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I'm puzzled by the following radial wave equation:

$$ \left(\frac{\hbar^2}{2m_r}\left(-\frac{d^2}{dr^2} -\frac{2}{r}\frac{d}{dr} + \frac{l(l+1)}{r^2}\right) + V(r)\right)R_{nl}(r) = ER_{nl}(r)$$

With $V(r) = \frac{-e^2}{4\pi \epsilon_0 r}$ and $l = 0$. This equation comes from the hydrogen atom.

What puzzles me is that there seem to be more than two linear independent solutions. While I thought that it was a standard result that a second order linear ODE has no more than two linear independent solutions.

What am I missing here?

The result about second order ODEs comes from here

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  • $\begingroup$ How many solutions are there to an infinite well? $\endgroup$ – Jon Custer Jun 15 at 14:50
  • $\begingroup$ The equation you wrote down has two linearly independent solutions. However, when solving the hydrogen atom, you let $n,l,m$ vary and this gives you many separate equations, all with different values for $n,l,m$, each of which has two linearly independent solutions. $\endgroup$ – J_P Jun 15 at 16:08
  • $\begingroup$ @JonCuster Is the solution to my puzzle that the result I stated holds only for second order ODEs with Cauchy boundary conditions, and that the infinite square well has Dirichlet boundary conditions? $\endgroup$ – Jens Wagemaker Jun 15 at 16:58
  • $\begingroup$ @J_P The problem was that the equation I wrote down has more than two linearly independent solutions (I think), that puzzled me. Can you confirm that? $\endgroup$ – Jens Wagemaker Jun 15 at 16:59
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    $\begingroup$ So the process is this: you fix $l$ ($0$ in your case) and now you're left with an uncountably infinite number of equations, each for its own $E$. You then solve each of these equations - they all have two linearly independent solutions. Then you find which of those $E$ give you solutions that satisfy the boundary conditions in the first place and only keep those, indexing them as $E_n$. That gives you infinitely many solutions. $\endgroup$ – J_P Jun 15 at 17:40

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