Hydrogen radial equation solution

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1. Apr 20, 2017

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!

2. Apr 20, 2017

stevendaryl

Staff Emeritus
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.

3. Apr 20, 2017

Boosh

Ok, thank you so much!

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