征函数的全部信息,同时当得到 $H$ 的全部本征值与本征函数， $G\left( z \right)$ 也就已知

复杂系统的格林函数处理

对于不能严格求解的系统，我们将 $H$ 分成两个部分

\begin{equation} H=H_{0}+H_{1} \end{equation}

相应对于 $H_{0}$ 的格林函数为

$G_{0}\left( z\right) =\frac{1}{z-H_{0}}$

$H$ 对应的格林函数为

\begin{align} G\left( z\right) & =\left( z-H_{0}-H_{1}\right) ^{-1}\nonumber\\ & =\left[ \left( z-H_{0}\right) \left( 1-\frac{H_{1}}{z-H_{0}}\right) \right] ^{-1}\nonumber\\ & =\left[ \left( z-H_{0}\right) \left( 1-G_{0}\left( z\right) H_{1}\right) \right] ^{-1}\nonumber\\ & =\left( 1-G_{0}\left( z\right) H_{1}\right) ^{-1}G_{0}\left( z\right) \\ & =G_{0}\left( z\right) \left( 1-H_{1}G_{0}\left( z\right) \right) ^{-1} \end{align}

也即

\begin{align} \left( 1-G_{0}\left( z\right) H_{1}\right) G\left( z\right) & =G_{0}\left( z\right) \\ G\left( z\right) \left( 1-G_{0}\left( z\right) H_{1}\right) & =G_{0}\left( z\right) \end{align}

可得

\begin{align} G\left( z\right) & =G_{0}\left( z\right) +G_{0}H_{1}G\\ G\left( z\right) & =G_{0}\left( z\right) +GH_{1}G_{0} \end{align}

可以看出式(11)(12)具有封闭性。

反复迭代^[1]

\begin{align} G\left( z\right) & =G_{0}\left( z\right) +GH_{1}G_{0}\nonumber\\ & =G_{0}\left( z\right) +\left( G_{0}\left( z\right) +GH_{1} G_{0}\right) H_{1}G_{0}\nonumber\\ & =G_{0}\left( z\right) +G_{0}\left( z\right) H_{1}G_{0}+G\left( H_{1}G_{0}\right) ^{2}\nonumber\\ & =...\nonumber\\ & =G_{0}\left( z\right) +G_{0}\left( z\right) H_{1}G_{0}+G_{0}\left( H_{1}G_{0}\right) ^{2}+...+G\left( H_{1}G_{0}\right) ^{m}\nonumber\\ & =G_{0}\sum_{n=0}^{\infty}\left( H_{1}G_{0}\right) ^{n}=\sum_{n=0} ^{\infty}\left( G_{0}H_{1}\right) ^{n}G_{0} \end{align}

我们定义一个新的算符 $T$

\begin{equation} G=G_{0}+G_{0}TG_{0} \end{equation}

与式(11)(12)比较可得

\begin{align} TG_{0} & =H_{1}G\\ G_{0}T & =GH_{1} \end{align}

其中

\begin{align} T & =H_{1}\sum_{n=0}^{\infty}\left( G_{0}H_{1}\right) ^{n}=H_{1}\left( 1+GH_{1}\right) \nonumber\\ & =H_{1}+H_{1}GH_{1}\\ & =H_{1}+TG_{0}H_{1}=H_{1}+H_{1}G_{0}T \end{align}

对于式(6),可以得到 $H_{0},H$
的本征方程

\begin{equation} H_{0}\left\vert \varphi_{n}\right\rangle =E_{n}\left\vert \varphi _{n}\right\rangle \end{equation} \begin{equation} H\left\vert \psi\right\rangle =E\left\vert \psi\right\rangle \end{equation}

我们可以从 $H_{0}$ 的本征函数 $\left\vert \varphi_{n}\right\rangle$ 以及 $G_{0}$ 求得 $H$ 的本征函数 $\left\vert \psi\right\rangle$ ，所以

\begin{equation} \left( E_{n}-H_{0}\right) \left\vert \varphi_{n}\right\rangle =G_{0} ^{-1}\left( E_{n}\right) \left\vert \varphi_{n}\right\rangle =0 \end{equation} \begin{equation} \left( E-H_{0}\right) \left\vert \psi\right\rangle =G_{0}^{-1}\left( E\right) \left\vert \psi\right\rangle =H_{1}\left\vert \psi\right\rangle \end{equation}

当 $E\neq E_{n}$ 时，由式(22)可得

\begin{equation} \left\vert \psi\right\rangle =G_{0}\left( E\right) H_{1}\left\vert \psi\right\rangle \end{equation}

当 $E=E_{n}$ 时，由式(21)(22)可得， $G_{0}^{-1}\left( E_{n}\right) \left\vert \psi\right\rangle -G_{0}^{-1}\left( E_{n}\right) \left\vert \varphi_{n}\right\rangle =H_{1}\left\vert \psi\right\rangle$

\begin{equation} \left\vert \psi\right\rangle =\left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) H_{1}\left\vert \psi\right\rangle \end{equation}

迭代可得

\begin{align} \left\vert \psi\right\rangle & =\left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) H_{1}\left( \left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) H_{1}\left\vert \psi\right\rangle \right) \nonumber\\ & =\left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) H_{1}\left\vert \varphi_{n}\right\rangle +\left( G_{0}\left( E_{n}\right) H_{1}\right) ^{2}\left\vert \psi\right\rangle \nonumber\\ & =\left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) H_{1}\left\vert \varphi_{n}\right\rangle +\left( G_{0}\left( E_{n}\right) H_{1}\right) ^{2}\left( \left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) H_{1}\left\vert \psi\right\rangle \right) \nonumber\\ & =\left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) H_{1}\left\vert \varphi_{n}\right\rangle +\left( G_{0}\left( E_{n}\right) H_{1}\right) ^{2}\left\vert \varphi_{n}\right\rangle +\left( G_{0}\left( E_{n}\right) H_{1}\right) ^{3}\left\vert \psi\right\rangle \nonumber\\ & =\sum_{i=0}^{\infty}\left( G_{0}\left( E_{n}\right) H_{1}\right) ^{i}\left\vert \varphi_{n}\right\rangle \nonumber\\ & =\left( 1+GH_{1}\right) \left\vert \varphi_{n}\right\rangle =\left( 1+G_{0}T\right) \left\vert \varphi_{n}\right\rangle \nonumber\\ & =\left\vert \varphi_{n}\right\rangle +G_{0}\left( E_{n}\right) T\left( E_{n}\right) \left\vert \varphi_{n}\right\rangle \end{align}

即有

\begin{equation} T\left( E_{n}\right) \left\vert \varphi_{n}\right\rangle =H_{1}\left\vert \psi\right\rangle \end{equation}

格林函数相关的物理量

由于格林函数与哈密顿量一样，含有系统的所有信息，因此凡是通过哈密顿量能够计算出来的物理量，从格林函数也可以求出。比如我们计算态密度^[2]

\begin{equation} \rho\left( E\right) =\sum_{n}\delta\left( E-E_{n}\right) \end{equation}

整个能量的积分

\begin{equation} \int_{-\infty}^{+\infty}\rho\left( E\right) dE=\sum_{n} \end{equation}

计算态密度时会用到以下的恒等式

$\lim_{\eta\rightarrow0^{+}}\frac{1}{x\pm i\eta}=P\frac{1}{x}\mp i\pi \delta\left( x\right)$

求态密度的动机：

$\begin{aligned} 1.&系统的热力学量需要态密度；跃迁几率，跃迁振幅......\\ 2.&态密度较易观测 \end{aligned}$

在具体表象的表示

坐标表象

格林函数表达

对式(1)取坐标表象的矩阵元

\begin{equation} \left\langle \mathbf{r}\right\vert \left( z-H\right) G\left( z\right) \left\vert \mathbf{r}\right\rangle =\left\langle \mathbf{r}\right\vert \mathbf{r\rangle} \end{equation}

已知

\begin{equation} \left\langle \mathbf{r}\right\vert \mathbf{r}^{\prime}\mathbf{\rangle=\delta }\left( \mathbf{r-r}^{\prime}\right) \end{equation}

完备关系

\begin{equation} \int d\mathbf{r}\left\vert \mathbf{r}\right\rangle \left\langle \mathbf{r} \right\vert =1 \end{equation}

将式(29)化为

\begin{equation} \int d\mathbf{r}^{\prime\prime}\left\langle \mathbf{r}\right\vert \left( z-H\right) \left\vert \mathbf{r}^{\prime\prime}\right\rangle \left\langle \mathbf{r}^{\prime\prime}\right\vert G\left( z\right) \left\vert \mathbf{r}^{\prime}\right\rangle =\mathbf{\delta}\left( \mathbf{r-r}^{\prime }\right) \end{equation}

记

\begin{equation} \left\langle \mathbf{r}^{\prime\prime}\right\vert G\left( z\right) \left\vert \mathbf{r}^{\prime}\right\rangle =G\left( \mathbf{r}^{\prime \prime},\mathbf{r}^{\prime};z\right) \end{equation}

为格林算符在坐标表象下的矩阵元，同理对于哈密顿量

\begin{equation} \left\langle \mathbf{r}\right\vert H\left\vert \mathbf{r}^{\prime }\right\rangle =H\left( \mathbf{r},\mathbf{r}^{\prime}\right) \end{equation}

哈密顿量是单体算符因此式(34)可以表示为^[3]

\begin{equation} H\left( \mathbf{r},\mathbf{r}^{\prime}\right) =H\left( \mathbf{r}\right) \mathbf{\delta}\left( \mathbf{r-r}^{\prime}\right) \end{equation}

因此 $H$ 的本征方程及式(35)可化为

\begin{align} \int\left\langle \mathbf{r}\right\vert H\left\vert \mathbf{r}^{\prime }\right\rangle \langle\mathbf{r}^{\prime}\left\vert \varphi_{n}\right\rangle d\mathbf{r}^{\prime} & =E_{n}\varphi_{n}\left( \mathbf{r}\right) \nonumber\\ \int H\left( \mathbf{r}\right) \mathbf{\delta}\left( \mathbf{r-r}^{\prime }\right) \langle\mathbf{r}^{\prime}\left\vert \varphi_{n}\right\rangle d\mathbf{r}^{\prime} & =E_{n}\varphi_{n}\left( \mathbf{r}\right) \nonumber\\ H\left( \mathbf{r}\right) \varphi_{n}\left( \mathbf{r}\right) & =E_{n}\varphi_{n}\left( \mathbf{r}\right) \end{align} \begin{equation} \left[ z-H\left( \mathbf{r}\right) \right] G\left( \mathbf{r} ,\mathbf{r}^{\prime};z\right) =\mathbf{\delta}\left( \mathbf{r-r}^{\prime }\right) \end{equation}

式(5)可以表示为

\begin{align} G\left( \mathbf{r},\mathbf{r}^{\prime};z\right) & =\left\langle \mathbf{r}\right\vert \sum_{n}\frac{\left\vert \varphi_{n}\right\rangle \left\langle \varphi_{n}\right\vert }{z-E_{n}}\left\vert \mathbf{r}^{\prime }\right\rangle \nonumber\\ & =\sum_{n}\frac{\varphi_{n}\left( \mathbf{r}\right) \varphi_{n}^{\ast }\left( \mathbf{r}^{\prime}\right) }{z-E_{n}} \end{align}

同时取共轭可得

\begin{equation} G^{\ast}\left( \mathbf{r},\mathbf{r}^{\prime};z\right) =G\left( \mathbf{r}^{\prime},\mathbf{r};z^{\ast}\right) \end{equation}

如果我们可以以一种方式计算出格林函数，那么它的一阶极点处就是能量本征值

在坐标表象中求解格林函数：

\begin{align*} 1.&由格林函数的本征方程(37)出发，结合边界条件得到格林函数\\ 2.&由格林函数的定义(38)出发,求出H的本征值与本征函数得到格林函数 \end{align*}

对于式(38)的求和是对分立谱求和以及连续谱积分。

其中连续谱，我们可以定义从上半平面或者下半平面无限趋于实轴的侧极限:

\begin{equation} G^{\pm}\left( \mathbf{r},\mathbf{r}^{\prime};E\right) =G\left( \mathbf{r},\mathbf{r}^{\prime};E\pm i0^{+}\right) =\sum_{n}\frac{\varphi _{n}\left( \mathbf{r}\right) \varphi_{n}^{\ast}\left( \mathbf{r}^{\prime }\right) }{E\pm i0^{+}-E_{n}} \end{equation}

式(11)在坐标表象下的表示为

\begin{equation} \left\langle \mathbf{r}\right\vert G\left( z\right) \left\vert \mathbf{r}^{\prime}\right\rangle =\left\langle \mathbf{r}\right\vert G_{0}\left( z\right) \left\vert \mathbf{r}^{\prime}\right\rangle +\left\langle \mathbf{r}\right\vert G_{0}H_{1}G\left\vert \mathbf{r}^{\prime }\right\rangle \end{equation}

即

\begin{equation} G\left( \mathbf{r},\mathbf{r}^{\prime};z\right) =G_{0}\left( \mathbf{r} ,\mathbf{r}^{\prime};z\right) \\ +\int d\mathbf{r}_{1}d\mathbf{r}_{2} G_{0}\left( \mathbf{r},\mathbf{r}_{1};z\right) H_{1}\left( \mathbf{r} _{1},\mathbf{r}_{2}\right) G\left( \mathbf{r}_{2},\mathbf{r}^{\prime };z\right) \end{equation}

其中 $H_{1}$ 是单体格林算符

\begin{equation} H_{1}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) =\left\langle \mathbf{r}_{1}\right\vert H_{1}\left\vert \mathbf{r}_{2}\right\rangle =V\left( \mathbf{r}_{1}\right) \mathbf{\delta}\left( \mathbf{r} _{1}\mathbf{-r}_{2}\right) \end{equation}

格林函数(42)可以写为

\begin{align*} G\left( \mathbf{r},\mathbf{r}^{\prime};z\right) & =G_{0}\left( \mathbf{r},\mathbf{r}^{\prime};z\right) +\int d\mathbf{r}_{1}d\mathbf{r} _{2}G_{0}\left( \mathbf{r},\mathbf{r}_{1};z\right) V\left( \mathbf{r} \right) \mathbf{\delta}\left( \mathbf{r}_{1}\mathbf{-r}_{2}\right) G\left( \mathbf{r}_{2},\mathbf{r}^{\prime};z\right) \\ & =G_{0}\left( \mathbf{r},\mathbf{r}^{\prime};z\right) +\int d\mathbf{r} _{1}G_{0}\left( \mathbf{r},\mathbf{r}_{1};z\right) V\left( \mathbf{r} _{1}\right) G\left( \mathbf{r}_{1},\mathbf{r}^{\prime};z\right) \end{align*}

波函数的表达

由完备关系，将式(24)(25)改写为

\begin{equation} \psi\left( \mathbf{r}\right) =\varphi_{n}\left( r\right) +\int d\mathbf{r}_{1}d\mathbf{r}_{2}G_{0}\left( \mathbf{r},\mathbf{r}_{1} ;E_{n}\right) H_{1}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) \psi\left( \mathbf{r}_{2}\right) \end{equation} \begin{equation} \psi\left( \mathbf{r}\right) =\varphi_{n}\left( \mathbf{r}\right) +\int d\mathbf{r}_{1}d\mathbf{r}_{2}G_{0}\left( \mathbf{r,r}_{1};E_{n}\right) T\left( \mathbf{r}_{1}\mathbf{,r}_{2}\mathbf{;}E_{n}\right) \varphi _{n}\left( \mathbf{r}_{2}\right) \end{equation}

当能级处于连续谱，只能使用侧极限

\begin{equation} \psi^{\pm}\left( \mathbf{r}\right) =\varphi_{n}\left( r\right) +\int d\mathbf{r}_{1}d\mathbf{r}_{2}G_{0}^{\pm}\left( \mathbf{r},\mathbf{r} _{1};E_{n}\right) H_{1}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) \psi^{\pm}\left( \mathbf{r}_{2}\right) \end{equation} \begin{equation} \psi^{\pm}\left( \mathbf{r}\right) =\varphi_{n}\left( \mathbf{r}\right) +\int d\mathbf{r}_{1}d\mathbf{r}_{2}G_{0}^{\pm}\left( \mathbf{r,r}_{1} ;E_{n}\right) T^{\pm}\left( \mathbf{r}_{1}\mathbf{,r}_{2}\mathbf{;} E_{n}\right) \varphi_{n}\left( \mathbf{r}_{2}\right) \end{equation}

用于求能级为 $H_{0}$ 的连续谱的波函数^[4]。

态密度

态密度，单位能量间隔内的状态数目^[5]。

由态密度的定义(27)可得

\begin{align} \rho\left( E\right) & =\sum_{n}\delta\left( E-E_{n}\right) \mathbf{1}\nonumber\\ & =\sum_{n}\delta\left( E-E_{n}\right) \int_{V}\sum_{m}\varphi_{m}\left( \mathbf{r}\right) \varphi_{m}^{\ast}\left( \mathbf{r}^{\prime}\right) d\mathbf{r}^{\prime}\nonumber\\ & =\int_{V}\sum_{n}\delta\left( E-E_{n}\right) \varphi_{n}\left( \mathbf{r}\right) \varphi_{n}^{\ast}\left( \mathbf{r}^{\prime}\right) d\mathbf{r}^{\prime}\\ & =\int_{V}\rho\left( \mathbf{r};E\right) d\mathbf{r} \end{align}

其中

\begin{equation} \rho\left( \mathbf{r};E\right) =\sum_{n}\delta\left( E-E_{n}\right) \varphi_{n}\left( \mathbf{r}\right) \varphi_{n}^{\ast}\left( \mathbf{r} ^{\prime}\right) \end{equation}

我们要用格林函数来表示态密度,对式(40)取对角元可知

\begin{align} G^{\pm}\left( \mathbf{r},\mathbf{r};E\right) & =G\left( \mathbf{r} ,\mathbf{r};E\pm i0^{+}\right) =\sum_{n}\frac{\varphi_{n}\left( \mathbf{r}\right) \varphi_{n}^{\ast}\left( \mathbf{r}\right) }{E\pm i0^{+}-E_{n}}\nonumber\\ & =\sum_{n}\varphi_{n}\left( \mathbf{r}\right) \varphi_{n}^{\ast}\left( \mathbf{r}\right) \left[ P\frac{1}{E-E_{n}}\mp i\pi\delta\left( E-E_{n}\right) \right] \end{align}

与式(50)对照^[6]

\begin{equation} \rho\left( \mathbf{r};E\right) =\mp\frac{1}{\pi}\operatorname{Im}G^{\pm }\left( \mathbf{r},\mathbf{r};E\right) \end{equation}

对全空间积分可得

\begin{equation} \rho\left( E\right) =\mp\frac{1}{\pi}\int d\mathbf{r}\operatorname{Im} G^{\pm}\left( \mathbf{r},\mathbf{r};E\right) \end{equation}

从 $G_{0}$ 与 $H_{1}$ 的对易关系也可以看出 ↩︎
求和是对态求和，相当于求 $E=E_{n}$ 的态有多少个，即简并度 ↩︎
Q：对所有的单体算符都成立吗？ ↩︎
$T^{\pm}$ 由 $G^{\pm}$ 决定 ↩︎
态密度是相对连续能谱的 ↩︎
$G^{+}$ 与 $G^{-}$ 互为共轭; $\rho\left( \mathbf{r},E\right)$ 称为局域态密度 ↩︎