What Are the Eigenfunctions for the 1D Infinite Square Well?

[andre220]

Homework Statement

Find the ground and first excited state eigenfunctions of for the 1D infinite square well with boundaries -L/2 and +L/2

Homework Equations

$$\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x) = E\psi(x)$$

The Attempt at a Solution

Okay so I know how to solve it and get that $$\psi_1(x) = \sqrt{\frac{2}{L}}\cos{\frac{\pi x}{L}}$$. Next, I also know that $$\psi_2(x) = \sqrt{\frac{2}{L}}\sin{\frac{2\pi x}{L}}$$ one could reason this by arguing that each eigenfunction should have $n-1$ nodes. However, what is a more mathematical reasoning to $\psi$ for a given excited state. I am sure it is quite simple, I just can't seem to see it.

 

[vela]

I'm not sure exactly what your question is. If you solve the Schrödinger equation with the appropriate boundary conditions — that is, solve the math problem — those are two of the solutions you get.