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Physical Review E

Ghost orbit spectroscopy

A. S. Bhullar, R. Blümel, Patrick M. Koch

Published Jan 13, 2006• 6 citations
Publisher

Abstract

Direct periodic-orbit expansions of individual spectral eigenvalues is a new direction in quantum mechanics. Using a unitary -matrix theory, we present exact, convergent, integral-free ghost orbit expansions of spectral eigenvalues for a step potential in the tunneling regime. We suggest an experiment to extract ghost orbit information from measured spectra in the tunneling regime (ghost orbit spectroscopy). We contrast our unitary, convergent theory with a recently published nonunitary, divergent theory [Yu. Dabaghian and R. Jensen, Eur. J. Phys. 26, 423 (2005)].

DOI
10.1103/physreve.73.016211

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Comment on ‘Quantum chaos in elementary quantum mechanics’ by Yu Dabaghian and R Jensen

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Algebraic, numerical and conceptual problems in a recently published paper by Dabaghian and Jensen ( 2005 Eur. J. Phys. 26 423–439 ) are discussed. A convergent scheme for summing periodic orbits in the tunnelling regime is suggested.

European Journal of Physics•Dec 22, 2005•7 citations
Direct periodic-orbit expansions of individual spectral eigenvalues is a new direction in quantum mechanics. Using a unitary -matrix theory, we present exact, convergent, integral-free ghost orbit expansions of spectral eigenvalues for a step potential in the tunneling regime. We suggest an experiment to extract ghost orbit information from measured spectra in the tunneling regime (ghost orbit spectroscopy). We contrast our unitary, convergent theory with a recently published nonunitary, divergent theory [Yu. Dabaghian and R. Jensen, Eur. J. Phys. 26, 423 (2005)].
In the preceding chapters we have learnt that random matrix theory is perfectly able to explain the universal properties of the spectra of chaotic systems, and this in spite of the oversimplifying assumptions applied. On the one hand it is very satisfactory that one single theory can cope with such a variety of systems as nuclei, mesoscopic structures, or microwave billiards, on the other hand this is a bit disappointing. If there is no possibility of discriminating between the spectra of a nucleus and a quantum dot, then there is little hope of learning anything of relevance about it. Fortunately, random matrix theory is only one side of the coin. We have already come across some examples demonstrating its limits of validity. Remember the spectral level dynamics where bouncing balls disturbed the otherwise universal Gaussian velocity distribution (see Section 5.2.2). Another example is the scarring phenomenon observed in many wave functions. Here obviously closed classical orbits have left their fingerprints in the amplitude patterns. We cannot expect that the universal random matrix theory can correctly account for individual features such as periodic orbits. We now come to an alternative approach to analysing the spectra. As we know from the correspondence principle, in the semiclassical limit quantum mechanics eventually turns into classical mechanics. That is why classical dynamics must be hidden somewhere in the spectra, at least in the limit of high quantum numbers. In the introduction we have already discussed this connection for a particle in a one-dimensional box.
The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrödinger equation with a potential, where the orbits depend on the energy.In special cases the simplicity has been restored by rescaling the size of the orbit, and perhaps a coupling constant at the same time.We point out that the goal can be achieved for any system of this class simply by rescaling the overall coupling constant.In each of these situations we inspect critically the relation between the oscillation frequency and the period of the orbit; in many cases it is observed that a characteristic length of the orbit is a better indicator.When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity.Finally, we comment on the alleged 'paradox' that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.