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Journal of Mathematical Physics

Investigation of the time evolution of Lane–Emden-type Kanai–Caldirola oscillator

Serhat Fevzi Özeren

Published Jan 1, 2009• 14 citations
Publisher

Abstract

In this study, we have investigated the time evolution of the first kind Lane–Emden-type Kanai–Caldirola oscillator. The total energy of the one dimensional classical harmonic oscillator with time-dependent mass is obtained. Then by using the SU(1,1) coherent states, the classical equation of motion of the oscillator has been analyzed and the effects of increasing mass have been clearly illustrated in the figures.

DOI
10.1063/1.3043790

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In this study, we have investigated the time evolution of the first kind Lane–Emden-type Kanai–Caldirola oscillator. The total energy of the one dimensional classical harmonic oscillator with time-dependent mass is obtained. Then by using the SU(1,1) coherent states, the classical equation of motion of the oscillator has been analyzed and the effects of increasing mass have been clearly illustrated in the figures.
We obtain the exact wave functions of two Lane-Emden-type Kanai-Caldirola (LE-KC) oscillators by using the dynamical invariant method. To do so, we analytically solve the respective Milne-Pinney equation for each oscillator. We also calculate the uncertainty product, the transition probability and the quantum-mechanical energy expectation value for each oscillator. We compare the results with those of the well-known KC oscillator. The quantum-mechanical energy expectation value of the KC oscillator goes to zero faster than that of the LE-KC oscillators, indicating that the KC system is more damped than the LE-KC ones.
We study the time evolution of SU(1,1) coherent states driven by a damped harmonic oscillator as described by the Kanai-Caldirola Hamiltonian. The coherence of these states is preserved and their time evolution is described by nonlinear ``classical'' equations. The trajectories of all SU(1,1) coherent states in phase space asymptotically approach a common point as t\ensuremath{\rightarrow}\ensuremath{\infty}. In all cases, this approach is accompanied by a damping of the associated energy expectation value to zero.
In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) m(t)=m0fractt0m(t)=m_0frac{t}{t_0}m(t)=m0​fractt0​, (II) m(t)=m0m(t)=m_0m(t)=m0​ and (III) m(t)=m0ajustfractt02m(t)=m_0ajust{frac{t}{t_0}}^2m(t)=m0​ajustfractt0​2. In all three cases omega(t)=omega0fract0tomega(t)=omega_0frac{t_0}{t}omega(t)=omega0​fract0​t. For class (b) we study the Caldirola-Kanai oscillator (IV)where omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0extexpajustgammatm(t)=m_0 ext{exp}ajust{gamma t}m(t)=m0​extexpajustgammat and the oscillator with omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0ajust1+fractt0alpham(t)=m_0ajust{1+frac{t}{t_0}}^alpham(t)=m0​ajust1+fractt0​alpha, for alpha=2alpha=2alpha=2 (V) and alpha=4alpha=4alpha=4 (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of q(t)q(t)q(t), p(t)p(t)p(t) as well as the phase diagram q(t)q(t)q(t) vs p(t)p(t)p(t). To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.
The evolution operator of a Caldirola-Kanai type quantum parametric oscillator with a generalized quadratic Hamiltonian is obtained using the Wei-Norman Lie algebraic approach, and time evolution of the eigenstates of a harmonic oscillator and Glauber coherent states is found explicitly. Behavior of this oscillator is investigated under the influence of the external mixed term B(t)(q^p^+p^q^)/2, which affects the squeezing properties of the wave packets, and linear terms D0(t)q^, E0(t)p^ responsible for their displacement in time. According to this, we construct all exact quantum models with different parameters B(t), for which the structure of the Caldirola-Kanai oscillator in position space is preserved. Then, for each model, we obtain explicit solutions and analyze the squeezing and displacement properties of the wave packets according to the frequency modification by B(t) and periodic forces in the corresponding classical equation of motion.