I'm a beginner in both quantum mechanics and computational physics, So please be as simple as possible. I know these topics can become so complicated.
I've been trying to solve the Schrödinger equation for an infinite potential well with an obstacle in it's middle section. Here's my potential as a function of x:
I wrote a program to solve this equation numerically and first tested it with an ordinary potential well and I got correct results with tolerance of 0.0001. Then added potential and increased $a$ by a fixed step each time and computed $\Psi$ for the lowest energy state (ground state) and plotted the wave function ($\Psi$) against $x$. Here's the results (note that $a$ has increased by a fixed step each time):
I noticed the first energy level and the second get close and closer each time. For example in
psi680, in dimensionless units, $E_0$ is equal to $71.166$ and $E_1$ is $71.180$ (so $E_1-E_0=0.014$).
I'm curious to find out where do they merge into one? I couldn't solve this numerically because of the method I used for finding energy eigenvalues and of course my code's accuracy.
- I wonder if there is an analytical way to predict this?
- An other question would be at what point this middle section becomes an infinite wall? (Where will the $\Psi$ be equal to zero in $x=0.5$ in this particular potential function?)
- Any other help about this problem would be very appreciated.