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Physical Review E

Radial rescaling approach for the eigenvalue problem of a particle in an arbitrarily shaped box

Erwin Lijnen, Liviu F. Chibotaru, Arnout Jozef Ceulemans

Published Jan 3, 2008• 16 citations
Publisher

Abstract

In the present work we introduce a methodology for solving a quantum billiard with Dirichlet boundary conditions. The procedure starts from the exactly known solutions for the particle in a circular disk, which are subsequently radially rescaled in such a way that they obey the new boundary conditions. In this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision. Test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy.

DOI
10.1103/physreve.77.016702

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In the present work we introduce a methodology for solving a quantum billiard with Dirichlet boundary conditions. The procedure starts from the exactly known solutions for the particle in a circular disk, which are subsequently radially rescaled in such a way that they obey the new boundary conditions. In this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision. Test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy.
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.
A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.
We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and NtoinftyN \\to \\inftyNtoinfty regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
Exact eigenfunctions for a particle in a box are obtained using Gaussian wavepacket dynamics. The eigenfunctions are obtained by propagating, without approximation, an infinite set of Gaussian wavepackets that collectively satisfy the boundary conditions of the problem, being coherent states appropriate to this problem. The method of images is applied to enforce these boundary conditions. This technique may be applied to the quantum billiard problem whenever the particle is confined to any open or closed region that tessellates space, regardless of the dimension of the region. Also, it is shown that the use of frozen Gaussians along with the De Leon-Heller spectral quantisation method gives the exact solution for the one-dimensional problem as well as for the above multi-dimensional problems, provided the components of the momentum of the wavepackets are chosen at random.
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.