Total Internal Reflection for Electron waves

Idea: Hamilton's opto-mechanical analogy makes it plausible that Snell's law can apply to `rays of electrons'.

Application: Build waveguides for electrons based on total internal reflection of de Broglie matter waves
Procedure: 

1. Take into account angle of incidence of electron beam in solving Schrodinger's equation for rectangular potential barrier.
2. Solve Schrodinger's equation for two unequal potential barriers.

$$V(x)=\begin{cases}V_1,  x>0 \\V_2,  x<0\end{cases}$$

$$\theta_c=\arcsin\sqrt{\frac{E-V_2}{E-V_1}}$$