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Journal of Mathematical Physics

Schrödinger equation for convex plane polygons. II. A no-go theorem for plane waves representation of solutions

V. Amar, Massimo Pauri, Alexandria Scotti

Published Aug 1, 1993• 11 citations
Publisher

Abstract

In the present article we discuss whether or not ansatzes more general than those considered hitherto, for the representation in terms of plane waves of eigensolutions of the Schrödinger equation for a free particle in a plane convex polygonal domain, lead to the construction of a complete set for a class of domains larger than the well-known one, containing the polygons for which such a complete set can be obtained in terms of a superposition of a finite number of plane waves. Our conclusion is in the negative. Comments are also made on some features of a representation in terms of analytic functions of two complex variables.

DOI
10.1063/1.530080

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In the present article we discuss whether or not ansatzes more general than those considered hitherto, for the representation in terms of plane waves of eigensolutions of the Schrödinger equation for a free particle in a plane convex polygonal domain, lead to the construction of a complete set for a class of domains larger than the well-known one, containing the polygons for which such a complete set can be obtained in terms of a superposition of a finite number of plane waves. Our conclusion is in the negative. Comments are also made on some features of a representation in terms of analytic functions of two complex variables.
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.
It is shown fields that cannot be represented over one complex plane can be further decomposed for representation over multiple complex planes. This finding is demonstrated here by solving of the Schrödinger equation for the hydrogen atom in a complex space containing three complex planes. The complex coordinate system is generated from real coordinates using an isometric transformation. One plane is applied to mix energy and time; the other two planes are used to represent the z-component of angular momentum of the electron. The eigensolutions of the Schrödinger equation are shown to be holomorphic in the complex planes.
A method for solution of the Schrödinger equation in momentum space is described. If the Schrodinger equation in momentum space is represented by Oχ=0, where O is an integral operator and χ is the momentum function, the method involves two steps: (i) the representation of χ as a rational function of P and (ii) expansion of Oχ in a power series in some appropriately chosen variable t (‖t‖≤1) which is a function of P. Equating to zero the coefficients of the first N powers of t (which determines the eigenvalues) makes Oχ differ from zero by terms of order tN. As N increases the eigenvalues approach the correct limiting values if the method converges. Within its circle of convergence a power series converges for both real and complex values of the variable and hence the Schrödinger equation, an integral equation, is forced to hold for both real and complex values of the variable. This is in contrast to variational methods which involve only real values of the independent variable. To test the method the motion of a particle in the field of a Yukawa potential is studied. Only the lowest energy state is considered. Two extreme cases, involving linear combinations of rational basis sets, are studied. Both basis sets are complete but the power series method converges only for one basis set. For the other set the power series method provides an eigenvalue which oscillates around the correct value without approaching a limit but a variation method using the same basis set converges rapidly to the correct eigenvalue. The variation method succeeds because the function is defined only on the real axis while the power series method fails because, in the complex plane, the basis set can only represent a meromorphic function (one whose only finite singular points are poles) and the momentum eigenfunction cannot be meromorphic. Thus the choice of a wrong basis set in a variation method can completely conceal the true analytic nature of an eigenfunction. It is shown that momentum eigenfunctions are not meromorphic because they possess branch points. This is true not only for the Yukawa potential but for several one-electron models including the Hartree–Fock equations.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.
The present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates. The ‘‘polygon ground-state theorem’’ is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the ‘‘elemental polygons’’ (defined in the text). The ‘‘polygon first excited-state theorem’’ is established which states that for every N-sided regular polygon, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the N-sided regular polygon is spanned by any two of these N nodal-diagonal eigenstates (i.e., the first excited state is twofold degenerate). At various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations. Thus, for example, correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive, nonoverlapping classes of polygons. Stemming from the preceding conclusions, a discussion is included on the possibility of employing transverse magnetic (TM) modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic.