A variational principle, wave-particle duality, and the Schr\"{o}dinger equation

TL;DR

This paper introduces a variational principle combining wave and particle aspects of quantum particles, deriving the Schrödinger equation as a condition where two functionals' variations are equal, thus formalizing wave-particle duality.

Contribution

It presents a novel variational principle that unifies wave and particle properties to derive the Schrödinger equation from first principles.

Findings

Derives Schrödinger equation from a variational principle involving two functionals.

Provides a mathematical formulation of wave-particle duality.

Links the phase of the wave function to the total energy of the particle.

Abstract

A principle is proposed according to which the dynamics of a quantum particle in a one-dimensional configuration space (OCS) is determined by a variational problem for two functionals: one is based on the mean value of the Hamilton operator, while the second one is based on the mean value of the total energy of the particle, which is determined through the phase of the wave function with help of the generalized Planck-Einstein relation. The first functional contains information about the corpuscular properties of a quantum particle, and the second one comprises its wave properties. The true dynamics is described by a wave function for which the variations of these two functionals are equal. This variational principle, which can also be viewed as a mathematical formulation of wave-particle duality, leads to the Schr\"{o}dinger equation.