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I'm second year undergrad in Physics and I've been studying the time-independent Schroedinger Equation (TISE), QM operators, etc and now application of this to a hydrogenic atom.

I've come to a bit of a dead-end though - I've got to the stage where I've separated the TISE into Radial and Angular parts. With the angular part, I've deduced that the corresponding eigenfunctions are spherical harmonics with corresponding eigenvalue [tex]\hbar l(l+1)[/tex] which I've put equal to [tex]2m(\lambda)[/tex].

Now, in solving the radial part of the TISE for the hydrogenic atom, the lecturer has considered the radial equation at very large distances from the nucleus, when the effective potential [tex]/V_eff[/tex] can be neglected. i.e. considering bound states of the atom. He has put [tex]\E = -\frac{\kappa^2}{2}[/tex], substituted into TISE (using the relation R = chi/r) and has obtained the solution in term of an unknown function, F(r), [tex]\chi(r) = F(r) \exp(-\kappa r)[/tex]. He has obtained a corresponding differential equation for F, resulting in [tex]\frac{d^2 F}{dr^2} = -frac{l(l+1)}{r^2} F = 2k \frac{dF}{dr} - \frac{2z}{r}F[/tex]. Solution of this is by series solution with condition that r is well behaved as it tends to 0. Now the problem is that I've lost notes on this particular derivation and am unsure how to find the energy for hydrogenic atom in the parameters I've described. I've looked at books, but they all show a different method, one that won't be examined in my course.

I was wondering if one of you could perhaps fill in the gaps for me, by using a series solution method.

Thanks

I'll have a go showing a rough derivation of where I've got to..

Hamiltonian in 3D in spherical polar coordinates with coulomb potential:

[tex]\hat{H} = -\frac{\hbar^2}{2m_e} [\frac{1}{r^2} \pd{}{r}{} (r^2 \pd{}{r}{}) - \frac{\hat{L^2}}{\hbar^2 f^2}] - \frac{Ze^2}{4 \pi \epsilon_0 r}[/tex]

Therefore TISE:

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