This labels the energy, E_n (Of course, the energy should depend on l, since l does appear in the radial equation, but magically it turns out not to affect the energies directly) The solution to the radial equation, R(r), does explicitly depend on both l and n. So, we label it R_nl(r).
Similarly, we labeled the angular solution Y_l^m(theta,phi). Thus, the complete wave function is given by
Beiser has a nice table with some of the R's, including normalization.
(On the next page, I also show a couple)
Here's one interesting feature of the ground state, psi_1,0,0:
The wave function is
The probability of finding an electron between r and r+Delta r is
This is maximum at the place where its derivative is zero:
The Bohr prediction for the radius of this level is now seen to be simply where the probability is the highest! However,
The expectation value of r is not quite the same as a0, because the wavefunction has a really long tail. We note, finally, that
(I guess this makes sense, in that <-Ze^2/r> will work out right)
(Short lecture today - review at start, plus FCQ's)
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