Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 62
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expressed in terms of the expansion coefficients of al(tN) by taking the inner product of Eq. (6.6)
with I m),
am(tN+1) - exp[-im2k/2] Y a](tN)Wml, (6.7)
where the matrix element is defined as Win1- (m exp[-ikV(k)] 1). From this recursion relation
we can find the state vector at any time tN if we know the initial state at time to.
We take the periodic potential to be V(x) - cosxfor numerical computation. From the generat-
ing function for Bessel functions, we have the formula
exp[-ikcosx] (-i)nJ(k)exp(-inx), (6.8)
where J is the Bessel function of order n and k- K/k.
Substituting Eq. (6.8) into Eq. (6.7) and using the orthonormality of the momentum eigenfunc-
tions, we obtain the recursion relation for the coefficients am(tN+1)
am(tN+1) - exp[-im2k/2] Y (-i)1-mJ-m(k)al(tN) (6.9)
6.3 Mapping of Bohmian Trajectories of the Kicked Rotor at Quantum Resonance
According to p= VS [10, 11, 12] and using the solution of the wavefuction at expansion, the
momentum and velocity of the trajectory can be expressed as
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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/m1/79/: accessed December 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .