Kirchhoff's Rule for Quantum Wires

TL;DR

This paper develops a quantum scattering theory for finite graphs with multiple channels, explicitly deriving the S-matrix based on boundary conditions and graph geometry, and demonstrating properties like unitarity and symmetry.

Contribution

It introduces a comprehensive quantum scattering framework on arbitrary finite graphs with explicit S-matrix formulas and a composition rule, extending classical Kirchhoff's law to quantum networks.

Findings

The S-matrix is explicitly expressed in terms of boundary conditions and line lengths.

The S-matrix is shown to be unitary, reflecting conservation of probability.

A duality transformation relates low and high energy behaviors of the system.

Abstract

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy $E>0$ is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.