Conical Quantum Billiard

The conical quantum billiard is examined. Wavefunctions are obtained in an open domain which excludes the polar axis, involving associated Legendre functions of the first and second kind. The first excited state is three-fold degenerate. One wavefunction is nonnodal. The nodal surface of either of the other states is a bisecting plane which includes the axis of the cone. These nodal properties maintain for 0 < θ0 ≤ π/2, where θ0 is the half vertex angle of the cone. At θ > π/2, the nonnodal state acquires a nodal at θ = π/2. Thus, as with the image problem in two dimensions, there is critical vertex angle about which the nodal structure of one of the eigenstates suffers a topological change. This nodal transition is accompanied by a geometrical transformation of the cone from convex to concave. Solutions obtained are valid for all conical quantum billiards to the limit of the spherical quantum billiard excluding the polar axis.