Toward a Quantum Definition of π in Molecular Systems: Original Formula, Mathematical Framework, and Foundational Implications

Abstract

I propose an operational definition of a quantum π for molecular systems — a dimensionless, system-specific invariant π_q that generalizes the classical constant π to include electronic-phase topology and density-weighted phase winding of delocalized electrons. I define π_q via a density-weighted winding-number of the complex electronic amplitude around chemically relevant cycles (rings or effective closed paths), and I show how π_q reduces to the classical π in canonical limits (simple particle-on-a-ring, uniform density). I derive the formula from the Madelung (polar) decomposition of molecular wavefunctions, demonstrate its mathematical properties (gauge invariance, additivity under non-overlapping cycles, continuity under weak perturbations), and relate π_q to observable quantities: energy spacing of ring modes, current (ring magnetism), and phase-sensitive spectroscopic signals. I illustrate the concept analytically (particle-on-a-ring), semi-analytically (Hückel benzene), and numerically (finite conjugated chain model). Finally I discuss implications for aromaticity, molecular electronics, and a program to test π_q experimentally.

Quantum pi Molecular topology Electronic phase continuity Electron delocalization Aromaticity Quantum chemistry Wavefunction periodicity Molecular orbital theory Topological descriptors Phase coherence Electronic structure Quantum-π series Conjugated systems Molecular resonance