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.
Best Answer
First, since the potential is unbounded from below at the left side, the particle has continuous spectrum. This means that what you just have to do is compute the coefficients $c_i$, taking $E$ (real, never complex!) as input parameter.
One of these coefficients, $c_3$, is in fact arbitrary, because it only influences normalization, not smoothness of wavefunction. Thus, to compute the connection coefficients you have to find $c_1,c_2,c_4,c_5$.
And finally, your equation is a simple system of $4$ algebraic equations with $4$ unknowns. If I take $c_3=1$ and use arbitrary letters to denote all those Airy functions and their derivatives, I get $$\left\{\begin{align} c_4A+c_5B&=c_1C+c_2D,\\ c_4G+c_5H&=c_1I+c_2J,\\ c_1K+c_2L&=M,\\ c_1N+c_2O&=P. \end{align}\right.$$
Solve it for $c_i$ and you've solved almost the whole of your problem. Now if you need normalized wavefunctions, you have to use some scheme for normalization of unbounded states, e.g. so called "normalization by Dirac delta", discussed e.g. in [1].
References: