Finding the Normalization Constant for the Hydrogen Radial Wave Function

[Patroclus]

1. Find the normalization constant for the radial wave function for Hydrogen.

I'm told that C = 1/(24a^5)^1/2
But how do I get that?

2.
n=2, l=1
R(2)(1)=Cr^(-r/2a。)
the integral from 0 to infinity of (x^4 * e^-"alpha"x) = 24 / alpha^5

3. I honestly don't know where to start

 

[CompuChip]

Depends what your starting point is.
If you're going from the Schrödinger equation, you can apply separation of variables:
$$\psi(r, \theta, \phi) = R(r) Y_{\ell m}(\theta, \phi)$$
and plug this into the time independent wave equation
$$\left( - \frac{\hbar^2}{2m} \nabla^2 + V(r) \right) \psi(r, \theta,\phi) = E \psi(r, \theta, \phi)$$
and derive the equation for R(r).
Then if you solve it and impose appropriate boundary conditions, you will find
$$R_{nl} (r) = \sqrt {{\left ( \frac{2 Z}{n a_{\mu}} \right ) }^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- Z r / {n a_{\mu}}} \left ( \frac{2 Z r}{n a_{\mu}} \right )^{l} L_{n-l-1}^{2l+1} \left ( \frac{2 Z r}{n a_{\mu}} \right )$$
which you can normalise using the properties of the Laguerre polynomials.

If you were looking for something simpler, please give us more information :-)