Normalizing Wave Functions

[theblender]

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

ψ_n(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]

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

 

[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.

 

[theblender]

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

 

[jewbinson]

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))