Section 13.8: Radial Representations of the Coulomb Solutions

n = | l =

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In Section 13.7 we found the radial solutions

Rnl(r) = Anl e-r/na0 [ (r/na0)l+1/r ] vn(r/na0) ,

where Anl is the normalization constant and vnl(ρ) = L2l+1n-l-1 (2r/na0) are the associated Laguerre polynomials.

In Animation 1 radial energy eigenfunctions corresponding to the Coulomb potential, −e2/r, are plotted versus distance given in Bohr radii. These are shown for n = 1, 2, 3, 4 with the appropriate l values. In Animation 2,  the quantum numbers are given in spectroscopic notation:

s p d f
l = 1 l = 2 l = 3 l = 4

and hence 4f corresponds to n = 4 and l = 3.  For the radial energy eigenfunction, notice how the number of crossings is related to the quantum numbers n and l.  You should see that the number of crossings is n l − 1.

In addition, for the same quantum numbers in Animation 2, Rnl2(r) and the probability density, Pnl(r) = Rnl2(r)r2, are shown. You can change the start and end of the integral for R2(r) and Rnl2(r)r2 as well as the range plotted in the graph by changing values and clicking the button associated with the state you are interested in. You should quickly convince yourself that while

Rnl2(r) dr ≠ 1, [integral from 0 to +∞]


Rnl2(r) r2 dr = ∫ unl2(r) dr = 1, [integrals from 0 to +∞]

and that indeed, Pnl(r) = Rnl2(r)r2.

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