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Aspects of Quantum Chaos

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Nonlinear Dynamics and Quantum Chaos

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Abstract

This chapter discusses two important ways of defining quantum chaoticity. One access to characterize dynamical quantum systems is offered by the powerful approach of semiclassics. Here we compare the properties of the quantum system with its classical analogue, and we combine classical intuition with quantum evolution. The second approach starts from the very heart of quantum mechanics, from the quantum spectrum and its properties. Classical and quantum localization mechanisms are presented, again originating either from the classical dynamics of the corresponding problem and semiclassical explanations, or from the quantum spectra and the superposition principle. The essential ideas of the theory of random matrices are introduced. This second way of characterizing a quantum system by its spectrum is reconciled with the first approach by the conjectures of Berry and Tabor and Bohigas, Giannoni, and Schmit, respectively.

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Notes

  1. 1.

    This relation between the hermitian operator \(\hat{H}\) and the unitary \(\hat{U}\) is due to a theorem of Stone [4]. We assume \(\hat{H}\) was time-independent, otherwise time-ordering must be used.

  2. 2.

    The validity of this approximation is similar to the conditions discussed at the beginning of Sect. 4.2.2.

  3. 3.

    They had to be treated separately in the WKB approximation. There, this was achieved by using the Airy functions close to the turning point.

  4. 4.

    Following [20].

  5. 5.

    A similar relation was actually shown in Sect. 4.2.4. Here one starts from the infinitesimal action \(R(\mathbf{r }',\mathbf{r })= \frac{m}{2 \delta t}(\mathbf{r }'-\mathbf{r })^2-\delta t V(\mathbf{r }')\) with momentum \(p = \frac{\mathbf{r }'-\mathbf{r }}{\delta t}\), and shows that \(\frac{\delta t}{m}\frac{\partial \mathbf{r }'}{\partial \mathbf{r }}|_{\mathbf{r }=\mathbf r' } = \frac{\partial \mathbf{r }'}{\partial \mathbf p }|_\mathbf{p (\mathbf{r }=\mathbf r' )}\), which in turn is the inverse of the matrix \(- \frac{\partial R}{\partial \mathbf{r }' \partial \mathbf{r }}\).

  6. 6.

    Please remember the phase convention right after Eq. (4.3.25).

  7. 7.

    In the notation of Sect. 3.8.4, the exponents are called \(\sigma \), see Eq. 3.8.36).

  8. 8.

    The unfolding from Eq. (4.3.125) makes the unit of time 1 corresponding to the inverse level spacing in dimensionless units. Hence only for energies larger than the inverse level spacing we can expect good correspondence.

  9. 9.

    We assume that the integral in the definition of the Weyl symbol and the time-derivative can be interchanged.

  10. 10.

    Please see the asymptotic expansion with number [9.3.1] in Chap. 9 of [13].

  11. 11.

    The notion dynamic or dynamical localization is unfortunately also used in different contexts, one example being the suppression of tunneling by a periodic driving force [81], for experimental realizations of this latter effect see, e.g., [82, 83].

  12. 12.

    A short overview over possible spectra of a quantum system is found in the appendix of this chapter.

  13. 13.

    To make the statement as clear as possible we abuse the notation here a bit.

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    Sandro Wimberger

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Wimberger, S. (2014). Aspects of Quantum Chaos. In: Nonlinear Dynamics and Quantum Chaos. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06343-0_4

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