Conical quantum billiard revisited

Liboff, Richard L.

doi:10.1090/qam/1828457

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 \[ {\varphi _G} = {P_\nu }\left ( \mu \right ){j_\nu }\left ( {x_{\nu 1}}r/a \right ), \qquad {E_G} = {\hbar ^2}{\left ( 6.4387 \right )^2}/\left ( 2M{a^2} \right )\] 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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