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Investigation of Quantum Waveguides (Solution of Helmholtz's Equation at any Arbitrary Cross sectional Area)

Ahmad Tabatabaeifar, Heydar Izadneshan

Published Jan 1, 2010• 0 citations

Abstract

Introduction: Nowadays extraordinary properties of nano world and economical problems are the reasons for directing techniques toward micro implements, electronic circuits dimensions have been reduced to 10 to 100 nano meters. Moreover microstructures in cells with nano meter dimensions have been investigated by biologists, there are structures in electronic and biology which are very similar to the billiards and the electromagnetic waveguides but their dimensions are in nano meter range, therefore they were called quantum waveguides and billiards. Aim: In this research billiards and quantum waveguides at any arbitrary cross sectional area have been investigated. Method: A considerable effort was devoted in solving Helmholtz’s equation with two dimensional cross sectional areas. Linear combinations of plane waves or Heller methods for several types of connective and no connective billiards with regular and irregular shapes have been used. Then eigenvalues and eigenvectors were calculated. At the final part by applying a simple method billiard type (chaotic or quiet) in a quantum scale has been realized. Results and Discussions: In this paper two types of circular and stadium billiards have been studied and by using Heller’s method and applying the numerical solution, energy levels together with the function forms have been attained. Conclusions: This paper shows that Heller’s method despite its quickness and simplicity is a very accurate method for finding eigenvalues and eigenvectors. Although in high eigenvalues the calculation is weighted, nevertheless this method is an acceptable computational method for the investigation of quantum nanobilliards such as quantum points.

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Introduction: Nowadays extraordinary properties of nano world and economical problems are the reasons for directing techniques toward micro implements, electronic circuits dimensions have been reduced to 10 to 100 nano meters. Moreover microstructures in cells with nano meter dimensions have been investigated by biologists, there are structures in electronic and biology which are very similar to the billiards and the electromagnetic waveguides but their dimensions are in nano meter range, therefore they were called quantum waveguides and billiards. Aim: In this research billiards and quantum waveguides at any arbitrary cross sectional area have been investigated. Method: A considerable effort was devoted in solving Helmholtz’s equation with two dimensional cross sectional areas. Linear combinations of plane waves or Heller methods for several types of connective and no connective billiards with regular and irregular shapes have been used. Then eigenvalues and eigenvectors were calculated. At the final part by applying a simple method billiard type (chaotic or quiet) in a quantum scale has been realized. Results and Discussions: In this paper two types of circular and stadium billiards have been studied and by using Heller’s method and applying the numerical solution, energy levels together with the function forms have been attained. Conclusions: This paper shows that Heller’s method despite its quickness and simplicity is a very accurate method for finding eigenvalues and eigenvectors. Although in high eigenvalues the calculation is weighted, nevertheless this method is an acceptable computational method for the investigation of quantum nanobilliards such as quantum points.

The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

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We introduce an additional method to solve Schr\"odinger's equation for a free particle in an infinite well of arbitrary shape (the Helmholtz equation with Dirichlet boundary conditions), a problem of interest in the area of quantum chaos. We expand the wave function in a basis of products of sine functions, then use the constraint operator to contain the wave function to a region within the domain of the basis functions. In this manner, a quantum billiard problem of arbitrary shape can be solved. Several methods exist to solve problems of this sort, but as recent work reviewing these methods has shown, all have shortcomings. Our work represents a different direction in the solution of these problems. Our method is different in that it provides a means of computing an eigenbasis. It is also interesting from a physical standpoint in that it can represent the Hamiltonian of a classically chaotic system in the basis of a classically regular system. \textcopyright{} 1996 The American Physical Society.

The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy. From the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains. In contrast, in the present work we demonstrate that eigenstates of non-convex billiards, in general, cannot be approximated by any solution of the Helmholtz equation regular everywhere in R 2 (in particular, by linear combinations of a finite number of plane waves having the same energy). From this we infer that PWDM cannot be applied to billiards in non-convex domains. Furthermore, it follows from our results that unlike the properties of integrable billiards, where each eigenstate can be extended into the billiard exterior as a regular solution of the Helmholtz equation, the eigenstates of non-convex billiards, in general, do not admit such an extension.

Until a short time ago, the study of the quantum-mechanical behaviour of classically chaotic systems was an exclusive domain of the theoretical physicist. Model systems amongst others were irregularly shaped billiards, such as the Sinai and the stadium billiard. An alternative experimental approach to study these systems has recently been demonstrated using the fact that the time-independent Schrödinger and wave equations are mathematically equivalent (though, in general, the boundary conditions are different). This chapter concentrates on the presentation of microwave studies of billiards, but experiments with vibrating plates and water surfaces waves are also briefly discussed. Topics are the statistical properties of spectra, level dynamics with respect to geometrical changes of the billiards, and the close connection between quantum-mechanical spectra and classical trajectories. Introduction One day in February 1808 Ernst F. Chladni who was then on a long stay in Paris received an invitation to the Tuileries to give a demonstration of his famous experiments with vibrating plates in the presence of Napoleon and an illustrious audience. The Emperor was impressed by the performance and offered a prize of 3000 francs for the correct mathematical explanation of the sound figures which was still lacking at that time. The prize was paid in 1816 to the French mathematician Sophie Germain, in spite of the fact that her solution was still incomplete. The correct explanation for circular plates was not given until 1850 by Robert Kirchhoff (all the above details are taken from the book of F. Melde on the life of Chladni [1]).

A simple one-dimensional differential equation in the centerline coordinate of an arbitrarily curved quantum waveguide with a varying cross section is derived using a combination of differential geometry and perturbation theory. The model can tackle curved quantum waveguides with a cross-sectional shape and dimensions that vary along the axis. The present analysis generalizes previous models that are restricted to either straight waveguides with a varying cross-section or curved waveguides, where the shape and dimensions of the cross section are fixed. We carry out full 2D wave simulations on a number of complex waveguide geometries and demonstrate excellent agreement with the eigenstates and energies obtained using our present 1D model. It is shown that the computational benefit in using the present 1D model to calculate both 2D and 3D wave solutions is significant and allows for the fast optimization of complex quantum waveguide design. The derived 1D model renders direct access as to how quantum waveguide eigenstates depend on varying cross-sectional dimensions, the waveguide curvature, and rotation of the cross-sectional frame. In particular, a gauge transformation reveals that the individual effects of curvature, thickness variation, and frame rotation correspond to separate terms in a geometric potential only. Generalization of the present formalism to electromagnetics and acoustics, accounting appropriately for the relevant boundary conditions, is anticipated.