论文标题
球形良好限制锥体的球形粒子
Quantum particle in a spherical well confined by a cone
论文作者
论文摘要
我们考虑球形盒中粒子的量子问题或有限的球形，该球形良好地限制在圆锥形圆锥体中，顶角$2θ_0$从球体中心发出，$ 0 <θ_0<π$。可以通过用于球形对称问题的技术的扩展来解决这种非中心电位。本征态的角部分取决于Azimuthal Angle $φ$和Polar Angle $θ$作为$P_λ^m（\cosθ）{\ rm e}^{imφ} $，其中$p_λ^m $是整数订单$ m $ and（通常是noninteneger）$ usande noninteger $ unteger unigendre的ligendre函数。有一组无限的离散值$λ=λ_i^m $（$ i = 0,1,3，\ dots $），它取决于$ m $和$θ_0$。每个$λ_i^m $都有一个无限的特征力序列$ e_n（λ_i^m）$，并具有相应的特征性径向部分。在球形盒中，离散的能谱取决于球形贝塞尔函数的零。对于几个$θ_0$，我们证明了Weyl连续估算的有效性$ {\ cal n} _W $，确切的状态$ \ cal n $ to energy $ e $，并评估$ \ cal n $的波动大约$ {\ cal n} _w $。我们检查了有限深度$ u_0 $的界面状态的行为，并在所有绑定状态消失时找到临界值$ u_c（θ_0）$。井外的零能量特征态的径向部分为$ 1/r^{λ+1} $，对于$λ\ le 1/2 $，这是不可正常的。 （$ 0 <λ\ le 1/2 $可以以$θ_0>θ_c\约0.726π$出现，并且在球形对称的电位上没有平行。
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 $P_λ^m(\cosθ){\rm e}^{imφ}$ where $P_λ^m$ is the associated Legendre function of integer order $m$ and (usually noninteger) degree $λ$. There is an infinite discrete set of values $λ=λ_i^m$ ($i=0,1,3,\dots$) that depend on $m$ and $θ_0$. Each $λ_i^m$ has an infinite sequence of eigenenergies $E_n(λ_i^m)$, 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 $θ_0$ 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 $U_c(θ_0)$ 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 $λ\le 1/2$. ($0<λ\le 1/2$ can appear for $θ_0>θ_c\approx 0.726π$ and has no parallel in spherically-symmetric potentials.) Bound states have spatial extent $ξ$ which diverges as a (possibly $λ$-dependent) power law as $U_0$ approaches the value where the eigenenergy of that state vanishes.
