Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 9
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dt(m) V- V(V+ Q) x x(t) (2.10)
d/dt = =/t+- .V (2.11)
m = -V(V+ Q) x=x(t) (2.12)
We see that what is new in the quantum equation of motion is the quantum potential and conse-
quently the quantum force in Newton's second law, a second order differential equation. However
this equation is a formal derivation. It is pointed out in this dissertation that the quantum potential
is actually part of the quantum kinetic energy, and therefore the quantum potential is a fictitious
potential. The quantum force is also a fictitious force.
2.3 Quantum Lyapunov Exponent
Quantum chaos is a field largely unsettled. Classical chaos is defined as a system with expo-
nential sensitivity to the initial conditions. Positive Lyapnov exponents are the signature of chaos
in classical mechanics. We take this condition as a criterion for chaos in quantum systems with
Bohmian trajectories [7, 13, 14]. Increasing attention is given to the de Broglie-Bohm approach
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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/m1/26/: accessed February 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .