African Journal Of Mathematical Physics Volume 8(2010)43-50 On The Memory Of Non-Locally Damped Harmonic Oscillator A. N. Ikot 1 , E. J. Uwah 2 , L. E. Akpabio 1 , I. O. Akpan 2 1 Dept. of Physics, University of Uyo, Uyo, Nigeria 2 Dept. of Physics, University of Calabar, Calabar, Nigeria. ndemikot2005@yahoo.com abstract We investigate the equation of motion for damped oscillator with arbitrary time mem- ory. We show that the classical dynamics which breaks down the local composition law still preserved the basic uncertainty relation. The propagator and the wavefunction of the system is also evaluated. I. INTRODUCTION Approximation methods such as perturbation theory have been used to evaluate solution time depen- dent Schr¨ odinger equation [1-9]. Khandekar and Lawande [10] had evaluated the exact quantum theory of a classical force oscillator with a time-dependent frequency and a velocity-dependent damping term using path integral approach. However, the damped harmonic oscillator is a system displaying energy dissipa- tion. There are different approach in describing these quantum dissipative system such as Caldirola-Kanai formation [11-12] and the canonical formation of classical mechanics to dissipative system through the non-Hamiltonian system using the double number of degrees of freedom proposed by Bateman [13.14]. Recently, the investigated the time-dependent Harmonic oscillator via a path integral approach with a modified Caldirola-Kanai Hamiltanian [15]. In the article, our aim will be to analyzed the equation of motion of this modified equation with an arbi- trary time memory to [16]. Unlike the result of D. Chruseinski and J. Jurkowski [16] which corresponds to the asymptotic regime γ t →∞, our analysis is of the short time regime γ t →0 and it corresponds to the notion of switch on the interaction adiabatically. In many body systems, one specified the Hamiltonian describing this system as [17]. ˆ H = ˆ H 0 + e −γt ˆ H 1 , (1) At the regime γ t →0, the Hamiltonian ˆ H in Eq (1) becomes the full Hamiltonian of the interaction system and when γ t →0, the system is switch on and off adiabatically. The modified Lagrangian of our model is [15]. L = e sin γt [ 1 2 m ˙ q 2 − 1 2 mω 2 (t)q 2 ] (2) 0 c ⃝ a GNPHE publication 2010, ajmp@fsr.ac.ma 43