[orgo814]

Hello,

For this unnormalized wave function I am fine with the integration but confused on one thing. To normalize a wave function we are supposed to take the integral of the square function. I have the function: e^(i)(phi).

My book then does the next step as integral(e^(-i)(phi) x e^(i)(phi)). I know how to integrate it from here, so that's not the question. My question is why are we doing a -(i)(phi) instead of just squaring it. What is the purpose of the negative sign?

Thanks a bunch for any help. Normalizing wave functions is taking me a lot of practice to get the hang of!

[Borek]

Rules for complex functions are slightly different. You need not just a square, but to multiply by conjugate. Actually it is the same for real functions. They just don't have imaginary part so squaring is enough.

[Corribus]

In mathematical speak, it's not the square of the wavefunction that matters, but the square modulus (or square of the absolute value). This is important because the probability of finding a particle in a volume segment of space has to be real.  Therefore the integrand has to be real. Squaring a complex function will lead to a complex function, so you must use the absolute value of the function. Multiplying a complex number by its complex conjugate will always yield a real function.

Maybe these links will help,

http://en.wikipedia.org/wiki/Complex_conjugate
http://en.wikipedia.org/wiki/Absolute_value

If you don't care about the rigors of it, just always multiple the wavefunction by its complex conjugate. The complex conjugate of a function is just the function with a -i substituted wherever there is an i in the function. If the wavefunction is real, then the complex conjugate is just the function itself - so you're integrating the function squared. In the normalization condition, the two functions being multipled together in the integrand are the same. But do note that in many applications (such as showing two wavefunctions are orthogonal, or, even more so, finding expectation values) the two functions in the integral are not the same, and in these occasions it's important to be sure you're using the complex conjugate of the correct function.  Convention is that the first function is always the complex conjugate.  That is, we take the integral of ψ₁*[A]ψ₂, where [A] is some operation.