How to Derive the General Solution for the Hydrogen Radial Equation?

[Boosh]

I am going through my Quantum textbook, just reviewing the material, i.e. this isn't a homework question. We are solving the radial equation for the Hydrogen Atom, first looking at the asymptotic behavior. My issue is I am completely blanking on how to solve the differential equation:

d^2u/dp^2 = [l(l+1)/p^2]u.

The general solution is:

u(p) = Cp^(l+1) + Dp^-l.

Can someone walk me through the steps of getting to this general solution? Thank you!

 

[stevendaryl]

With differential equations, solving them often just means guessing a solution, and then tweaking parameters to get the equations to work out.

You have the equation: $\frac{d^2 u}{dp^2} = \frac{\mathcal{l}(\mathcal{l}+1)}{p^2} u$.
You guess: $u = p^\alpha$.

Then $\frac{du}{dp} = \alpha p^{\alpha-1}$ and $\frac{d^2 u}{dp^2} = \alpha (\alpha -1) p^{\alpha - 2}$. Plugging this into the differential equation gives:

$\alpha (\alpha - 1) p^{\alpha - 2} = \frac{\mathcal{l}(\mathcal{l} + 1)}{p^2} p^\alpha$

For the equation to be true, $\alpha (\alpha - 1) = \mathcal{l} (\mathcal{l} + 1)$

So two possibilities are: $\alpha = -\mathcal{l}$ and $\alpha = \mathcal{l} + 1$

The general solution is a linear combination of the solutions.

 

[Boosh]

Ok, thank you so much!