# Expectation Values Question

1. Jan 30, 2012

### aglo6509

1. The problem statement, all variables and given/known data
A wave function ψ is A(eix+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.

2. Relevant equations

3. The attempt at a solution

So to normalize the function, I multiplied it by its complex conjugate (A(e-ix+eix) and got:
∫A2[(eix+e-ix)(e-ix+eix)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)(eix+e-ix)dx From 0 to π/8
(sqrt(1/4π))∫(eix+e-ix)dx
(sqrt(1/4π))(ieix-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

2. Jan 30, 2012

### ehild

Hi aglo,

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

ehild

Last edited: Jan 31, 2012
3. Jan 31, 2012

### 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 $\int \psi ^* \psi dx$