QM: Infinite Square Well -a/2 to a/2

[chrisd]

I have read similar threads about this problem but I wasn't able to make progress using them.

Homework Statement

Consider an infinite square-well potential of width a, but with the coordinate system shifted so that the infinite potential barriers lie at x=$\frac{-a}{2}$ and x=$\frac{a}{2}$.

Solve the Schrödinger equation for this case to calculate the normalized wave function $\psi$_n(x) and the corresponding energies E_n

Homework Equations

Time Independent Schrödinger Equation (V(x)=0 within the well)
-\hbar²$\frac{1}{2m}$ ($\frac{∂}{∂x}$)²$\psi$=E$\psi$

The Attempt at a Solution

Skipping ahead to the general solution for $\psi$(x) , I get:

$\psi$(x) = Ae^ikx + Be^-ikx, k = $\frac{\sqrt{2mE}}{\hbar}$

Using the boundary conditions,

$\psi$($\frac{a}{2}$)=Ae^ika/2 + Be^-ika/2=0
$\psi$($\frac{-a}{2}$)=Ae^-ika/2 + Be^ika/2=0,

together with some substitution I am able to prove that,

k =$\frac{\pi}{a}$n

E_n =($\frac{n\pi\hbar}{a}$)²$\frac{1}{2m}$

which I believe to be correct for infinite square well.

I run into trouble trying to solve for the constants A and B. My approach was to try and normalize the wavefunction and then use substitution to solve for A or B, but I can't seem to get anywhere after a certain point.
I would expect to get cosines for n=1,3,5... and sines for n=2,4,6... based on what I know about the shape of the wavefunction, but I am unsure how to prove this mathemetically.
Any tips would be appreciated.

 

[vela]

Try plugging in your allowed values of k into the boundary condition equations. You'll get a different relationship between A and B depending on whether n is odd or even.

 

[chrisd]

Ah, that's the ticket. It's probably easier without using exponential form, but I was able to prove that A=B for odd n (giving me a cosine) and A=-B for even n (giving me a sine). Thanks!