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Infinite Spherical Potential Well

Consider a particle of mass and energy moving in the following simple central potential:
 (647)

Clearly, the wavefunction is only non-zero in the region . Within this region, it is subject to the physical boundary conditions that it be well behaved (i.e., square-integrable) at , and that it be zero at (see Sect. 5.2). Writing the wavefunction in the standard form
 (648)

we deduce (see previous section) that the radial function satisfies
 (649)

in the region , where
 (650)

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

 (651)

The two independent solutions to this well-known second-order differential equation are called spherical Bessel functions,and can be written
 (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 (i.e., they are not square-integrable) at , whereas the functions are well behaved everywhere. It follows from our boundary condition at that the are unphysical, and that the radial wavefunction is thus proportional to only. In order to satisfy the boundary condition at [i.e., ], the value of must be chosen such that corresponds to one of the zeros of . Let us denote the th zero of as . It follows that
 (658)

for . Hence, from (650), the allowed energy levels are
 (659)

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

Table 1: The first few zeros of the spherical Bessel function .
 3.142 6.283 9.425 12.566 4.493 7.725 10.904 14.066 5.763 9.095 12.323 15.515 6.988 10.417 13.698 16.924 8.183 11.705 15.040 18.301

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)

when . Given that the are mutually orthogonal (see Sect. 8), this ensures that wavefunctions (648) corresponding to distinct sets of values of the quantum numbers , , and are mutually orthogonal.

Next: Hydrogen Atom Up: Central Potentials Previous: Derivation of Radial Equation
Richard Fitzpatrick 2010-07-20