线性谐振子在坐标表象中表示

哈密顿算符

$\hat{H}=\hat{p}^2/(2\mu)+\mu\omega^2x^2/2$

定态薛定谔方程

$\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2}\psi+(E-\frac{\mu \omega^2x^2}{2})\psi=0$

渐近解

做代换 $\zeta=\sqrt{\mu\omega/\hbar}x\equiv\alpha,\alpha=\sqrt{\mu\omega/\hbar},\lambda\equiv2E/\hbar\omega$
使方程变为 $\frac{d^2}{d\zeta^2}\psi(\zeta)+(\lambda-\zeta^2)\psi(\zeta)=0$
在\zeta \approx \pm \infty 时，有渐进解

$\psi(\zeta)=e^{-\zeta^2/2}H(\zeta)$

$\begin{matrix} H(\zeta)=\sum_{v=0}^{\infty}a_v\zeta^v\\ a_{v+2}=\frac{(2v-\lambda+1)}{(v+1)(v+2)}a_v \end{matrix}$

两端条件要求阶段，因此有

$\begin{matrix} \lambda = 2n+1,n=0,1,2,...\\ \frac{d^2}{d\zeta^2}H_n-2\zeta\frac{d}{dz}H_n+2nH_n=0\\ H_n(\zeta)=(-1)^ne^{\zeta^2}\frac{d^n}{d\zeta^n}(e^{-\zeta^2})\\ E_n=(n+\frac{1}{2})\hbar\omega,n=0,1,2... \end{matrix}$

则线性谐振子的定态波函数为：

$\begin{matrix} \psi_n(\zeta)=N_ne^{-\zeta^2/2}H_n(\zeta)\\ N_n=(\alpha/\pi^{1/2}2^\pi n!)^{1/2} \end{matrix}$

粒子数表象中线性谐振子能级和波函数

引入算符与其共轭算符

$\begin{matrix} a=(\frac{\mu\omega}{2\hbar})^{1/2}(\hat{x}+\frac{i}{\mu}\hat{p})\\ a^+=(\frac{\mu\omega}{2\hbar})^{1/2}(\hat{x}-\frac{i}{\mu}\hat{p}) \end{matrix}$

基本对易关系 $[x,p]=i\hbar$
消灭与产生算符 $[a,a^+]=1$
消灭与产生算符构成粒子数算符
- 粒子数算符的本征值为0,1,2,3....,N的本征表象为粒子数表象

粒子数表象

升算符与降算符(raising operator and lowering operator)

$\begin{matrix} \hat{a}^+|n\rangle=\sqrt{n+1}|n+1\rangle\\ \hat{a}|n\rangle=\sqrt{n}|n-1\rangle\\ a|0\rangle=0 \end{matrix}$

求矩阵元

$\begin{matrix} \langle n|\hat{a}|n\rangle=\sqrt{n}\delta_{n,n-1}\\ \langle n|\hat{a}^+|n\rangle=\sqrt{n+1}\delta_{n,n+1} \end{matrix}$

$a=\begin{bmatrix} 0 &\sqrt{1} &0 &0 &\cdots \\ 0& 0 &\sqrt{2} & 0 &\cdots \\ 0& 0 &0 &\sqrt{3} &\cdots \\ 0&0 &0 & 0 & \cdots\\ \vdots & \vdots & \vdots & \vdots &\vdots \end{bmatrix}$

$a^+= \begin{bmatrix} 0 & 0 &0 & 0 &\cdots \\ \sqrt{1} &0 &0 &0 & \cdots\\ 0 & \sqrt{2} & 0 & 0 & \cdots \\ 0 & 0 & \sqrt{3} &0 & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots \end{bmatrix}$

坐标算符与动量算符的改写

$\begin{matrix} \hat{x}=(\frac{\hbar}{2\mu\omega})^{1/2}(\hat{a}+\hat{a}^+)\\ \hat{p}=i(\frac{\hbar\mu}{2\omega})^{1/2}(\hat{a}^+-\hat{a}) \end{matrix}$