Lagrangian form of Schr\"odinger equation

TL;DR

This paper develops a Lagrangian formulation of the Schr"odinger equation, connecting it to classical field equations like Klein-Gordon, and illustrating its application in different bases.

Contribution

It introduces a Lagrangian framework for quantum mechanics, providing a second order real vector space equation and linking to Klein-Gordon for real fields.

Findings

Lagrangian form of Schr"odinger equation is derived in general.

The Klein-Gordon equation is shown to be the Lagrangian form for real fields.

Application demonstrated in eigenbasis and coordinate representation.

Abstract

Lagrangian formulation of quantum mechanical Schr\"odinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein-Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schr\"odinger equation.