Normalizing Wave Functions

1. Nov 6, 2011

theblender

Pretty basic question here, but I was wondering if someone could explain to me how to normalize a wave function. Specifically:

ψn(x) = A * √x * sin (n∏x2/L2), n = 1, 2, 3, ...

Normalized in the range 0 to L.

Thanks for the help, a little bit of a walk through would be much appreciated.

2. Nov 6, 2011

Staff: Mentor

What have you tried so far, and where are you stuck?

3. Nov 6, 2011

theblender

Well, I'm working it now, its been shed under some new light, so what I am doing is just integrating the square of psi and setting it equal to one, then solving for A. Which is what I tried initially, but I just got confused with multiplying the original by the complex conjugate.

4. Nov 6, 2011

theblender

Also having trouble integrating the statement cos ((2*pi*x^2)/L^2) dx.

5. Nov 8, 2011

jewbinson

Hint: you need to substitute x for some variable y so that you end up with cos(y^2) in the integrand. Then you can try messing around with trig, for example cos(y^2) = cos(y*y) = ?

You realize there is no easy way to expand that. In fact, wolframalpha gives:

http://www.wolframalpha.com/input/?i=cos(x^2)

which uses the "Fresnel C Integral" which I haven't even come across until this example.

So the integral is not representable by standard elementary functions...

Edit: However integrating the square of your original wavefunction is quite straightforward once you calculate the integral of x(sin^2)(x^2).

Use the fact that (sin^2)(y) = (1/2)(1-cos(y))

Last edited: Nov 8, 2011