Quantum Particle in a Part of a Spherical Potential Well Confined by a Cone

  Raz Halifa- Levi  ,  Yacov Kantor  
Tel Aviv University

We consider the problem of a quantum particle in a spherical potential well confined by a cone. By varying the apex angle 2θ0 of the cone we modify the confinement of the particle from being inside a needle-like space via hemisphere and to an almost complete sphere with an excluded needle-like volume.  In an infinite spherical well the energy spectrum is discrete and is determined by the zeros of the spherical Bessel functions, whose order is determined by θ0. We numerically demonstrated, for several values of θ0, that the total number of eigenstates up to energy E is correctly given by Weyl’s formula [1].

For the spherical well of a finite depth -U, there is a discrete energy spectrum of bound states with eigenenergies E<0 and a continuous spectrum of states with E>0, as long as the well is deep enough. For each θ0 there is a critical depth Uc at which the last bound state disappears. The bound states are localized inside the well such that outside the well the wavefunctions decay over a length scale (localization length) ξ. Close to the critical depth Uc the localization length of the last remaining bound state diverges as ξ~(U-Uc)-ν, with exponent ν=1/2 for small and moderate cone apex semi-angles.  For θ0 >0.726π, the exponent ν depends on θ0, reaching 1 for θ0 = π.


[1] V. Ivrii, Bull. Math. Sci. 6 (2016).