Quantum Mechanics in a Spherical Wedge: Complete Solution and Implications for Angular Momentum Theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915686467502080 |
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| author | Bakr, Mustafa Amari, Smain |
| author_facet | Bakr, Mustafa Amari, Smain |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Mechanics in a Spherical Wedge: Complete Solution and Implications for Angular Momentum Theory Bakr, Mustafa Amari, Smain Quantum Physics 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. |
| title | Quantum Mechanics in a Spherical Wedge: Complete Solution and Implications for Angular Momentum Theory |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2512.17558 |