Quantum Mechanics in a Spherical Wedge: Complete Solution and Implications for Angular Momentum Theory

TL;DR

This paper provides an exact solution for a particle in a spherical wedge, revealing how boundary conditions affect angular momentum quantisation and demonstrating that angular momentum projection can be a quantum observable with inherent uncertainty.

Contribution

It introduces a solvable model of a particle in a spherical wedge, analyzing the impact of boundary conditions on angular momentum quantisation and spectral structure.

Findings

Angular momentum projection has quantum uncertainty and is not a good quantum number.

Effective azimuthal quantum number can be non-integer, affecting wavefunction regularity.

Boundary conditions modify the hydrogen spectrum, breaking the usual integer angular momentum quantisation.

Abstract

We solve the stationary Schr\"odinger equation for a particle confined to a 3D spherical wedge -- the region $\{(r,\theta,\phi): 0 \leq r \leq R,\, 0 \leq \theta \leq \pi,\, 0 \leq \phi \leq \Phi\}$ 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 $\Delta L_z = \hbar n_\phi\pi/\Phi \neq 0$, demonstrating that angular momentum projection becomes an observable with genuine quantum uncertainty rather than a good quantum number. Second, the effective azimuthal quantum number $\mu = n_\phi\pi/\Phi$ is generically non-integer, and square-integrability of the polar wavefunctions at both poles requires the angular eigenvalue parameter $\nu$ to satisfy $\nu - \mu \in \mathbb{Z}_{\geq 0}$. This regularity constraint yields a hierarchy: sectoral solutions ($\nu = \mu$, satisfying the first-order highest-weight condition) exist for any real $\mu > 0$, while tesseral and zonal solutions require integer steps, appearing only when $\mu$ itself is integer. Third, application to a Coulomb potential shows that the familiar integer angular momentum spectrum of hydrogen arises from the periodic identification $\phi \sim \phi + 2\pi$ 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.