### D.78 Impenetrable spherical shell

To solve the problem of particles stuck inside an impenetrable shell of radius , refer to addendum {A.6}. According to that addendum, the solutions without unacceptable singularities at the center are of the form

 (D.53)

where the are the spherical Bessel functions of the first kind, the the spherical harmonics, and is the classical momentum of a particle with energy . is the constant potential inside the shell, which can be taken to be zero without fundamentally changing the solution.

Because the wave function must be zero at the shell , must be one of the zero-crossings of the spherical Bessel functions. Therefore the allowable energy levels are

 (D.54)

where is the -th zero-crossing of spherical Bessel function (not counting the origin). Those crossings can be found tabulated in for example [1], (under the guise of the Bessel functions of half-integer order.)

In terms of the count of the energy levels of the harmonic oscillator, 1 corresponds to energy level , and each next value of increases the energy levels by two, so