Infinite Well with Sinusoidal Potential

[jyoung11509]

Homework Statement

Assume a potential of the form V(x)=V_{0}sin({\frac{\pi x}{L}}) with 0<x<L and V(x)=\infty outside this range. Assume \psi = \sum a_{j} \phi_{j}(x), where \phi_{j}(x) are solutions for the infinite square well. Construct the ground state wavefunction using at least 10 basis functions.

Homework Equations

I obtained the solution \phi_{n} = \sqrt{\frac{2}{L}} \sin({\frac{n\pi x}{L}}) for the infinite square well with zero potential inside.

The Attempt at a Solution

After obtaining the solution shown above, I attempted to expand the summation. I know that I want to do a linear combination of energy eigenstates, however, I am not sure what to do about the leading coefficients. I have found by searching that a_{j} = \int_{-\infty}^{\infty} \phi_{j}^{*}(x) \psi dx, but how can I solve this if I'm using the a_{j}'s to find \psi?

Any help or advice would be greatly appreciated. Thanks!

 

[fzero]

You need to apply the Schrodinger equation to your linear combination, since the particle in a box basis are not solutions once we turn on this sinusoidal potential. You'll probably be able to use a trig identity to simplify the potential terms with V(x) \phi_k(x), then group like orders of \sin(n\pi x/L). This will give you some recursive relations among the a_j. To obtain the ground state energy, you might have to minimize something.