Quantum Mechanics for Engineers |
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© Leon van Dommelen |
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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
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(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
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(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