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Europhysics Letters (EPL)

Critique and correction of the currently accepted solution of the infinite spherical well in quantum mechanics

Young‐Sea Huang, Hans-Rudolf Thomann

Published Sep 1, 2016• 7 citations
Publisher

Abstract

An error in the currently accepted solution of the problem of the infinite spherical well is pointed out.The problem is then solved by considering the self-adjointness of the Hamiltonian operator.In contrast to the currently accepted solution, the radial probability density for finding the particle at the center of the spherical well is not necessarily zero, in accordance with the solutions obtained.

DOI
10.1209/0295-5075/115/60001

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An error in the currently accepted solution of the problem of the infinite spherical well is pointed out.The problem is then solved by considering the self-adjointness of the Hamiltonian operator.In contrast to the currently accepted solution, the radial probability density for finding the particle at the center of the spherical well is not necessarily zero, in accordance with the solutions obtained.
Recently, the problem of the infinite spherical well was solved by the group-theoretical method to resolve all the peculiarities in the currently accepted solution [DOI: 10.13140/RG.2.2.18172.44162 (Researchgate, 2017)]. With a view to further justifying the group-theoretical method, the problem is first studied from the viewpoint of classical mechanics. Then the radial probability densities predicted by classical mechanics are compared with those predicted from solutions of the problem obtained by the group-theoretical method. The comparisons clearly indicate the convergence of predictions of quantum mechanics and classical mechanics in the limit of large eigen-energies. Therefore, the group-theoretical method is justified as the right way to solve the problem of the infinite spherical well.
Abstract We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone with an apex angle 2θ 0 emanating from the center of the sphere, with 0 < θ 0 < π. This non-central potential can be solved by an extension of techniques used in spherically-symmetric problems. The angular parts of the eigenstates depend on azimuthal angle φ and polar angle θ as P λ m ( cos θ ) e im φ where P λ m is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values λ = λ i m (i = 0, 1, 3, ...) that depend on m and θ 0. Each λ i m has an infinite sequence of eigenenergies E n ( λ i m ) , with corresponding radial parts of eigenfunctions. In a spherical box the discrete energy spectrum is determined by the zeros of the spherical Bessel functions. For several θ 0 we demonstrate the validity of Weyl’s continuous estimate  W for the exact number of states  up to energy E, and evaluate the fluctuations of  around  W . We examine the behavior of bound states in a well of finite depth U 0, and find the critical value U c (θ 0) when all bound states disappear. The radial part of the zero energy eigenstate outside the well is 1/r λ+1, which is not square-integrable for λ ≤ 1/2. (0 < λ ≤ 1/2) can appear for θ