Quantum particle in a spherical well confined by a cone

Abstract

We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone with an apex angle $2{\theta }_0$ emanating from the center of the sphere, with $0<{\theta }_0<\pi$. This non-central potential can be solved by an extension of techniques used in spherically-symmetric problems. The angular parts of the eigenstates depend on azimuthal angle $\phi$ and polar angle $\theta$ as $P_{\lambda }^m(\cos \theta ){\mathrm{e}}^{im\phi }$ where $P_{\lambda }^m$ is the associated Legendre function of integer order $m$ and (usually noninteger) degree $\lambda$. There is an infinite discrete set of values $\lambda ={\lambda }_i^m$ ($i=0,1,3,\dots$) that depend on $m$ and ${\theta }_0$. Each ${\lambda }_i^m$ has an infinite sequence of eigenenergies $E_n({\lambda }_i^m)$, with corresponding radial parts of eigenfunctions. In a spherical box the discrete energy spectrum is determined by the zeros of the spherical Bessel functions. For several ${\theta }_0$ we demonstrate the validity of Weyl's continuous estimate ${\mathcal{N}}_W$ for the exact number of states $\mathcal{N}$ up to energy $E$, and evaluate the fluctuations of $\mathcal{N}$ around ${\mathcal{N}}_W$. We examine the behavior of bound states in a well of finite depth $U_0$, and find the critical value $U_c({\theta }_0)$ when all bound states disappear. The radial part of the zero energy eigenstate outside the well is $1/r^{\lambda +1}$, which is not square-integrable for $\lambda \le 1/2$. ($0<\lambda \le 1/2$ can appear for ${\theta }_0>{\theta }_c\approx 0.726\pi$ and has no parallel in spherically-symmetric potentials.) Bound states have spatial extent $\xi$ which diverges as a (possibly $\lambda$-dependent) power law as $U_0$ approaches the value where the eigenenergy of that state vanishes.

arXiv commentLaTeX, 21 pages, 6 figures

References in corpus (18)

• 100 years of Weyl's law
• New Methods for Two-Dimensional Schrödinger Equation: SUSY-separation of Variables and Shape Invariance
• Bound States in one and two Spatial Dimensions
• Supersymmetry,Shape Invariance and Exactly Solvable Noncentral Potentials
• Casimir force between sharp-shaped conductors
• Acoustic Scattering from Corners, Edges and Circular Cones
• Exactly solvable Schrödinger operators
• Quantum Bound States with Zero Binding Energy
• Existence of Bound States in Continuous 0<D<\infty Dimensions
• Polymer-mediated entropic forces between scale-free objects
• Entropic force of polymers on a cone tip
• A class of exactly solvable rationally extended non-central potentials in Two and Three Dimensions
• Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics
• Ideal Polymers near Scale-Free Surfaces
• Diffusion in the Presence of Scale-Free Absorbing Boundaries
• Comment on "Quantum bound states with zero binding energy"
• Pinning and unbinding of (ideal) polymers from a wedge corner
• Localization of random walks to competing manifolds of distinct dimensions