Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.77 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 zerocrossings of
the spherical Bessel functions. Therefore the allowable energy levels
are

(D.54) 
where is the th zerocrossing 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
halfinteger 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