正交多项式与一维谐振子
2024-9-28
正交多项式
厄密多项式
权重函数 \(e^{-x^2}\)，积分区间 \((-\infty,\infty)\). 正交多项式的微分表示似乎被称为 Rodrigues representation.
$$\begin{equation} \begin{split} & H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2} \\ & \int_{-\infty}^\infty H_m H_n e^{-x^2}dx = \delta_{mn}2^n n!\sqrt{\pi} \\ & H_n(-x) = (-1)^n H_n(x) \\ & H_{n+1}(x) = 2xH_n(x) -2nH_{n-1}(x) \\ & H^\prime(x) = 2n H_{n-1}(x) \\ \end{split} \end{equation}$$
勒让德多项式
权重函数 \(1\), 正交区间\([−1, 1]\).
$$\begin{equation} \begin{split} & P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n \\ & P_n(x) = \sum_{k=0}^N \frac{(-1)^k (2n-2k)!}{2^n k! (n-k)! (n-2k)!} x^{n-2k},\:\:N=\begin{cases}\frac{n}{2},\:\:\text{n is even}\\\frac{n-1}{2},\:\:\text{n is odd}\end{cases} \\ & \int_{-1}^1 P_n P_m dx = \frac{2}{2n+1}\delta_{nm} \\ & P_n(x) = \frac{1}{2\pi i 2^n}\oint_C \frac{(z^2-1)^n}{(z-x)^{n+1}} dz,\:\:\text{Schlaefli integral} \\ & (2n+1)x P_n(x) - nP_{n-1}(x) = (n+1)P_{n+1}(x) \\ \end{split} \end{equation}$$
连带勒让德多项式
$$\begin{equation} \begin{split} & P_l^m(x) = (-1)^m (1-x^2)^{\frac{m}{2}}\frac{d^m}{dx^m}P_l(x) \\ & P_l^m(x) = \frac{(-1)^m}{2^l l!}(1-x^2)^{\frac{m}{2}}\frac{d^{l+m}}{dx^{l+m}} (x^2-1)^l \\ & P_l^0(x) = P_l(x) \\ & (l-m)P_l^m(x) = x(2l-1)P_{l-1}^m(x) - (l+m-1)P_{l-2}^m(x) \\ & \int_{-1}^1 P_l^m P_{l^\prime}^m dx = \frac{2}{2l+1}\frac{(l+m)!}{(l-m)!}\delta_{ll^\prime} \\ & \int_{-1}^{1} P_l^m P_l^{m^\prime} \frac{dx}{1-x^2} = \frac{(l+m)!}{m(l-m)!}\delta_{mm^\prime} \\ \end{split} \end{equation}$$
连带勒让德多项式在上指标为 0 时退化为勒让德多项式. 连带勒让德多项是球谐函数的一部分, 球谐函数是球坐标下的拉普拉斯方程的解
$$\begin{equation} \begin{split} & Y_l^m(\theta,\phi) = (-1)^m \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos{\theta}) e^{im\phi} \\ & \int_{0}^\pi \int_0^{2\pi} (Y_l^m)^\ast Y_{l^\prime}^{m^\prime} \sin{\theta}d\theta d\phi = \delta_{ll^\prime}\delta_{mm^\prime} \\ \end{split} \end{equation}$$
拉瓜尔多项式
权重函数 $e^{-x}$, 正交区间 $[0,\infty)$ $$\begin{equation} \begin{split} & L_n(x) = \frac{e^x}{n!}\frac{d^n}{dx^n}\left(x^n e^{-n}\right) \\ & (n+1) L_{n+1}(x) = (2n+1-x)L_n(x) -nL_{n-1}(x) \\ & xL^\prime_{n}(x) = nL_n(x) -nL_{n-1}(x) \\ & \int_0^\infty L_n(x)L_{n^\prime}e^{-x} dx = \delta_{nn^\prime} \\ \end{split} \end{equation}$$
连带拉瓜尔多项式
权重函数 $x^k e^{-x}$, 正交区间 $[0,\infty)$
$$\begin{equation} \begin{split} & L_n^k(x) = \frac{e^x x^{-k}}{n!}\frac{d^n}{dx^n}\left(e^{-x}x^{n+k}\right) = (-1)^k \frac{d^k}{dx^k}L_{n+k}(x) \\ & \int_0^\infty L_n^k L_{m}^k x^k e^{-x} dx = \frac{(n+k)!}{n!}\delta_{mn} \\ \end{split} \end{equation}$$
连带拉瓜尔多项式对于不同的上指标 \(k\) 不正交. 因此类氢原子径向函数 $R_{nl}(\vec{\boldsymbol{r}})$ 对于不同的角量子数 $l$ 不正交, 正交性需要在相同的上指标情况下讨论. 单数连带勒让德多项式对于不同的上指标与下指标都是正交的.
一维谐振子
$V(x)=\frac{k}{2}x^2=\frac{m}{2}\omega^2x^2$, $\omega=\sqrt{\frac{k}{m}}$. 作变量代换
$$\begin{cases} \xi = \sqrt{\frac{m\omega}{\hbar}}x = \alpha x \\ \alpha = \sqrt{\frac{m\omega}{\hbar}} \\ \lambda = \frac{2\epsilon}{\hbar\omega} \\ \end{cases}$$
$$\begin{equation} \begin{split} & \frac{\hbar^2}{2m}\frac{d^2}{dx^2}f+\left(\epsilon-\frac{m\omega^2}{2}x^2\right)f = 0 \\ & \frac{d^2}{d\xi^2}f + \left(\lambda-\xi^2\right)f = 0 \\ \end{split} \end{equation}$$
特征函数与特征值为 ($H_n$ 为厄密多项式)
$$\begin{equation} \begin{split} & \epsilon_n = \hbar\omega\left(n+\frac{1}{2}\right) \\ & f_n(\xi) = \sqrt{\frac{\alpha}{\pi^{1/2}2^n n!}}e^{-\frac{\xi^2}{2}} H_n(\xi) \\ & f_n(x) = \sqrt{\frac{\alpha}{\pi^{1/2} 2^n n!}}e^{-\frac{\alpha^2}{2}x^2} H_n(\alpha x) \\ & \xi f_n(\xi) = \sqrt{\frac{n}{2}} f_{n-1}(\xi) + \sqrt{\frac{n+1}{2}} f_{n+1}(\xi) \\ & \frac{d}{d\xi} f_n(\xi) = \sqrt{\frac{n}{2}} f_{n-1}(\xi) - \sqrt{\frac{n+1}{2}} f_{n+1}(\xi) \\ \end{split} \end{equation}$$