Quantum Mechanics in a Spherical Wedge: Complete Solution and Implications for Angular Momentum Theory
Mustafa Bakr, Smain Amari
TL;DR
The paper analyzes a quantum particle confined to a spherical wedge with Dirichlet boundaries, solving the Schrödinger equation by separation of variables to reveal three key effects of boundary conditions on angular momentum. The azimuthal sector yields non-integer effective quantum numbers μ = nφπ/Φ, producing standing waves that are not eigenstates of Lz and exhibit zero mean but nonzero variance for Lz. The polar part enforces a sectoral-tesseral hierarchy via ν−μ ∈ Z_{≥0}, with sectoral states existing for any μ>0 and tesseral states arising only when μ is effectively an integer offset. Applying the framework to a Coulomb potential shows that the familiar integer angular-momentum spectrum of hydrogen arises from the 2π periodic domain, while the wedge physically reorganises the spectrum and breaks the SO(4) symmetry, offering a clear, exact demonstration of how domain constraints shape quantum observables. The model provides conceptual insight into angular-momentum quantisation and its dependence on the Hilbert space boundary conditions, with connections to quantum dots, cold-atom traps, and topological defects.
Abstract
We solve the stationary Schrödinger equation for a particle confined to a 3D spherical wedge -- the region $\{(r,θ,φ): 0 \leq r \leq R,\, 0 \leq θ\leq π,\, 0 \leq φ\leq Φ\}$ with Dirichlet BCs on all surfaces. This exactly solvable constrained-domain model exhibits spectral reorganisation under symmetry-breaking BCs and provides an operator-domain viewpoint on angular momentum quantisation. We obtain three main results. First, the stationary states are standing waves in the azimuthal coordinate and consequently are \emph{not} eigenstates of $\hat{L}_z$; we prove $\langle L_z \rangle = 0$ with $ΔL_z = \hbar n_φπ/Φ\neq 0$, demonstrating that angular momentum projection becomes an observable with genuine quantum uncertainty rather than a good quantum number. Second, the effective azimuthal quantum number $μ= n_φπ/Φ$ is generically non-integer, and square-integrability of the polar wavefunctions at both poles requires the angular eigenvalue parameter $ν$ to satisfy $ν- μ\in \mathbb{Z}_{\geq 0}$. This regularity constraint yields a hierarchy: sectoral solutions ($ν= μ$, satisfying the first-order highest-weight condition) exist for any real $μ> 0$, while tesseral and zonal solutions require integer steps, appearing only when $μ$ itself is integer. Third, application to a Coulomb potential shows that the familiar integer angular momentum spectrum of hydrogen arises from the periodic identification $φ\sim φ+ 2π$ that defines the full-sphere Hilbert space domain; modified boundary conditions yield a reorganised spectrum with non-integer effective angular momentum. The model clarifies the distinct roles of single-valuedness (selecting integer $m$ via azimuthal topology) and polar regularity (selecting integer $\ell \geq |m|$ via analytic constraints) in the standard quantisation of orbital angular momentum.