17 物质波¶
文本统计：约 1567 个字
17.1 波函数概述¶
物质波沿着 \(x\) 正方向传播
\[
\Psi(x, t) = \psi_0 e^{i(kx - \omega t)} = \psi_0 e^{\frac{i}{\hbar}(px - Et)}
\]
其中 \(\Psi_0\) 为波的振幅，第二个等号的公式的依据是
\[
\begin{aligned}
\hbar \omega = h \nu = E\\
\hbar k = \frac{h}{\lambda} = p
\end{aligned}
\]
波函数的物理解释¶
The product \(\Psi\Psi^*\text{d}x\) gives the probability that the particle in question will be found between positions \(x\) and \(x+\text{d}x\).
- Probability density \(P(x)\)
\[
P(x) = \Psi\Psi^*
\]
- The probability in range from \(x_1\) to \(x_2\), 在 \(x_1\) 和 \(x_2\) 之间找到的几率
\[
\int_{x_1}^{x_2} P(x) \text{d}x = \int_{x_1}^{x_2} \Psi\Psi^* \text{d}x
\]
- Normalization (归一化)
\[
\int_{-\infty}^{\infty} P(x) \text{d}x = \int_{-\infty}^{\infty} \Psi\Psi^* \text{d}x = 1
\]
自由粒子的概率密度，对于一个自由粒子，概率密度为：
\[
P(x) = [ \psi_0 e^{i(kx - \omega t)} ][ \psi_0^* e^{-i(kx - \omega t)} ] = \left| \psi_0 \right|^2
\]
Note
对于自由粒子来说，每一处的概率均为常数，物理意义就是该粒子在空间中的每一个位置被找到的概率相同，又由归一化条件，我认为这里的 \(\psi_0^2\) 非常小
预期值¶
预期值的意思就是某一个变量的期望
\[
\bar{x} = \frac{\int_{-\infty}^{\infty} x \Psi^* \Psi \, dx}{\int_{-\infty}^{\infty} \Psi^* \Psi \, dx} = \int_{-\infty}^{\infty} \Psi^* x \Psi \, dx = \langle \psi | x | \psi \rangle = \langle |x| \rangle,
(\int_{-\infty}^{\infty} \Psi^* \Psi \, dx = 1)
\]
The average value of the potential energy \(U(x)\) is
\[
\bar{U} = \int_{-\infty}^{\infty} \Psi^* U(x) \Psi \, dx = \langle \psi | U(x) | \psi \rangle
\]
17.2 动量算符与能量算符¶
动量算符¶
Consider the free particle wave-function
\[
\Psi = \psi_0 e^{i(kx - \omega t)}, \quad p = \hbar k, \quad E = \hbar \omega,
\]
\[
\frac{\partial \Psi}{\partial x} = ik \psi_0 e^{i(kx - \omega t)}
\]
\[
-i\hbar \frac{\partial \Psi}{\partial x} = (-i\hbar)ik \psi_0 e^{i(kx - \omega t)} = \hbar k \psi_0 e^{i(kx - \omega t)}
\]
\[
\therefore -i\hbar \frac{\partial \Psi}{\partial x} = p \psi_0 e^{i(kx - \omega t)} = p \Psi
\]
We can see there is an association between the dynamical quantity \(p\) and the operator.
\[
p \leftrightarrow -i\hbar \frac{\partial}{\partial x}
\]
计算平均动量 \(\overline{p}\)
\[
\therefore \bar{p} = \int_{-\infty}^{\infty} \Psi^* \left( -i\hbar \frac{\partial}{\partial x} \right) \Psi \, dx
\]
\[
\begin{aligned}
\overline{p} &= \int_{-\infty}^{\infty} \Psi^* \left( -i\hbar \frac{\partial}{\partial x} \right) \psi_0 e^{i(kx - \omega t)} \, \text{d}x\\
&= \int_{-\infty}^{\infty} \Psi^* \left( -i\hbar ik \psi_0 e^{i(kx - \omega t)} \right) \, \text{d}x = \hbar k \int_{-\infty}^{\infty} \Psi^* \psi_0 e^{i(kx - \omega t)} \, \text{d}x\\
&= \hbar k \int_{-\infty}^{\infty} \Psi^* \Psi \, \text{d}x = \hbar k
\end{aligned}
\]
能量算符¶
运用同样的方法，我们可以得到相应的能量算符
\[
\begin{aligned}
&\frac{\partial \Psi}{\partial t} = -i\omega \psi_0 e^{i(kx - \omega t)}\\
&i\hbar \frac{\partial \Psi}{\partial t} = (i\hbar)(-i\omega) \psi_0 e^{i(kx - \omega t)} = \hbar \omega \psi_0 e^{i(kx - \omega t)} = E \Psi
\end{aligned}
\]
相应的能量算符为
\[
E \leftrightarrow i\hbar \frac{\partial}{\partial t}
\]
计算平均能量 \(\overline{E}\)
\[
\begin{aligned}
\overline{E} &= \int_{-\infty}^{\infty} \Psi^* \left( i\hbar \frac{\partial}{\partial t} \right) \Psi \, \text{d}x = \int_{-\infty}^{\infty} \Psi^* \left( i\hbar \frac{\partial}{\partial t} \right) \psi_0 e^{i(kx - \omega t)} \, \text{d}x\\
&= \int_{-\infty}^{\infty} \Psi^* \left( i\hbar (-i\omega) \psi_0 e^{i(kx - \omega t)} \right) \, \text{d}x = \hbar \omega \int_{-\infty}^{\infty} \Psi^* \psi_0 e^{i(kx - \omega t)} \, \text{d}x\\
&= \hbar \omega \int_{-\infty}^{\infty} \Psi^* \Psi \, \text{d}x = \hbar \omega
\end{aligned}
\]
17.3 薛定谔方程¶
Schrödinger’s Equation tell us how to obtain the wave-function \(\Psi(x,t)\) associated with a particle that is not free.
一个物体的总体能量为
\[
E=\frac12 mv^2+U=\frac{p^2}{2m}+U
\]
\[
\begin{aligned}
&E \Psi = \frac{p^2}{2m} \Psi + U \Psi\\
&i \hbar \frac{\partial \Psi}{\partial t} = \frac{1}{2m} (-i \hbar \frac{\partial}{\partial x})(-i \hbar \frac{\partial}{\partial x}) \Psi + U \Psi\\
&i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + U \Psi
\end{aligned}
\]
一维含时薛定谔方程
\[
i \hbar \frac{\partial \Psi(x, t)}{\partial t} = \left[ -\frac{\hbar^2}{2m} \frac{\text{d}^2}{\text{d}x^2} + U(x, t) \right] \Psi(x, t)
\]
\[
i \hbar \frac{\partial \Psi(x, t)}{\partial t} = \hat{H} \Psi(x, t)
\]
- 一维(1D):
\[
\hat{H} = -\frac{\hbar^2}{2m} \frac{\text{d}^2}{\text{d}x^2} + U(x, t)
\]
- 三维(3D):
\[
\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + U(x, t)
\]
\(\hat{H}\) is energy operator (能量算符).
如果势能 \(U\) 是已知的，这个方程可以在原则上解出来，并且解出可能与粒子相关的波函数。
对于自由粒子来说, \(U(x) = 0\)
\[
i \hbar \frac{\partial \Psi(x, t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\text{d}^2 \Psi(x, t)}{\text{d}x^2}
\]
如果波函数为 \(\Psi = A \cos(kx - \omega t)\) 或者 \(\Psi = A \sin(kx - \omega t)\)
\[
\frac{\partial \Psi}{\partial t} = -\omega A \sin(kx - \omega t)
\]
\[
\frac{\partial^2 \Psi}{\partial x^2} = -k^2 A \cos(kx - \omega t)
\]
显然两者都不满足Schrödinger方程。然而，这两个函数的特定组合确实满足Schrödinger方程。
\[
\Psi = A \cos(kx - \omega t) + Ai \sin(kx - \omega t)= Ae^{i(kx - \omega t)}
\]
\[
\begin{aligned}
\frac{\partial^2 \Psi}{\partial x^2} = -k^2 Ae^{i(kx - \omega t)}\\
\frac{\partial \Psi}{\partial t} = -i\omega Ae^{i(kx - \omega t)}
\end{aligned}
\]
带入薛定谔方程，我们得到
\[
\begin{aligned}
&\frac{\hbar^2 k^2}{2m} Ae^{i(kx - \omega t)} = \hbar \omega Ae^{i(kx - \omega t)}\\
&\frac{\hbar^2 k^2}{2m} = \hbar \omega
\end{aligned}
\]
这样我们得到了能量与动量关系
\[
E=\frac{p^2}{2m}
\]
定态薛定谔方程¶
If the potential independent of \(t\): \(U(x)\)
\[
i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + U \Psi
\]
如果势能 \(U\) 与 \(t\) 无关，则可以将波函数 \(\Psi(x,t)\) 分解为位置依赖部分 \(\psi(x)\) 和时间依赖 \(e^{-i\omega t}\) 的乘积
\[
\Psi(x, t) = \psi(x) e^{-i \omega t}
\]
将波函数带入到薛定谔方程中
\[
\begin{aligned}
i \hbar \frac{\partial}{\partial t} (\psi e^{-i \omega t}) &= -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} (\psi e^{-i \omega t}) + U \psi e^{-i \omega t}\\
(-i \omega) i \hbar \psi e^{-i \omega t} &= -\frac{\hbar^2}{2m} \frac{\text{d}^2 \psi}{\text{d}x^2} e^{-i \omega t} + U \psi e^{-i \omega t}\\
\hbar \omega \psi &= -\frac{\hbar^2}{2m} \frac{\text{d}^2 \psi}{\text{d}x^2} + U \psi
\end{aligned}
\]
这样我们得到了定态薛定谔方程
\[
-\frac{\hbar^2}{2m} \frac{\text{d}^2 \psi(x)}{\text{d}x^2} + U(x) \psi(x) = E \psi(x)
\]
其中 \(E\) 为物体的总能量，\(U(x)\) 为相应的势能
\[
\frac{\text{d}^2 \psi(x)}{\text{d}x^2} + \frac{2m}{\hbar^2} (E - U) \psi(x) = 0
\]
结合之前的 \(\hat{H}\) 算符，我们可以得到
\[
\hat{H} \psi = E \psi
\]
特征方程：
\[
\hat{H} \psi_n = E_n \psi_n
\]
其中 \(E_n\) 是特征值（能量本征值），\(\psi_n\) 是特征函数（波函数）。
特征值 (Eigenvalue):
- 特征值 \(E_n\) 表示系统的能量水平。
特征函数 (Eigenfunction):
- 特征函数 \(\psi_n\) 描述了对应于特定能量水平 \(E_n\) 的波函数。
特征态 (Eigenstate):
- 特征态是由特征函数 \(\psi_n\) 描述的状态。
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