I asked this problem in Physics SE but I did not get any useful answers except one. I believe asking this question here would be more beneficial owing to the Mathematical nature of the problem.
I am working on a problem which looks like this.
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 $+ \infty$.
My approach to the problem: 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 : $-(\hbar^2/2m) \ d^2\psi/dx^2 + qFx =E\psi $
The Schrodinger equation for region $I$ & $III$is : $-(\hbar^2/2m) \ d^2\psi/dx^2 + qFx +V_0=E\psi $
$\hbar = h/2\pi$
Solutions :
Region $I$ : $\psi(x) = c_4 A_i[\alpha(V_0 + Fqx - E_n)] + c_5 B_i[\alpha(V_0 + Fqx - E_n)]$
Region $II$ : $\psi(x) = c_1 A_i[\alpha( Fqx - E_n)] + c_2 B_i[\alpha(Fqx - E_n)]$
Region $III$ : $\psi(x) = c_3 A_i[\alpha( V_0 + Fqx - E_n)]$ (The $Bi$ part is excluded because it blows up on $+\infty$.)
where $c_1 , c_2 , c_3 , c_4 , c_5$ are constants, $\alpha = (2^{1/3}m/\hbar^2 (Fmq/\hbar^2)^{2/3})$, $A_i$ and $B_i$ are Airy functions of first and second kind respectively, and $E_n$ are energy eigenvalues.
Applying the boundary conditions gives the following four equations:
$ c_4 A_i[\alpha(V_0 - E_n)] + c_5 B_i[\alpha(V_0 - E_n)] = c_1 A_i[\alpha( - E_n)] + c_2 B_i[\alpha(- E_n)]$
$ c_4 A_i'[\alpha(V_0 - E_n)] + c_5 B_i'[\alpha(V_0 - E_n)] = c_1 A_i'[\alpha( - E_n)] + c_2 B_i'[\alpha(- E_n)]$
$c_1 A_i[\alpha( FqL - E_n)] + c_2 B_i[\alpha(FqL - E_n)] = c_3 A_i[\alpha( V_0 + FqL - E_n)]$
$c_1 A_i'[\alpha( FqL - E_n)] + c_2 B_i'[\alpha(FqL - E_n)] = c_3 A_i'[\alpha( V_0 + FqL - E_n)]$
My query is 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.