Solving for a particle in a triangular well

Question

I am currently trying to solve the problem of a particle confined within an infinite well subject to a linear electric field (i.e. triangular well). This entails solving the Schrodinger equation $$-\frac{\hbar^2}{2m}\partial^2_x\psi_n(x) + (V_l(x) +V_w(x)) \psi_n(x) = E_n \psi_n(x)$$ where $V_l(x) = q \varepsilon x $ (with $q$ and $\varepsilon$ being the elementary charge and amplitude of the electric field respectively) and $V_w(0>x>L) \to \infty$ (0 otherwise).

By defining the variable $z \equiv \left(\frac{2mq\varepsilon}{\hbar^2} \right)^{1/3}\left(x-\frac{E_n}{q\varepsilon} \right)$, the above Schrodinger equation (for $0<x<L$) becomes of the form

$$\partial_z^2\psi_n-z\psi_n=0.$$ The above has the general solution $\psi_n(z) = a Ai(z) + b Bi(z)$.

If the particle was not confined within a well (i.e. $V_w(x) = 0$ everywhere), one could set $b=0$ since $Bi(z)\to \infty$ as $z\to \infty$, and the wave function would not be normalizable otherwise. In this case, the energies $E_n$ can be easily found by solving for $Ai(z)=0$. After these are found, it is then easy to compute the normalization condition that would return an expression for $a$.

However, I am not sure that the argument $b=0$ can be made for the case in which the particle is indeed confined within an infinite well of finite length. This complicates the problem and I am not sure how I should proceed to find the energies $E_n$ and normalization constants $a$ and $b$. Can anyone help me with this?

Answer 1

Here's how I would approach it. I'll use units where $m = q\epsilon = \hbar = 1$ for notational simplicity.

First, use a temporary normalization where $a = 1$. We'll fix the overall normalization later.

The boundary conditions on your function are \begin{align*} x &= 0: && \text{Ai}(-E_n) + b \,\text{Bi}(-E_n) = 0 \\ x &= L: && \text{Ai}(L-E_n) + b \,\text{Bi}(L-E_n) = 0 \end{align*} This is a set of two equations and two unknowns in $b$ & $E_n$, and so should have a discrete solution set. It may be numerically hard to find, but it should exist. For example, here's what it looks like for $L = 4$; the blue contours represent the solutions to the first condition, while the yellow contours represent the solutions to the second condition.

We can see that these contours intersect at points corresponding to $E_n \approx 2.4, 4.5$, and $7.6$, with corresponding values of $b$. The values read off from the graphs can then be refined with numerical root-finding techniques.

Finally, once $b$ has been found, you would normalize the overall wavefunctions in the usual way. This will probably require numerical integration.

EDIT: A better parametrization scheme would be to set $a = c \cos \theta$ and $b = c \sin \theta,$ with $0 \leq \theta < \pi$. If you set $a = 1$ by fiat, then my original method risks missing solutions where $b \gg a$. The equations above then become a system in $\theta$ and $E_n$, and $c$ can be fixed by the normalization constraint once these values are found.