Infinite potential well with V(x)=Asin(x pi/a)

[bk00]

Homework Statement

Hi, this is basically a particle in a box problem but V(x) is sinusoidal inside the box.

V(x)=-Asin(pi x/a) 0<x<a
V(x)=Infinity otherwise

A: depth of the well

In the question, it asks me to plot the first 4 eigenfunctions.

The Attempt at a Solution

After plugging the sinusoidal potential function into the Schrodinger equation, the equation looks like:

y''(x)-(E+Asin(Cx)y(x)=0

I don't have a clue about solving this. I've been looking at the quantum mechanics books and diff. eqs. books for hours but I can't seem to find anything useful. I'm using Mathematica I cannot solve it numerically since I don't know "E".

Is there any place that I can look at for a possible solution or direction? Thank you.

 

[Jhero]

Thank you for your question. I understand that you are trying to solve the Schrodinger equation for a particle in a box with a sinusoidal potential inside the box. This is a challenging problem, but I can suggest some steps that may help you find a solution.

First, it is important to note that the Schrodinger equation you have written is not quite correct. The correct equation for this problem is:

y''(x) + (E + Asin(pi x/a))y(x) = 0

Where E is the energy of the particle. This equation is known as the time-independent Schrodinger equation and is used to find the energy eigenvalues and eigenfunctions for a given potential.

To solve this equation, you will need to use boundary conditions. In this case, the boundary conditions are that the wavefunction y(x) must be continuous and its derivative must be continuous at the edges of the box (x=0 and x=a). This will result in a discrete set of energy eigenvalues and corresponding eigenfunctions.

Unfortunately, this equation cannot be solved analytically for arbitrary values of E. However, there are some techniques that can be used to approximate the solutions. One approach is to use perturbation theory, which involves expanding the potential term in terms of a small parameter and solving iteratively. Another approach is to use numerical methods, such as the finite difference method, to approximate the solutions.

I would suggest consulting a quantum mechanics textbook for more information on these techniques. Additionally, there may be resources available online or at your university that can provide further guidance on solving this type of problem.

I hope this helps and best of luck with your research.