Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 1
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CHAPTER 1
INTRODUCTION
The original definition of classical chaos is a system with exponential sensitivity to initial condi-
tions. However, In quantum mechanics because of the linear property of the Schrodinger equation,
variation on initial condition of the wave function will result in no change in the correlation func-
tion, which will result in no chaos in quantum mechanics. Berry's correspondence theorem [1] says
that all classical chaotic systems also have quantum chaos. Reference [2] calls quantum chaos
the quantum mechanics of classically chaotic systems. Currently the study of quantum chaos is
certainly going beyond this point. However, the field remains largely unsettled [3, 4].
The concept of chaos is strong sensitivity to initial conditions. For quantum chaos, a variation
of the Hamiltonian can be introduced to test the stability of the quantum system to the Hamiltonian
[5]. A correlation function of the unperturbed and perturbed is studied and a signature of quantum
chaos is given [5]. Other criteria are also introduced [5, 6, 7, 8, 9]. Among these approaches
to quantum chaos, only the de Broglie-Bohm approach [10, 11, 12] of quantum mechanics [6] is
analogous to classical chaos.
Positive Lyapunov exponents or KS entropy is the signature of classical deterministic chaos,
and this is also taken as the case for Bohmian trajectories [6, 13, 14]. The use of the trajectories
of de Broglie-Bohm theory to quantum chaos is a break-through in the application of the theory.
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Zheng, Yindong. Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor, dissertation, August 2005; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc4824/m1/18/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .