How Do Boundary Conditions Affect Wave Functions in a Particle in a Box Problem?

[Nio]

Homework Statement

Assume we have an infinitely deep square well of length L, with the left edge of the well at -L/2. Assume U = 0 at the bottom and infinity at the top. A) Using the time independent Schrödinger Equation, derive the wave equation for a particle trapped in this well. Make sure your equation is properly normalized

Homework Equations

shrodingers equation, the general solution to it which is Asinkx +Bcoskx=psi(0)

The Attempt at a Solution

I know that sin is and odd function and cos is even so I am supposed to arrive at two wave functions and have different values of n, even values for cos and odd for sin. I am supposed to get Psi(x)= (2/L)^.5sin(npix/L) where n is odd values and the same thing except cos and even values. I am having trouble deriving this using my boundary conditions, and I am also having trouble normalizing it.

 

[nickjer]

I would save normalization for the end. What are your boundary conditions so far?

 

[Easty]

Why is it that you can get sine and cosine solutions? i thought all you needed was the sine solution because it would always satify the boundary conditions of haveing a node at each boundary.

 

[nickjer]

Well both satisfy Schrodingers equations. If the boundary conditions were Psi(0) = 0 and Psi(L) = 0, then sine would be your only solution because cos(0) = 1. But now your BC's are Psi(-L/2) = 0 and Psi(L/2) = 0. Therefore both can be solutions. You just need to find the 'k' that satisfies those BC's.