2.1 Parametrically driven harmonic oscillator A harmonic oscillator subject to a parametric driving can be defined...
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2.1 Parametrically driven harmonic oscillator A harmonic oscillator subject to a parametric driving can be defined by the Hamiltonian H(p, q;t) = m p 2m 2 = + 1/72 (w(t)) q, w(t) = wo (1+a sin(t)), |a| <1. a Solve the equations of motion for this system in the adiabatic limit < < wo. Hint: Rescale p and q by factors depending on m and w(t) so that both assume the same dimension Vaction (check that the transformation is canonical), so that the trajectories of the system would be circles if the frequency did not change. Apply the Adiabatic Theorem to find the time- dependence of their radii and finally transform back to the original unscaled phase-space coordinates p, q. b Using the results of part a, calculate the total energy of the system as a function of time and the action accumulated over a single period T = 2/w to compare how strongly they depend on time. C As a case of the opposite extreme, a rapid, hence non-adiabatic driving, consider a sudden change of the frequency at t = to, w(t) = w(to-t)+w(t-to) = tto, w t>to. Swi wX(t) w car 0 Solve the equations of motion assuming that at t = to, the trajectory remains continuous, that is, does not jump, lim,, to+0=r(t) = limto+0 r(t), and calculate the difference in total energy and in action (phase-space area enclosed in the orbit) across to. How does the difference in action (if any) depend on the phase of the oscillator at to when the frequency changes? Sketch the trajectory before and after to for different values of this phase. 2.1 Parametrically driven harmonic oscillator A harmonic oscillator subject to a parametric driving can be defined by the Hamiltonian H(p, q;t) = m p 2m 2 = + 1/72 (w(t)) q, w(t) = wo (1+a sin(t)), |a| <1. a Solve the equations of motion for this system in the adiabatic limit < < wo. Hint: Rescale p and q by factors depending on m and w(t) so that both assume the same dimension Vaction (check that the transformation is canonical), so that the trajectories of the system would be circles if the frequency did not change. Apply the Adiabatic Theorem to find the time- dependence of their radii and finally transform back to the original unscaled phase-space coordinates p, q. b Using the results of part a, calculate the total energy of the system as a function of time and the action accumulated over a single period T = 2/w to compare how strongly they depend on time. C As a case of the opposite extreme, a rapid, hence non-adiabatic driving, consider a sudden change of the frequency at t = to, w(t) = w(to-t)+w(t-to) = tto, w t>to. Swi wX(t) w car 0 Solve the equations of motion assuming that at t = to, the trajectory remains continuous, that is, does not jump, lim,, to+0=r(t) = limto+0 r(t), and calculate the difference in total energy and in action (phase-space area enclosed in the orbit) across to. How does the difference in action (if any) depend on the phase of the oscillator at to when the frequency changes? Sketch the trajectory before and after to for different values of this phase.
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Microeconomics An Intuitive Approach with Calculus
ISBN: 978-0538453257
1st edition
Authors: Thomas Nechyba
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