What is the Probability Density for a Wave Function?

[aglo6509]

Homework Statement

A wave function ψ is A(e^ix+e^-ix) in the region -π<x<π and zero elsewhere. Normalize the wave function and find the probability of the particle being (a) between x=0 and x=π/8, and (b) between x=0 and x=π/4.

Homework Equations

The Attempt at a Solution

So to normalize the function, I multiplied it by its complex conjugate (A(e^-ix+e^ix) and got:
∫A²[(e^ix+e^-ix)(e^-ix+e^ix)dx=1 From -π to π
∫2A²dx=1
2A²x(from -π to π)=1
2A²π+2A²π=1
4A²π=1

A=sqrt(1/4π)

Now that I have the function normalized, I can find the probability the question asks for. The problem I'm having is however do you take the integral of complex numbers the same way as a real number?

The best attempt I can get is:
∫(sqrt(1/4π)(e^ix+e^-ix)dx From 0 to π/8
(sqrt(1/4π))∫(e^ix+e^-ix)dx
(sqrt(1/4π))(ie^ix-ie^-ix)

Would I now just plug in 0 and π/8 and leave my answers in terms of i?

Thanks for taking the time to look at this.
Aglo6509

 

[ehild]

Hi aglo,

The probability density is ψψ* ("*" for complex conjugate). The wave function can be complex, but probability can not!

ehild

 

[genericusrnme]

It might be helpful for you if you note that
$\frac{e^{ix}+e^{-ix}}{2} = Cos(x)$
so your answers won't even involve any i since those trig functions are real!

also, the density you're looking for is $\int \psi ^* \psi dx$