## Section 13.8: Radial Representations of the Coulomb Solutions

Animation 1: Radial Energy Eigenfunction | Animation 2: Other Combinations of R(r)

Please wait for the animation to completely load.

In Section 13.7 we found the radial solutions

*R*_{nl}(*r*) = *A*_{nl }e^{-r/na}0 [ (*r*/*na*_{0})^{l+1}/*r* ] *v*_{n}(*r*/*na*_{0}) ,

where *A*_{nl} is the normalization constant and *v*_{nl}(ρ) = *L*^{2l+1}_{n-l-1 }(2*r*/*na*_{0}) are the associated Laguerre polynomials.

In Animation 1 radial energy eigenfunctions corresponding to the Coulomb potential, −*e*^{2}/*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, *R*_{nl}^{2}(*r*) and the probability density, *P*_{nl}(*r*) = *R*_{nl}^{2}(*r*)*r*^{2}, are shown. You can change the start and end of the integral for *R*^{2}(*r*) and *R*_{nl}^{2}(*r*)*r*^{2} 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

∫ *R*_{nl}^{2}(*r*) *dr* ≠ 1, [integral from 0 to +∞]

that

∫ *R*_{nl}^{2}(*r*) *r*^{2} *dr* = ∫ *u*_{nl}^{2}(*r*) *dr* = 1, [integrals from 0 to +∞]

and that indeed, *P*_{nl}(*r*) = *R*_{nl}^{2}(*r*)*r*^{2}.

*Right click on the applet to make a copy of t**he image. The mouse coordinates may be observed by left-clicking within the graph.*