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Journal of Physics A Mathematical and General

Quantum mechanics on graphs

Jonathan Gratus, Colin John Lambert, S. J. Robinson, R.W. Tucker

Published Oct 21, 1994• 30 citations
Publisher

Abstract

We analyse the problem of one-dimensional quantum mechanics on arbitrary graphs as idealized models for quantum systems on spaces with non-trivial topologies. In particular we argue that such models can be made to accommodate the physical characteristics of wavefunctions on a network of wires and offer several derivations of a particular junction condition. Throughout we adopt a continuity condition for the wavefunction at each primitive node in the network. Results are applied to the problem of the energy spectrum of a system containing one and infinitely many junctions.

DOI
10.1088/0305-4470/27/20/023

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We analyse the problem of one-dimensional quantum mechanics on arbitrary graphs as idealized models for quantum systems on spaces with non-trivial topologies. In particular we argue that such models can be made to accommodate the physical characteristics of wavefunctions on a network of wires and offer several derivations of a particular junction condition. Throughout we adopt a continuity condition for the wavefunction at each primitive node in the network. Results are applied to the problem of the energy spectrum of a system containing one and infinitely many junctions.
Abstract One usually assumes that the Schrödinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrödinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrödinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrödinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.
States of matter which are described by concepts in topology are of great interest in the community of condensed matter physics. Both from a pure theoretical perspective, where many non-intuitive and mathematically rich results can be found from seemingly simple models, and also since recent experimental developments allows for explicit construction of devices where the underlying topology has observable consequences. This thesis provides a conceptual introduction to the framework describing such topological states of matter (TSM). It starts by explaining ordinary band theory which is followed by a discussion of quantum mechanical symmetries. These two concepts provides a method of classifying all non-interacting fermionic TSM. Following are explicit applications to a few models in one spatial dimension. In the accompanying paper, we study one-dimensional superconductors with Josephson-junctions allowing phase shifts of π. We make a detailed analysis of the junction bound states and their properties in some different settings and compare the behaviour of trivial and topological superconducting junctions. In addition, we provide a phenomenological topological field theory in the low energy limit.
In this article we continue our investigations of one particle quantum scattering theory for Schroedinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff's rule for quantum wires'' [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary n×nn\times nn×n matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian.
Assembling large-scale quantum networks is a key goal of modern physics research with applications in quantum information and computation. Quantum wires and waveguides in which massive particles propagate in tailored confinement is one promising platform for realizing a quantum network. In the literature, such networks are often treated as quantum graphs, that is, the wave functions are taken to live on graphs of one-dimensional edges meeting in vertices. Hitherto, it has been unclear what boundary conditions on the vertices produce the physical states one finds in nature. This paper treats a quantum network from a physical approach, explicitly finds the physical eigenstates and compares them to the quantum-graph description. The basic building block of a quantum network is an X-shaped potential well made by crossing two quantum wires, and we consider a massive particle in such an X well. The system is analyzed using a variational method based on an expansion into modes with fast convergence and it provides a very clear intuition for the physics of the problem. The particle is found to have a ground state that is exponentially localized to the center of the X well, and the other symmetric solutions are formed so to be orthogonal to the ground state. This is in contrast to the predictions of the conventionally used so-called Kirchoff boundary conditions in quantum graph theory that predict a different sequence of symmetric solutions that cannot be physically realized. Numerical methods have previously been the only source of information on the ground-state wave function and our results provide a different perspective with strong analytical insights. The ground-state wave function has a spatial profile that looks very similar to the shape of a solitonic solution to a nonlinear Schr\"odinger equation, enabling an analytical prediction of the wave number. When combining multiple X wells into a network or grid, each site supports a solitonlike localized state. These localized solutions only couple to each other and are able to jump from one site to another as if they were trapped in a discrete lattice.
Here I present a new discrete model of quantum mechanics for relativistic 1-electron systems, in which particle movement is described by a directed space-time graph with attached 4-spinors, but without any continuous wave functions. These graphs only consist of few space-like edges, e.g. the ground state of atoms is described by two nodes and one edge, and interactions only take place at the nodes. The fundament is an extremal principle for a relativistic invariant "Lagrangian sum", from which "field-equations" and "equations of motion" are derived, so the states (including the graph nodes) are completely determined. As important validations of the model, the corresponding graphs for the stationary Dirac-equation for the atom are drawn and the correct spectra are computed (Sommerfeld-levels). Also a discrete Schroedinger approximation and an associated "Hamiltonian sum" are derived and the correct equation of a classical moving particle under Lorentz-force is presented. I hope, that this new approach will help, to overcome some problems of current quantum mechanics by making the wave function superfluous.
Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs ” — in other words, suitable families of shrinking manifolds — and discuss convergence of the spectra and resonances in such a setting. 1.