Infinite Spherical Potential Well

(647) |

we deduce (see previous section) that the radial function satisfies

(649) |

Defining the scaled radial variable , the above differential
equation can be transformed into the standard form

(651) |

(652) | |||

(653) |

Thus, the first few spherical Bessel functions take the form

(654) | |||

(655) | |||

(656) | |||

(657) |

These functions are also plotted in Fig. 20. It can be seen that the spherical Bessel functions are oscillatory in nature, passing through zero many times. However, the functions are badly behaved (

(658) |

The first few values of are listed in Table 1. It can be seen that is an increasing function of both and .

We are now in a position to interpret the three quantum numbers--, ,
and --which determine the form of the wavefunction
specified in Eq. (648). As is clear from Sect. 8, the
azimuthal quantum number determines the number of nodes in the
wavefunction as the azimuthal angle varies between 0 and . Thus,
corresponds to no nodes, to a single node, to two nodes,
*etc*. Likewise, the polar quantum number determines the
number of nodes in the wavefunction as the polar angle varies between 0 and .
Again, corresponds to no nodes, to a single node,
*etc*. Finally, the radial quantum number determines
the number of nodes in the wavefunction as the radial
variable varies between 0 and (not counting any
nodes at or ). Thus, corresponds to no nodes,
to a single node, to two nodes, *etc*. Note that,
for the
case of an infinite potential well,
the only restrictions on the values that the various quantum numbers can take are that must be a positive integer, must be
a non-negative integer, and must be an integer lying between and . Note, further,
that the allowed energy levels (659) only depend on the
values of the quantum numbers and . Finally, it is
easily demonstrated that the spherical Bessel functions are mutually
orthogonal: *i.e.*,

(660) |