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Quantum particle in a spherical well confined by a cone
Quantum Physics · Submitted 4 Jul 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim ($0<\lambda\le 1/2$ 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.
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The abstract, sentence by sentence
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 $\varphi$ and polar angle $\theta$ as $P_\lambda^m(\cos\theta){\rm e}^{im\varphi}$ 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 ${\cal N}_W$ for the exact number of states $\cal N$ up to energy $E$, and evaluate the fluctuations of $\cal N$ around ${\cal N}_W$. 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $0<\theta_0<\pi$. | contribution 0.63 | 0.71 | 0.04 | asserted 0.76 |
| s1 | This non-central potential can be solved by an extension of techniques used in spherically-symmetric problems. | method 0.93 | 0.27 | 0.06 | asserted 1.00 |
| s2 | The angular parts of the eigenstates depend on azimuthal angle $\varphi$ and polar angle $\theta$ as $P_\lambda^m(\cos\theta){\rm e}^{im\varphi}$ where $P_\lambda^m$ is the associated Legendre function of integer order $m$ and (usually noninteger) degree $\lambda$. | result 0.79 | 0.66 | 0.06 | asserted 1.00 |
| s3 | There is an infinite discrete set of values $\lambda=\lambda_i^m$ ($i=0,1,3,\dots$) that depend on $m$ and $\theta_0$. | result 0.95 | 0.76 | 0.07 | asserted 1.00 |
| s4 | Each $\lambda_i^m$ has an infinite sequence of eigenenergies $E_n(\lambda_i^m)$, with corresponding radial parts of eigenfunctions. | result 0.92 | 0.69 | 0.06 | asserted 1.00 |
| s5 | In a spherical box the discrete energy spectrum is determined by the zeros of the spherical Bessel functions. | result 0.64 | 0.34 | 0.05 | asserted 1.00 |
| s6 | For several $\theta_0$ we demonstrate the validity of Weyl's continuous estimate ${\cal N}_W$ for the exact number of states $\cal N$ up to energy $E$, and evaluate the fluctuations of $\cal N$ around ${\cal N}_W$. | result 0.69 | 0.94 | 0.15 | asserted 1.00 |
| s7 | 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. | result 0.63 | 0.96 | 0.15 | asserted 1.00 |
| s8 | The radial part of the zero energy eigenstate outside the well is $1/r^{\lambda+1}$, which is not square-integrable for $\lambda\le 1/2$. | result 0.99 | 0.77 | 0.06 | asserted 1.00 |
| s9 | ($0<\lambda\le 1/2$ 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. | result 0.93 | 0.79 | 0.16 | hedged possibility 0.55 |
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Original 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 $\varphi$ and polar angle $\theta$ as $P_\lambda^m(\cos\theta){\rm e}^{im\varphi}$ 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 ${\cal N}_W$ for the exact number of states $\cal N$ up to energy $E$, and evaluate the fluctuations of $\cal N$ around ${\cal N}_W$. 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.
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