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Nodal portraits of quantum billiards: Domains, lines, and statistics

Sudhir Ranjan Jain and Rhine Samajdar

Sudhir Ranjan Jain

  • Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

Rhine Samajdar

  • Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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Rev. Mod. Phys. 89, 045005 – Published 26 December, 2017

DOI: https://doi.org/10.1103/RevModPhys.89.045005

Abstract

This is a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. How a random superposition of plane waves can model chaotic eigenfunctions is discussed and the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution are highlighted. Various approaches to counting the nodal domains—using trace formulas, graph theory, and difference equations—are also illustrated with examples. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wave functions of a “particle in a box” as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity and far-reaching implications of the problem.

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Rev. Mod. Phys. 89, 045005– Published 26 December, 2017
  • Received 27 March 2017

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