Lagrangian Densities and Principle of Least Action in Nonrelativistic Quantum Mechanics

TL;DR

This paper demonstrates how the Principle of Least Action applied to a simple Lagrangian density can derive the Schrödinger and Gross-Pitaevskii equations, connecting variational principles with quantum dynamics.

Contribution

It introduces a simple Lagrangian density involving second-order derivatives that reproduces key quantum equations and extends to time-dependent many-boson systems.

Findings

Derives Schrödinger equation from a simple Lagrangian density.

Shows Hamiltonian density yields Schrödinger equation via Hamilton's equations.

Obtains Gross-Pitaevskii equation for Bose-Einstein condensates using variational principles.

Abstract

The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.