Show that the (1,0,0) and (2,0,0) wave functions are properly normalized.
We know that:
Ψ(1,0,0) = (2/(a0^(3/2))*e^(-r/a0)*(1/sqrt(2))*(1/sqrt(2*pi))
R(r) = (2/(a0^(3/2))*e^(-r/a0)
Θ(θ) = (1/sqrt(2))
Φ(φ) = (1/sqrt(2*pi))
(1) ∫|Ψ|^2 dx = 1
(2) |R(r)|^2*|Θ(θ)|^2*|Φ(φ)|^2*r^2*sinθ dr dθ dφ ... (? might be useful)
The Attempt at a Solution
So I know that in order for a function to be properly normalized, it has to have the absolute value of the sine wave squared equal to 1. I originally integrated from negative inf to positive inf, but that did not give me 1 = 1. I tried looking to see where I went wrong but I found equation (2) in my book, but wasn't sure how to integrate that. What am I doing wrong? I think my limits of integration might be wrong because I was doing:
(3) ∫ ((2/(a0^(3/2))*e^(-r/a0)*(1/sqrt(2))*(1/sqrt(2*pi)))^2 dr = 1 from - inf to + inf
and not worrying about that sin θ thing I mentioned earlier.
Do I need to be using equation (2)? How do I find limits of integration? I know that - infinity doesn't make sense for r, so maybe it is just from 0 to infinity with equation 3 maybe?