How to Normalize a Wave Function in a Potential Well?

[Safder Aree]

Homework Statement

I have the wave function Ae^(ikx)*cos(pix/L) defined at -L/2 <= x <= L/2. and 0 for all other x.

The question is:
A proton is in a time-independent one-dimensional potential well.What is the probability that the proton is located between x = − L/4 and x = L/4 ?

Homework Equations

∫ψψ* = 1

The Attempt at a Solution

I know i have to normalize this first. Should I be normalizing with the bounds being -L/2 and L/2 or should the bounds be -L/4 and L/4.
A^2∫cos^2(πx/L)dx =1
(1/2)A^2[x+ L/2π (sin(2πx/L)] evaluated at some bounds.

I actually evaluated it both ways one answer gives me A=√2/L at L/2,-L/2 and the other one gives me A = √2
π/L which makes more sense (at -L/4,L/4)

But i just want to make sure i am approaching this right way.
Thank you.

 

[kuruman]

Actually, the normalization in one dimension is formally over all space, i.e. from -∞ to +∞, but since the wavefunction is zero outside the box, you would be adding a whole bunch for zeroes if you went outside it. Therefore it suffices to integrate from -L/2 to +L/2, i.e. over all space where there is non-zero probability.

 

[Safder Aree]

I actually re did it and got A=√2/L. Thank you for the clarification. Now to find the probability I can just integrate over the other bounds right?

 

[kuruman]

Right. Make sure the number you get is less than 1.