TY - JOUR

T1 - Quantum particle in a spherical well confined by a cone

AU - Halifa Levi, Raz

AU - Kantor, Yacov

N1 - Publisher Copyright:
© 2022 The Author(s). Published by IOP Publishing Ltd.

PY - 2022/5

Y1 - 2022/5

N2 - 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 2θ 0 emanating from the center of the sphere, with 0 < θ 0 < π. 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 φ and polar angle θ as Pλm(cosθ)eimφ where Pλm is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values λ=λim (i = 0, 1, 3, ...) that depend on m and θ 0. Each λim has an infinite sequence of eigenenergies En(λim), 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 θ 0 we demonstrate the validity of Weyl's continuous estimate NW for the exact number of states N up to energy E, and evaluate the fluctuations of N around NW . We examine the behavior of bound states in a well of finite depth U 0, and find the critical value U c (θ 0) when all bound states disappear. The radial part of the zero energy eigenstate outside the well is 1/r λ+1, which is not square-integrable for λ ≤ 1/2. (0 < λ ≤ 1/2) can appear for θ 0 > θ c ≈ 0.726π and has no parallel in spherically-symmetric potentials. Bound states have spatial extent ζ which diverges as a (possibly λ-dependent) power law as U 0 approaches the value where the eigenenergy of that state vanishes.

AB - 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 2θ 0 emanating from the center of the sphere, with 0 < θ 0 < π. 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 φ and polar angle θ as Pλm(cosθ)eimφ where Pλm is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values λ=λim (i = 0, 1, 3, ...) that depend on m and θ 0. Each λim has an infinite sequence of eigenenergies En(λim), 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 θ 0 we demonstrate the validity of Weyl's continuous estimate NW for the exact number of states N up to energy E, and evaluate the fluctuations of N around NW . We examine the behavior of bound states in a well of finite depth U 0, and find the critical value U c (θ 0) when all bound states disappear. The radial part of the zero energy eigenstate outside the well is 1/r λ+1, which is not square-integrable for λ ≤ 1/2. (0 < λ ≤ 1/2) can appear for θ 0 > θ c ≈ 0.726π and has no parallel in spherically-symmetric potentials. Bound states have spatial extent ζ which diverges as a (possibly λ-dependent) power law as U 0 approaches the value where the eigenenergy of that state vanishes.

KW - density of states

KW - quantum

KW - spherical well

UR - http://www.scopus.com/inward/record.url?scp=85133707799&partnerID=8YFLogxK

U2 - 10.1088/2399-6528/ac6bdc

DO - 10.1088/2399-6528/ac6bdc

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AN - SCOPUS:85133707799

SN - 2399-6528

VL - 6

JO - Journal of Physics Communications

JF - Journal of Physics Communications

IS - 5

M1 - 055017

ER -