Nodal algebraic curves and entropy diagnostics in degenerate two-dimensional harmonic-oscillator shells
Degeneracy allows the nodal structure of a quantum state to vary without changing its energy. We study this effect for real superpositions in the fixed-energy shells of the two-dimensional isotropic harmonic oscillator. In Cartesian coordinates $(x,y)$, any such state in the $N$th shell has the parametrized form $ψ_N(x,y;\mathbf{c})=ψ_0(x,y)\,P_N(x,y;\mathbf{c})$, where $ψ_0$ is the non-vanishing ground state and the real coefficients $\mathbf{c}$ determine the polynomial factor $P_N$. The nodal set is exactly the algebraic curve $P_N=0$. Thus, varying $\mathbf{c}$ reshapes this curve at fixed energy. We show that its topology can change only when it develops a finite singular point, $P_N=\nabla P_N=0$, or when two asymptotic nodal directions merge. We characterize this geometry using three diagnostics: the nodal-domain entropy $S_{\rm dom}$, the Cartesian mutual information $I(x;y)$, and the entropic uncertainty sum $S_r+S_p$, which respectively probe probability redistribution among nodal domains, coordinate correlations, and global position-momentum delocalization. The lowest shells reveal a clear hierarchy. For $N=1$, mixing only rotates a nodal line. Along a representative $N=2$ path, a closed conic passes through parallel lines into a hyperbola-type curve; $S_{\rm dom}$ clearly detects this transition, while $S_r+S_p$ remains smooth. For $N=3$, merging asymptotic directions brings smooth cubic branches close together and enhances both $S_{\rm dom}$ and $I(x;y)$. These signatures are accessible in Hermite-Gaussian structured light and nearly isotropic trapped oscillators.