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Scars in the wavefunction

  1. Apr 17, 2015 #1
    For a particle in a stadium billiard, it is observed that so-called 'scars' in the wavefunction appear at particular eigenvectors. These scars correspond to classical orbits, for example in the case shown below, which corresponds to a classical orbit of period 6


    The paths which can be distinguished in the wavefunction indicate that the particle has a high probability of being located along particular trajectories corresponding to classical orbits.

    A quantum particle does not travel in straight lines and bounce off of boundaries in the way that a classical particle does, so why do these scars corresponding to classical orbits appear in the wavefunction?
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  3. Apr 17, 2015 #2


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  4. Apr 17, 2015 #3


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    As the link above mentions, it is related to the Ehrenfest theorem.
  5. Apr 17, 2015 #4
    I've looked at the Wikipedia pages for the Ehrenfest theorem and Scars. I still don't really understand how the existence of the classical orbits in the wavefunction comes about.

    I know that the correspondence principle roughly says that quantum physics reproduces classical physics in large systems, but this phenomenon seems to be the case even for just one particle in the stadium.

    Also, after reading the Ehrenfest theorem I still don't see how it implies the existence of the scars. It says that "quantum mechanical expectation values obey Newton's classical equations of motion" but I can't really see how this relates to the wavefunction itself taking the form of classical orbits.
  6. Apr 17, 2015 #5


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    I'm not sure the scarring of the wave function is fully understood, eg. its relation to whether the classical system is chaotic or not. For example, Lecian, http://arxiv.org/abs/1311.0488 says "The appearance of scars or the wavefunction for the quantum version of billiard systems has been related with the hypothesis that the eigenvalues for the problem associated with the Laplace-Beltrami operator on the UPHP be generated by random-matrix theoretical models. ... Within the present work, the appearance of periodic orbits has been connected to the BKL statistics. From the quantum point of view, the presence of scars in correspondence to periodic orbits has also been motivated with different hypotheses."

    From what I understand, the major hypothesis is that scars in the wave function have something to do with periodic orbits in systems whose classical counterparts are chaotic. Here are some references which may be helpful.

    http://www.ericjhellergallery.com/index.pl?page=image;iid=22 [Broken]
    http://www.ericjhellergallery.com/index.pl?page=image;iid=23 [Broken]
    Kaplan and Heller, http://arxiv.org/abs/chao-dyn/9809011
    Provost and Baranger, http://arxiv.org/abs/chao-dyn/9305003
    Keating and Prado, http://arxiv.org/abs/nlin/0010022
    Last edited by a moderator: May 7, 2017
  7. Apr 17, 2015 #6

    Thanks for the response. I have been trying to gain a deeper understanding of the phenomenon of wavefunction scarring and I'll share what I have found in case others searching for this find it useful (most of this explanation is taken from papers I've linked below):

    It was first noticed by McDonald (1983) that individual eigenfunctions can have enhanced intensity along periodic orbits in classically chaotic systems. This phenomenon was later studied by Heller (1984), who called such structures scars. He developed a theory of scarring, based on wave packet dynamics.
    Keating and Prado: http://www2.maths.bris.ac.uk/~majpk/papers/49.pdf

    It seems that Heller's paper (1984) shows that if one were to consider a Gaussian wavepacket (which serves as a solution to the Schrodinger equation) being launched along a classical periodic orbit, then the autocorrelation of the wavepacket (its overlap at some later time with the initial wavepacket) has 'peaks' at multiples of the period of that classical orbit. It follows that certain unstable periodic orbits permanently scar some quantum eigenfunctions, in the sense that extra density surrounds the region of the periodic orbit (as h-bar tends to zero i.e. in the semi-classical limit).
    Heller: http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.53.1515 [Broken]

    Its important to note that the classical trajectories do not cause the scarring, because reference to classical structures can be eliminated from the explanation, since wavefunction scarring is essentially a quantum mechanical, phase interference phenomenon. However, philosopher of science Alisa Bokulich argues that a purely quantum mechanical explanation obtained by numerically solving the Schrodinger equation, without reference to classical structures at all, is deficient because it fails to provide an adequate understanding of the phenomenon.
    Bokulich: http://bjps.oxfordjournals.org/content/59/2/217.full.pdf+html

    She argues that knowledge of the classical periodic orbits completes the understanding of wavefunction scarring because it allows you to answer a wider variety of so called w-questions (what-if-things-had-been-different questions) regarding the phenomenon, and although it is possible to explain wavefunction scarring without reference to these classical orbits, it comes at a high cost of understanding.

    All this being said, I've also come across a statement that the classical trajectories are "dictated by Ehrenfest's theorem" and so that seems to be what the fundamental 'link' is classical and quantum behaviour which connects the wavefunction to the classical orbits.
    Wisniacki, Borondo and Benito: http://arxiv.org/pdf/quant-ph/0309083.pdf
    Last edited by a moderator: May 7, 2017
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