WKB approach applied to 1D time-dependent nonlinear Hamiltonian oscillators

We consider 1D time-dependent Hamiltonian systems and their statistical properties, namely the time evolution of microcanonical distributions, whose properties are very closely related to the existence and preservation of the adiabatic invariants. We review the elements of the recent developments by Robnik and Romanovski (during 2006–2008) for the entirely general 1D time-dependent linear oscillator and try to generalize the results to the 1D nonlinear Hamilton oscillators, in particular the power-law potentials like e.g. the quartic oscillator. Furthermore, we consider the limit opposite to the adiabatic limit, namely parametrically kicked 1D Hamiltonian systems. Even for the linear oscillator, interesting properties are revealed: an initial kick disperses the microcanonical distribution to a spread one, but an anti-kick at an appropriate moment of time can annihilate it, kicking it back to the microcanonical distribution. The case of periodic parametric kicking is also interesting. Finally, we propose that in the parametric kicking of a general 1D Hamilton system, the average value of the adiabatic invariant always increases, which we prove for the power-law potentials. We find that the approximation of kicking is good for quite long times of the parameter variation, up to the order of not much less than one period of the oscillator. We also look at the behavior of the quartic oscillator for the case of the kick and anti-kick, and also the periodic kicking.

Section: Numerical Calculations For the Quartic Oscillatormentioning

confidence: 99%

Statistical properties of 1D time-dependent Hamiltonian systems: from the adiabatic limit to the parametrically kicked systems

Papamikos¹,

Robnik²

2011

“…where a(t) is an unbounded monotonically increasing function of time. The general solution using the nonlinear WKB-like method [8] to the first order approximation is,…”

Section: Quartic Oscillatormentioning

confidence: 99%

“…is the potential as a function of the coordinate q and time t, a(t) is a time-dependent function and m is an integer = … m 1, 2, . In a recent work, Papamikos and Robnik [8] have developed the first-order nonlinear WKB-like method for such homogeneous power law potentials as an approximation of the general solution, which can be used successfully to generalize a series of studies on the time-dependent linear oscillator by Robnik and Romanovski [9][10][11][12][13], where the rigorous linear WKB method (to all orders) has been employed [14]. Using these tools we shall analyze the statistical properties of the energies of systems (1).…”

Section: Introductionmentioning

confidence: 97%

Statistical properties of the energy in time-dependent homogeneous power law potentials

Andresas¹,

Robnik²

2014

Self Cite

We study classical 1D Hamilton systems with homogeneous power law potential and their statistical behaviour, assuming the microcanonical distribution of the initial conditions and describing its change under monotonically increasing timedependent function a(t) (prefactor of the potential). Using the nonlinear WKB-like method by Papamikos and Robnik J. Phys. A: Math. Theor. 44 (2012) 315102 and following a previous work by Papamikos G and Robnik M J. Phys. A: Math. Theor. 45 (2011) 015206 we specifically analyze the mean energy, the variance and the adiabatic invariant (action) of the systems for large time t → ∞ and we show that the mean energy and variance increase as powers of a(t), while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is excellent which demonstrates the usefulness of the method in such applications.

“…Time-dependent Hamiltonian systems [1][2][3][4], in particular time periodic systems [3,5], have been the subject of very intense recent study, both classically [6][7][8][9][10] and quantally [11][12][13]. In particular, we should mention works on one-dimensional quantum billiards [14][15][16][17][18], especially the work by Šeba [19], and two-dimensional quantum billiards [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Quantum Fermi acceleration in the resonant gaps of a periodically driven one-dimensional potential box

Grubelnik¹,

Logar²,

Robnik³

2014

We study numerically the quantum mechanics of a point particle in the one-dimensional potential box, whose boundary oscillates periodically according to the sawtooth driving law. We perform very accurate numerical calculations over up to about 500 periods. Unlike in smooth driving, this system admits classical Fermi acceleration, because the Kolmogorov–Arnold–Moser theorem does not apply, and surprisingly also admits quantum Fermi acceleration, but only in the extremely narrow resonant gaps located at the values of the driving parameter that corresponds to either one-photon transitions or multi-photon transitions. In the gaps the energy of the particle increases indefinitely, quadratically with time, as predicted by our quite general two-level theory derived for general periodic quantum systems, whilst outside the gaps it oscillates and displays beatings with long or very long periods which are theoretically unexplained, but very well and clearly manifested.