Nodal algebraic curves and entropy diagnostics in degenerate two-dimensional harmonic-oscillator shells

TL;DR

This paper investigates how algebraic curves and entropy measures reveal topology changes in nodal patterns of degenerate quantum harmonic oscillator states, with implications for experimental systems and quantum information.

Contribution

It introduces algebraic criteria and entropy diagnostics to understand topology transitions in degenerate eigenspaces of the 2D harmonic oscillator, linking geometry to information measures.

Findings

N=1 shell only rotates a nodal line.

N=2 shell exhibits a conic transition detected by entropy.

N=3 shell shows cubic regimes with enhanced entropy responses.

Abstract

Degenerate quantum eigenspaces can support substantial changes in nodal geometry at fixed energy. We show that, for the two-dimensional isotropic harmonic oscillator, this restructuring is organized by the Hermite-constrained algebraic curve \(P_N(x,y)=0\) associated with each real shell state, $\psi_N(x,y)=e^{-\alpha r^2/2}P_N(x,y)$. Finite singularities, \(P_N=\nabla P_N=0\), together with projective degeneracies of the leading homogeneous part, identify the strata where topology-changing events can occur. We combine these algebraic criteria with three information diagnostics: the nodal-domain entropy \(S_{\rm dom}\), the Cartesian mutual information \(I(x;y)\), and the entropic uncertainty sum \(S_r+S_p\). The first three shells reveal a clear hierarchy. The \(N=1\) shell only rotates a nodal line; the \(N=2\) shell exhibits a conic transition at \(b^2=2ac\), sharply detected by \(S_{\rm dom}\) but not by global entropies; and the \(N=3\) shell supports cubic close-branch regimes organized by the projective discriminant, with enhanced responses in \(S_{\rm dom}\) and \(I(x;y)\). Thus algebraic stratification, rather than spectral ordering, organizes nodal geometry inside a degenerate eigenspace, while entropy diagnostics quantify the associated probability redistribution and coordinate correlations. The same stratification defines experimentally testable signatures in real-phase Hermite--Gaussian structured light and approximately isotropic trapped motional systems, and suggests a geometry-sensitive verification primitive for fixed-shell bosonic-qudit gates.