Schroedinger equation for current carrying states

TL;DR

This paper formulates a modified Schr"odinger equation incorporating a known current density using a Lagrange multiplier, enabling analysis of electronic transport and current-voltage characteristics in molecular devices.

Contribution

It introduces a novel non-linear Schr"odinger equation with a subsidiary current constraint, advancing the theoretical modeling of current-carrying quantum states and transport phenomena.

Findings

Derived a non-linear self-consistent Schr"odinger equation for current states.

Applied the approach to molecular electronic transport and derived a new current-voltage relation.

Validated the method through modeling of current in a one-dimensional harmonic oscillator.

Abstract

Schr\"odinger equation with given, {\it a priori} known current is formulated. A non-zero current density is maintained in the quantum system via a subsidiary condition imposed by vector, local Lagrange multiplier. Constrained minimization of the total energy on the manifold of an arbitrary current density topology results into a non-linear self-consistent Schr\"odinger equation. The applications to electronic transport in two-terminal molecular devices are developed and new macroscopic definition of a molecular current-voltage characteristic is proposed. The Landauer formula for the conductance of an ideal one-dimensional lead is obtained within the approach. The method is examined by modeling of current carrying states of one-dimensional harmonic oscillator.