Generalized versus selected descriptions of quantum LC - circuits

TL;DR

This paper explores generalized quantum descriptions of LC circuits using a flexible discretization rule for electric charge, leading to new equations and insights into charge conservation and inductance variations.

Contribution

It introduces a generalized discretization approach for quantum LC circuits, extending previous models and analyzing the effects on charge conservation and inductance.

Findings

Generalized discretization rule F(n) modifies the Schrödinger equation.

Site-dependent inductances emerge from periodic solutions of F(n).

Charge and current densities are extended to many-charge systems.

Abstract

Proofs are given that the quantum-mechanical description of the LC -circuit with a time dependent external source can be readily established by starting from a more general discretization rule of the electric charge. For this purpose one resorts to an arbitrary but integer-dependent real function F(n) instead of n. This results in a nontrivial generalization of the discrete time dependent Schrodinger-equation established before via F(n)=n, as well as to modified charge conservation laws. However, selected descriptions can also be done by looking for a unique derivation of the effective inductance. This leads to site independent inductances, but site dependent ones get implied by accounting for periodic solutions to F(n) in terms of Jacobian elliptic functions. Many-charge generalizations of quantum circuits, including the modified continuity equation for total charge and current densities, have also been discussed.