Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Conical quantum billiard revisited


Author: Richard L. Liboff
Journal: Quart. Appl. Math. 59 (2001), 343-351
MSC: Primary 81Q50; Secondary 33C45, 35J25, 35Q40
DOI: https://doi.org/10.1090/qam/1828457
MathSciNet review: MR1828457
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Abstract: Eigenstates of a particle confined to a cone of finite length capped by a spherical surface element are derived. A countable infinite set of solutions is obtained corresponding to integer azimuthal and orbital quantum numbers $ \left( m, l \right)$. These solutions apply to a discrete subset of the domain of half vertex angles, $ 0 \le {\theta _0} \le \pi /2$. For arbitrary real orbital quantum numbers, $ l \to \nu $, solutions are given in terms of the hypergeometric function, with $ \nu = \nu \left( {\theta _0} \right)$, and are valid in the $ {\theta _0}$ domain, $ 0 \le {\theta _0} < \pi /2$. Eigenstates are either nondegenerate or two-fold degenerate. Numerical examples of both classes of solutions are included. For the case $ \mu = \cos \pi /4$, the ground-state wavefunction and eigenenergy are

$\displaystyle {\varphi _G} = {P_\nu }\left( \mu \right){j_\nu }\left( {x_{\nu 1... ...ght), \qquad {E_G} = {\hbar ^2}{\left( 6.4387 \right)^2}/\left( 2M{a^2} \right)$

where $ \nu = 2.54791, {P_\nu }\left( \mu \right)$ are Legendre functions, $ {x_{\nu 1}}$ is the first finite zero of the spherical Bessel function $ {j_\nu }\left( x \right)$, $ M$ is the mass of the confined particle and $ a$ is the edgelength of the cone. Solutions constructed also represent the scalar $ \hat r \cdot E$ electric field, where $ \hat r$ is the unit radius from the vertex of the cone. The first excited state of the conical quantum billiard has the nodal surface $ \mu = 1$ for all $ 0 \le {\mu _0} \le 1$.

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DOI: https://doi.org/10.1090/qam/1828457
Article copyright: © Copyright 2001 American Mathematical Society


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