薛定谔方程的实数形式

$f_1=cos(kx-\omega t)$

$f_2=sin(kx-\omega t)$

$\frac{\partial f_1}{\partial t} = \omega \sin(kx - \omega t)$

$\frac{\partial^2 f_2}{\partial x^2} = -k^2 \cos(kx - \omega t)$

$\frac{\partial f_2}{\partial t} = -\omega \cos(kx - \omega t) $

$\frac{\partial^2 f_2}{\partial x^2} = -k^2 \sin(kx - \omega t)$

可得：

$k^2 \frac{\partial f_1}{\partial t}=-\omega \frac{\partial^2 f_2}{\partial x^2}$

$k^2 \frac{\partial f_2}{\partial t}=\omega \frac{\partial^2 f_1}{\partial x^2}$

即：

$\frac{\partial f_1}{\partial t} = -\frac{\omega}{k^2} \frac{\partial^2 f_2}{\partial x^2} = -\frac{\hbar}{2m} \frac{\partial^2 f_2}{\partial x^2} $

$\frac{\partial f_2}{\partial t} = \frac{\omega}{k^2} \frac{\partial^2 f_1}{\partial x^2} = \frac{\hbar}{2m} \frac{\partial^2 f_1}{\partial x^2}$

时间和空间部分可分离时：

$f_1 = \operatorname{Re}\psi \cos\omega t + \operatorname{Im}\psi \sin\omega t$

$f_2 = -\operatorname{Re}\psi \sin\omega t + \operatorname{Im}\psi \cos\omega t$

含势能时：

$\frac{\partial f_1}{\partial t} = -\frac{\hbar}{2m} \frac{\partial^2 f_2}{\partial x^2} + \frac{V}{\hbar} f_2$

$\frac{\partial f_2}{\partial t} = \frac{\hbar}{2m} \frac{\partial^2 f_1}{\partial x^2} - \frac{V}{\hbar} f_1$

通解为线性平面波的叠加：

$f_1 = \int_{-\infty}^{\infty} \left[ A(k) \cos(kx - \omega_k t) + B(k) \sin(kx - \omega_k t) \right] dk $

$f_2 = \int_{-\infty}^{\infty} \left[ A(k) \sin(kx - \omega_k t) - B(k) \cos(kx - \omega_k t) \right] dk$

$\omega_k = \frac{\hbar k^2}{2m}$

这等价于复波函数的傅里叶展开：

$\psi(x,t) = f_1 + i f_2 = \int_{-\infty}^{\infty} C(k) e^{i(kx - \omega_k t)} dk$

$C(k) = A(k) + i B(k)$

附电磁场：

$\frac{\partial E_y}{\partial x} = -\frac{\partial B_z}{\partial t} $

$\frac{\partial B_z}{\partial x} = -\mu_0 \varepsilon_0 \frac{\partial E_y}{\partial t}$

有源形式：

$\frac{\partial E}{\partial t} = -c^2 \frac{\partial B}{\partial x} - \frac{J}{\varepsilon_0}$

$\frac{\partial B}{\partial t} = -\frac{\partial E}{\partial x}$

慢变包络近似 + 外源：

对于光在介质中传播，考虑包络 $\mathcal{E}(x,y,z)$ 满足：

$i \frac{\partial \mathcal{E}}{\partial z} = -\frac{1}{2k} \nabla_\perp^2 \mathcal{E} + \frac{1}{2k} \rho_{\text{eff}}(x,y,z)$