Experimental determination of billiard wave functions

Abstract

Wave functions of a stadium billiard are determined in a microwave analog experiment. It is shown that Gutzwiller's semiclassical representation of the Green's function in terms of classical trajectories can account not only for eigenvalue spectra but also for eigenfunction patterns.PACS numbers: 05.45.+bThe perhaps most impressive way to demonstrate the qualitative difference between integrable and nonintegrable systems is the presentation of eigenfunction patterns of billiards. Whereas for a rectangular or circular billiard the nodal lines form a regular grid with a large number of crossings, for nonintegrable billiards the node lines perform meandric walks while avoiding any crossing (in the presence of degeneracies the crossings can be destroyed by suitable superpositions of eigenfunctions also in integrable billiards; for details see Chap. of Ref. [l]). This was first demonstrated by McDonald and Kaufman[2] who calculated wave functions in a stadium billiard (a more detailed account of this work can be found in Ref.[3]). A surprise was then the discovery by Heller that many wave functions are not distributed more or less uniformly over the whole billiard but form so-called scars, regions of extra high amplitudes near classical periodic orbits [4,5]. An approach to explain these scars was made by Bogomolny [6]. He used the fact that the Green's function,can be expressed semiclassically as a sum over classically allowed trajectories [7], G(q A ,qB,E)Him) 1/2 XlA, I 1/2 exp -rS r (q A ,q B ,E) -im rn l(2)In Eq. (1) E n and <p"(q) denote eigenenergy and eigenfunction of the Afth eigenvalue, respectively. In Eq. (2) the sum is over all trajectories starting at q A and ending at q B . S r (q Ay q B ,E) is the classical action for the trajectory, and m r is the Maslov index counting the number of conjugated points. For billiards the action is given by S(q Ay q B ,E)=hkq.Here q denotes the total length of the trajectory and k =J2mE/h is the de Broglie wave number of the particle. The Maslov index is given by m r ss 2/i r , where n r is the number of reflections (for a billiard with hard walls every reflection gives rise to a phase jump of K). The prefactor |A r | ,/2 entering into Eq. (2) takes into account the stability of the trajectory. A r can be written as a Jacobian determinant, A r Hd/>3(^,<7*,£)/9?!|, M-U2,where the p A are the two components of the momentum at the starting point, written as functions of q A ,q B ,E y and ql are the two components of the end position. Classically, A r is proportional to the density of trajectories at point qs starting isotropically from point q A (a very readable account of these questions can be found in Ref. [8]). Taking q A =q B =<7 in Eq.(2) and integrating over q, one gets the Gutzwiller trace formula establishing a correspondence between periodic orbits and the quantum-mechanical spectrum. This correspondence could be demonstrated experimentally by the present authors for electromagnetic eigenfrequency spectra of billiard-shaped microwave resonators (Ref.[9], hereafter...

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