Most classical many-body systems are chaotic in the sense that trajectories that start off nearby end up in completely different -- essentially independent -- places. (This is more or less intuitive, consider what happens when you fuck up a pool shot.) Since a quantum trajectory is, intuitively, just a smeared-out quantum trajectory, one might expect the quantum mechanical evolution of a chaotic system to spread out a narrowly focused initial wavefunction all over trajectory space: it would appear to follow from this logic that the steady-state wavefunctions, i.e. the eigenfunctions, should be a featureless average over trajectories.
In fact this isn't true, and it turns out that closed classical trajectories, even if they're unstable to small perturbations, leave "scars" of high density on the eigenfunctions, and therefore on the steady-state and equilibrium density profiles of various systems. The scarring effect was first really explained by Eric Heller in 1984:
Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits
Phys. Rev. Lett. 53, 1515 - 1518 (1984)
Eric J. Heller
In the latest PRL, Huang et al. extend this work to relativistic quantum systems described by the Dirac equation. Among other things, the new work offers a nice example of how the electron gas in graphene, which obeys the Dirac equation, can be used to study features of relativistic quantum mechanics that would otherwise be experimentally inaccessible.
Relativistic Quantum Scars
Phys. Rev. Lett. 103, 054101 (2009)
Liang Huang et al.
Abstract. The concentrations of wave functions about classical periodic orbits, or quantum scars, are a fundamental phenomenon in physics. An open question is whether scarring can occur in relativistic quantum systems. To address this question, we investigate confinements made of graphene whose classical dynamics are chaotic and find unequivocal evidence of relativistic quantum scars. The scarred states can lead to strong conductance fluctuations in the corresponding open quantum dots via the mechanism of resonant transmission.
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Disclaimer: I didn't read the article. I always thought of the scars as paths whose quantum trajectories (in the path integral sense) interfered constructively. If you consider any closed classical path, you should be able to choose some wavelength to construct it that obeys the BCs. This will give you the higher probability of the path in the places where the classical path lies up to some random quantum interference from other paths of that same energy (frequency).
ReplyDeleteInterfered constructively with what? Classically allowed paths that start off near the scar don't stay near the scar for very long by hypothesis. Presumably there are classically _disallowed_ paths -- quantum decorations of the scar -- that stay roughly in phase with it, because the scar's a saddle point, but I have the sense that this remark proves too much.
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