Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 60
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of the quantum to the classical Lyapunov exponents is also made. Both chaotic and non-chaotic
regimes are found for the quantum kicked rotor.
Only trajectories at quantum resonance are studied since the quantum potential is zero at quan-
tum resonance. The momentum diffusion for the average of the Bohmian trajectories is consistent
to the traditional quantum mechanics .
6.2 Solution of the Schrbdinger Equation
In terms of scaled variables  the dimensionless Hamiltonian of the kicked rotor is
H 2 2+KV(i) Y 6(t -n), (6.1)
where t is in units of the period T of the kicks, V(k) is a periodic potential and K is the control
(stochasticity) parameter. The displacement and canonical momentum operators are denoted by x
and p, respectively. The Schrbdinger equation for the state vector W(t) is
ikatty(t) = Ht(t), (6.2)
where the dimensionless scaled Planck's constant k hk T/M is given in terms of Planck's
constant h = 2nF, the wave number k0 of the periodic potential, the period of the kicks T, and
the mass M of the kicked rotor. The displacement and canonical momentum satisfy canonical
commutation relations with Planck's constant replaced by the scaled Planck's constant k.
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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/m1/77/: accessed November 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .