# Normalizing Wave Functions

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.

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

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.

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

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

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))

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