量子力学

量子力学期中作业：

-------------秦大粤【2014301020039】（物基一班）

(1)

首先，由

$$r=\sqrt{x^2+y^2+z^2}\tag 1\\ \theta=arctan\sqrt\frac{x^2+y^2}{z}\\ \varphi=arctan\frac xy$$
得
$$\Delta U=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial u}{\partial r})+\frac{1}{r^2sin\theta} \frac{\partial(sin\theta \partial u/\partial\theta)}{partial\theta}+\frac{1}{r^2sin^2\theta} \frac{\partial^2 u}{\partial \varphi^2}$$
球坐标系下亥姆霍兹方程为$\Delta u+\lambda u=0$
令$u(,r,\theta,\varphi)=R(r)Y(\theta,\varphi)$，得：
$$\frac1R \frac{d(r^2dR/dr)}{dr}+k^2r^2=-\frac1Y[\frac{1}{sin\theta} \frac{\partial(sin\theta \partial Y/\partial\theta)}{\partial\theta}+\frac{1}{sin^2\theta}\frac{\partial^2Y}{\partial\varphi^2}]$$
由于左边=右边=常数，令之为$l(l+1)$,有
$$\begin{align} r^2\frac{d^2R}{dr^2}+2r\frac{dR}{dr}+[k^2r^2-l(l+1)]R=0\tag{1}\\ \frac{1}{sin\theta}\frac{\partial(sin\theta\partial Y/\partial\theta)}{\partial\theta}+\frac{1}{sin^2\theta} \frac{\partial^2Y}{\partial\varphi^2}+l(l+1)Y=0\tag{2}\\ \end{align}$$
再令$Y(\theta,\varphi)=\Theta(\theta)\Phi(\varphi)$,带入（2)中
$$\begin{align} \Phi\prime\prime+m^2\Phi=0\space,m=0,1,2...\tag{}\\ \frac{1}{sin\theta} \frac{sin\theta d\Theta/d\theta}{d\theta}+[l(l+1)-\frac{m^2}{sin^2\theta}]\Theta=0\tag{3}\\ \end{align}$$
作变换$X=cos\theta,Y(x)=\Theta(\theta)$,(3)式变为：
$$(1-x^2)y\prime\prime-2xy\prime+[l(l+1)-\frac{m^2}{1-x^2}]y=0 \space\space\space\space\space\text{(关联勒让德方程)}$$

(2)求关联勒让德方程的解

先求$m=0$时的特解

$$\begin{align} （1-x^2)y\prime\prime-2xy\prime+l(l+1)y=0\tag{4}\\ 设特解为y=\sum_{k=0}^\infty C_kx^k,代入（4）中有\tag{}\\ (1-x^2)\sum_{k=2}^\infty k(k-1)C_kx^{k-2}-2x\sum_{k=1}^\infty kC_kx^{k-1}+l(l+1)\sum_{k=0}^{\infty}C_kx^k=0\tag{}\\ \end{align}$$

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不想写了直接上图