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 Schrödinger 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.