Conical quantum billiard revisited

Eigenstates of a particle confined to a cone of finite length capped by a s pherical surface element are derived. A countable infinite set of solutions is obtained corresponding to integer azimuthal and orbital quantum numbers (m, l). These solutions apply to a discrete subset of the domain of half v ertex angles, 0 less than or equal to theta (0) less than or equal to pi /2 . For arbitrary real orbital quantum numbers, l --> nu, solutions are given in terms of the hypergeometric function, with nu = nu(theta (0)), and are valid in the theta (0) domain, 0 less than or equal to theta (0) less than or equal to pi /2. Eigenstates are either nondegenerate or two-fold degener ate. Numerical examples of both classes of solutions are included. For the case mu = cos pi /4 the ground-state wavefunction and eigenenergy are phi (G) = P-nu(mu )j(nu)(x(nu)1r/a), E-G = h(2) (6.4387)(2)/(2Ma(2)) where nu = 2.54791, P nu(mu) are Legendre functions, x(nu1) is the first fi nite zero of the spherical Bessel function j(nu)(x), M is the mass of the c onfined particle and a is the edge-length of the cone. Solutions constructe d also represent the (r) over cap .E electric field, where (r) over cap 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 less than or equal to mu (0) less than or equal to 1.