Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 8
The following text was automatically extracted from the image on this page using optical character recognition software:
(1) "An individual physical system comprises a wave propagating in space and time together
with a point particle that moves continuously under the guidance of the wave."
(2) "The wave is mathematically described by 'Y(x, t), a solution to Schrodinger's wave equa-
(3) "The particle motion is obtained as the solution x(t) to the equation
x = (1/m) V S(x,t) Ixx(t) (2.8)
where S is the phase of '. To solve this equation we have to specify the initial condition
x(0) = xo while the initial velocity has already been defined. The initial position constitutes the only
extra information introduced by the theory that is not contained in '(x, t). An ensemble of possible
motions associated with the same wave is generated by varying xo."
(4) The probability that a particle in the ensemble lies in an element of volume d3x at x at time t
is given by
R2(x, t) d3x (2.9)
where R2 'P 12
From the above postulates, the momentum of the particle is p = VS and the velocity of the
particle is v= (1/m)VS. By applying the operator V to Eq. (2.5) and rearranging, the equation of
motion has the same form as Newton's second law. It is 
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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/m1/25/: accessed December 14, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .