Conical quantum billiard revisited

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 ( m , l ) \left ( m, l \right ) . These solutions apply to a discrete subset of the domain of half vertex angles, 0 ≤ θ 0 ≤ π / 2 0 \le {\theta _0} \le \pi /2 . For arbitrary real orbital quantum numbers, l → ν l \to \nu , solutions are given in terms of the hypergeometric function, with ν = ν ( θ 0 ) \nu = \nu \left ( {\theta _0} \right ) , and are valid in the θ 0 {\theta _0} domain, 0 ≤ θ 0 > π / 2 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 μ = cos ⁡ π / 4 \mu = \cos \pi /4 , the ground-state wavefunction and eigenenergy are \[ φ G = P ν ( μ ) j ν ( x ν 1 r / a ) , E G = ℏ 2 ( 6.4387 ) 2 / ( 2 M a 2 ) {\varphi _G} = {P_\nu }\left ( \mu \right ){j_\nu }\left ( {x_{\nu 1}}r/a \right ), \qquad {E_G} = {\hbar ^2}{\left ( 6.4387 \right )^2}/\left ( 2M{a^2} \right ) \] where ν = 2.54791 , P ν ( μ ) \nu = 2.54791, {P_\nu }\left ( \mu \right ) are Legendre functions, x ν 1 {x_{\nu 1}} is the first finite zero of the spherical Bessel function j ν ( x ) {j_\nu }\left ( x \right ) , M M is the mass of the confined particle and a a is the edgelength of the cone. Solutions constructed also represent the scalar r ^ ⋅ E \hat r \cdot E electric field, where r ^ \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 μ = 1 \mu = 1 for all 0 ≤ μ 0 ≤ 1 0 \le {\mu _0} \le 1 .