Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 59
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QUANTUM CHAOS OF THE KICKED ROTOR:
DE BROGLIE-BOHM APPROACH
Quantum chaos of the kicked rotor is found for the first time. The dependence of the Lyapunov
exponents corresponding to the classical control parameter Kfor K from 0 to 50 is shown. Stability
regions repeated with period n in the chaotic sea is similar to the classical case.
Classical deterministic chaos is characterized by strong sensitivity to the initial conditions of
trajectories. For quantum chaos, directly applying this concept to standard quantum mechanics
will lead to the result of no chaos in quantum mechanics because of the linear property of the
Schrodinger equation . However, the de Broglie-Bohm approach [10, 11, 12] of trajectories for
quantum mechanics made it possible to study quantum chaos in analogy to classical mechanics.
Positive quantum Lyapunov exponent is taken as the signature of quantum chaos .
For bounded systems, the Benettin et al. approach in Eq. (2.2) should be used for the Lyapunov
exponents. This can be applied to Bohmian quantum trajectories of the kicked rotor since the angle
is limited from 0 to 21r.
Results of the numerical computation using the Benettin et al. approach for the kicked rotor
Lyapunov exponents and their dependence on the control parameter are given. The comparison
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Zheng, Yindong. Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor, dissertation, August 2005; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc4824/m1/76/: accessed December 13, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .