Consider a charged particle with charge $q$ trapped in a box of length $L$ with finite constant potential $ V_0 $ on both ends. A constant (static) electric field of magnitude $F$ is applied from $- \infty $ to $+ \infty$.
I have divided the whole domain in three regions
- from $-\infty $ to $0$ as region I
- from $0 $ to $L$ as region II
- from $L$ to $+\infty$ as region III
Equations:
- The Schrodinger equation for region II is $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + qFx =E\psi \, .$$
- The Schrodinger equation for regions I and III is $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + qFx +V_0=E\psi \, .$$
Solutions:
Region I $$ \newcommand{\Ai}{\operatorname{Ai}} \newcommand{\Bi}{\operatorname{Bi}} \psi(x) = c_4 \Ai[\alpha(V_0 + Fqx - E_n)] + c_5 \Bi[\alpha(V_0 + Fqx - E_n)]$$
Region II $$\psi(x) = c_1 \Ai[\alpha( Fqx - E_n)] + c_2 \Bi[\alpha(Fqx - E_n)]$$
Region III $$\psi(x) = c_3 \Ai[\alpha( V_0 + Fqx - E_n)]$$ (The $\Bi$ part is excluded because it blows up on $+\infty$.)
where $c_1 , c_2 , c_3 , c_4 , c_5$ are constants, $\alpha = \left( 2^{1/3}m /\hbar^2 \right) \left(Fmq / \hbar^2 \right)^{2/3}$ , $\Ai$ and $\Bi$ are Airy functions of first and second kind respectively, and $E_n$ are energy eigenvalues.
Applying the boundary conditions gives the following four equations:
$$ \begin{align} c_4 \Ai[\alpha(V_0 - E_n)] + c_5 \Bi[\alpha(V_0 - E_n)] &= c_1 \Ai[\alpha( - E_n)] + c_2 \Bi[\alpha(- E_n)]\\ c_4 \Ai'[\alpha(V_0 - E_n)] + c_5 \Bi'[\alpha(V_0 - E_n)] &= c_1 \Ai'[\alpha( - E_n)] + c_2 \Bi'[\alpha(- E_n)]\\ c_1 \Ai[\alpha( FqL - E_n)] + c_2 \Bi[\alpha(FqL - E_n)] &= c_3 \Ai[\alpha( V_0 + FqL - E_n)]\\ c_1 \Ai'[\alpha( FqL - E_n)] + c_2 \Bi'[\alpha(FqL - E_n)] &= c_3 \Ai'[\alpha( V_0 + FqL - E_n)] \end{align} $$
How do I calculate the bound states $E_n$ from these equations? Also, I am bogged down by the fact that on both ends of the box $\psi$ behaves differently from what is seen in trivial problems? Also, I can use computational software like MATLAB, so if someone can help me with the computational technique to find $E_n$, that is perfectly fine.