薛定谔方程¶

一维自由粒子的薛定谔方程¶

一维自由粒子的波函数

\[ \Psi(x,t) = \Psi_0 e^{\frac{i}{\hbar}(p_x x - E t)} \]

一维自由粒子的薛定谔方程：

\[ i\hbar\frac{\partial}{\partial t} \Psi(x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi (x,t) \]

能量算符：\(\hat E \equiv i\hbar \frac{\partial}{\partial t}\)

动量算符：\(\hat p_x \equiv -i\hbar \frac{\partial}{\partial x}\)

坐标算符：\(\hat x \equiv x\)

满足本征方程：

\[ \hat E \Psi(x,t) = E \Psi(x,t) \]

\[ \hat p_x \Psi(x,t) = p_x \Psi(x,t) \]

\[ \hat x \Psi(x,t) = x \Psi(x,t) \]

\(E = \frac{p^2}{2m} + U(x,t)\) → \(\hat E = \frac{\hat p_x}{2m} U(x,t)\)

算符等式：

\[ i \hbar \frac{\partial}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + U \]

一维势场中的薛定谔方程：

\[ i \hbar \frac{\partial}{\partial t}\Psi = [- \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + U]\Psi \]

三个维度下：

\[ i\hbar \frac{\partial \Psi}{\partial t} = [- \frac{\hbar^2}{2m} \nabla^2 + U(\vec r, t)]\Psi \]

哈密顿算符：\(\hat H = - \frac{\hbar^2}{2m}\nabla^2 + U(\vec r,t)\)，代表粒子的总能量

非定态的薛定谔方程：

\[ i\hbar \frac{\partial \Psi}{\partial t} = \hat H \Psi \]

定态，\(U\) 与 \(t\) 无关时：

定态薛定谔方程：

\[ E \Phi(\vec r) = [-\frac{\hbar^2}{2m}\nabla^2 +U(\vec r)] \Phi(\vec r) \]

一维自由粒子，定态薛定谔方程：

\[ -\frac{\hbar^2}{2m} \cdot \frac{\text d^2 \Phi(x)}{\text d x^2} = E\Phi(x) \]

根据势能列出薛定谔方程
- \(|x| > \frac{a}{2} \to U(x) = \infty, \Phi = 0\)
- \(|x| \leq \frac{a}{2} \to U(x) = 0, \hat H = - \frac{\hbar^2}{2m} \cdot \frac{\text d^2}{\text d x^2}\)
  - 根据 \(\hat H \Phi = E \Phi \implies \frac{d^2 \Phi}{dx^2} = -\frac{2mE}{\hbar^2} \Phi = -k^2 \Phi\)
解出薛定谔方程 \(\Phi(x) = A \sin(kx + \varphi)\)
根据单值、有限、连续条件给出具体解
- \(\Phi(-a/2) = 0 \implies A \sin(-ka/2+\varphi) = 0\)
- \(\Phi(a/2) = 0 \implies A \sin(ka/2+\varphi) = 0\)
- 解得：\(\varphi = l \frac{\pi}{2}, l \in \mathbb N\)
归一化得到系数

根据不确定性，动能 \(E > 0 \implies k = \sqrt{2mE} / \hbar > 0\)

根据边值，解得 \(ka = n\pi, n \in \mathbb N^+\)。

则能量量子化，\(E_n = \frac{\pi^2 \hbar^2}{2ma^2} n^2 , n \in \mathbb N^+\)

根据德布罗意关系：\(\lambda_n = \frac{2a}{n}\)，是驻波，边界为波节

\[ U(x) = \begin{cases}0, &x < 0\\ U_0, &x>0\end{cases} \]

\[ \Psi(x) = \begin{cases} Ae^{ik_1x} + Be^{-ik_1x},& x < 0\\ C e^{-k_2x},& x>0\end{cases} \]

在 \(x > 0\) 处的概率密度：

\[ |\Psi(x)|^2 \propto e^{-\frac{2x}{\hbar} \sqrt{2m(U_0-E)}} \]

随着 \(x\) 增大，概率指数衰减

穿透系数：

\[ T \propto e^{-\frac{2a}{\hbar} \sqrt{2m(U_0-E)}} \]

概率密度的特点