SOLVING OPERATOR DIFFERENTIAL EQUATIONS IN TERMS OF THE WEYL-ORDERED POLYNOMIALS

2002 ◽  
Vol 17 (30) ◽  
pp. 2009-2017 ◽  
Author(s):  
ZENG-BING CHEN ◽  
HUAI-XIN LU ◽  
JUN LI
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Systematic Approach ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Adiabatic Invariants ◽  
Formal Aspect ◽  
First Order ◽  
Heisenberg Equations ◽  
Heisenberg Equations Of Motion

A systematic approach to integrate the Heisenberg equations of motion is proposed by using the Weyl-ordered polynomials. The solutions of the Heisenberg equations of motion, i.e. P(t) and Q(t), are expanded as a sum over the Weyl-ordered polynomials Tm,n(P(t),Q(t)) at time t = 0. The coefficients of the expansions satisfy two sets of first-order ordinary differential equations resulting from the Heisenberg equations of motion for time-independent systems. This general approach for time-independent systems is also tractable in obtaining the adiabatic invariants of the time-dependent systems. In this paper, interest is mainly focused on the formal aspect of the approach.

QUANTUM TREATMENT OF A TIME DEPENDENT SINGLE-TRAPPED ION INTERACTING WITH A BIMODAL CAVITY FIELD

2003 ◽  
Vol 17 (31n32) ◽  
pp. 5925-5941 ◽  
Author(s):  
MAHMOUD ABDEL-ATY ◽  
A.-S. F. OBADA ◽  
M. SEBAWE ABDALLA
Keyword(s):  
Cross Correlation ◽  
Analytic Solution ◽  
Coupling Parameter ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Field Interaction ◽  
Trapped Ion ◽  
Heisenberg Equations ◽  
Quantum Treatment ◽  
Heisenberg Equations Of Motion

In the present communication we consider a time dependent ion-field interaction. Here we discuss the interaction between a single trapped ion and two fields taking into account the coupling parameter to be time dependent and allowing for amplitude modulation of the laser field radiating the trapped ion. At exact resonances the analytic solution for the Heisenberg equations of motion is obtained. We examine the effect of the velocity and the acceleration on the Rabi oscillations by studying the second order correlation function. The phenomenon of squeezing for single and two fields cases is considered. The cross correlation between the fields is discussed.

Dynamics of a Basketball Rolling Around the Rim

2005 ◽  
Vol 128 (2) ◽  
pp. 359-364
Author(s):  
C. Q. Liu ◽  
Fang Li ◽  
R. L. Huston
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Coaching Strategies ◽  
Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

2D Saddle Potential Schrödinger Green's Function in the Presence of an Arbitrary Time-Dependent Electric Field

1997 ◽  
Vol 11 (26n27) ◽  
pp. 1193-1196 ◽  
Author(s):  
Norman J. M. Horing ◽  
Kashif Sabeeh
Keyword(s):  
Electric Field ◽  
Green’S Function ◽  
Green's Function ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Arbitrary Time ◽  
Heisenberg Equations ◽  
Time Variables ◽  
Arbitrary Time Dependence ◽  
Heisenberg Equations Of Motion

We derive the retarded nonrelativistic Green's function for a Schrödinger electron confined to a plane and subject to a parabolic saddle potential and an electric field having arbitrary time dependence and orientation on the plane. This derivation is carried out using Heisenberg equations of motion for position and momentum operators, following an earlier analysis of Schwinger, to obtain the retarded Green's function in closed form, as an explicit function of position and time variables.

scholarly journals Nonclassical Effects of Light in Fifth Harmonic Generation up to First-Order Hamiltonian Interaction

ISRN Optics ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Rajendra Pratap ◽  
D. K. Giri ◽  
Ajay Prasad
Keyword(s):  
Harmonic Generation ◽  
Fundamental Mode ◽  
Equations Of Motion ◽  
Field Amplitude ◽  
Harmonic Mode ◽  
Squeezed Light ◽  
First Order ◽  
Heisenberg Equations ◽  
Fifth Order ◽  
Heisenberg Equations Of Motion

The nonclassical effects of light in the fifth harmonic generation are investigated by quantum mechanically up to the first-order Hamiltonian interaction. The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established. The occurrence of amplitude squeezing effects in both quadratures of the radiation field in the fundamental mode is investigated and found to be dependent on the selective phase values of the field amplitude. The photon statistics in the fundamental mode have also been investigated and found to be sub-Poissonian in nature. It is observed that there is no possibility to produce squeezed light in the harmonic mode up to first-order Hamiltonian interaction. Further, we have found that the normal squeezing in the harmonic mode directly depends upon the fifth power of the field amplitude of the initial pump field up to second-order Hamiltonian interaction. This gives a method of converting higher-order squeezing in the fundamental mode into normal squeezing in the harmonic mode and vice versa. The analytic expression of fifth-order squeezing of the fundamental mode in the fifth harmonic generation is established.

Ordinary differential equations and systems with time-dependent discontinuity sets

2004 ◽  
Vol 134 (4) ◽  
pp. 617-637 ◽  
Author(s):  
J. Ángel Cid ◽  
Rodrigo L. Pouso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Existence Results ◽  
Time Dependent ◽  
Nonlinear Part ◽  
First Order ◽  
Discontinuity Sets

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

Sound Synthesis Based on Ordinary Differential Equations

2015 ◽  
Vol 39 (3) ◽  
pp. 46-58 ◽  
Author(s):  
Nikolaos Stefanakis ◽  
Markus Abel ◽  
André Bergner
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Systematic Approach ◽  
External Input ◽  
Sound Synthesis ◽  
Order Complex ◽  
First Order ◽  
Filter Synthesis ◽  
Modal Synthesis ◽  
Model Complex

Ordinary differential equations (ODEs) have been studied for centuries as a means to model complex dynamical processes from the real world. Nevertheless, their application to sound synthesis has not yet been fully exploited. In this article we present a systematic approach to sound synthesis based on first-order complex and real ODEs. Using simple time-dependent and nonlinear terms, we illustrate the mapping between ODE coefficients and physically meaningful control parameters such as pitch, pitch bend, decay rate, and attack time. We reveal the connection between nonlinear coupling terms and frequency modulation, and we discuss the implications of this scheme in connection with nonlinear synthesis. The ability to excite a first-order complex ODE with an external input signal is also examined; stochastic or impulsive signals that are physically or synthetically produced can be presented as input to the system, offering additional synthesis possibilities, such as those found in excitation/filter synthesis and filter-based modal synthesis.

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