Least constraint approach to non-relativistic quantum mechanics

TL;DR

This paper introduces a variational principle for non-relativistic quantum mechanics based on a least constraint approach, leading to quantum Euler equations equivalent to Schrödinger's equation.

Contribution

It presents a novel variational formulation inspired by Gauss's principle, unifying geometric constraints and dissipative forces in quantum dynamics.

Findings

Derives quantum Euler equations from the least constraint principle.

Shows equivalence of the variational formulation to Schrödinger's equation.

Provides a differential, instantaneous characterization of quantum evolution.

Abstract

We formulate a variational principle for non-relativistic quantum mechanics inspired by Gauss's principle of least constraint. We define a quantum constraint functional as the probability-weighted square deviation between the actual motion and the unconstrained motion that would arise from external forces alone. In this functional, the quantum potential plays the role of an intrinsic constraint that modifies the acceleration. Minimizing this quantum constraint functional with respect to the acceleration field yields the quantum Euler equations, which together with the continuity equation are equivalent to the Schr\"{o}dinger equation. The principle is instantaneous and provides a differential characterization of quantum evolution. We demonstrate that this formulation is not a mere rewriting of existing dynamics: it provides a unified and technically economical treatment of geometric constraints and velocity-dependent dissipative forces, neither of which admits a straightforward global variational formulation. Potential applications to a broad range of quantum phenomena are also indicated.