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Udon Thani Rajabhat University Journal of Science and Technology

การแกไขปญหาสมการการเคลอนทของไฮเซนเบรกของอนภาคทถกบงคบดวยแรงฮารโมนกออสซเลเตอรทขนอยกบเวลาโดยใชตวดำเนนการสรางและทำลายสถานะ

Artit Hutem, Piyarut Moonsri, Mantana Deena

Published Apr 30, 2021• 0 citations
OpenAlex

Abstract

For this study, the researchers purposed to evaluate the Heisenberg equation of motion for the creation and annihilation operator under the force vibration simple harmonic oscillation. Instead of using the first-order ordinary differential equation to solve the Heisenberg equation of motion, the researchers got the time-dependent creation and annihilation operator under the time-dependent force vibration simple harmonic oscillation. The behavior of wave group in the creation and annihilation operator for image part is destroyed by increasing of the parameter value of .

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In this research paper, we developed a theory and model of the forced simple harmonic oscillators in quantum mechanics system in a framework of a general approach to the Heisenberg picture. We can be used the time-dependent Hamiltonian for the forced simple harmonic oscillators. We use first-order ordinary linear differential equation to solve the time-dependent lowering operator and raising operator. We evaluate the lowering operator and raising operator as the function of time for particle mass depend on the parameter frequency of oscillation , the damping coefficient of particle bound in the simple harmonic oscillator potential, the initial force .

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รายละเอยดโครงการวจยเรอง การแกไขปญหาสมการการเคลอนทของไฮเซนเบรกของอนภาคทถกบงคบดวยแรงภายนอกทขนอยกบเวลาโดยใชตวดำเนนการสรวงและทำลายสถานะ (Solving the Heisenberg Equation of Motion of the Time-Dependent Forced Harmonic Oscillator Using Time-Dependent Annihilation and Creation Qperators.)

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รายละเอยดโครงการวจยเรอง การแกไขปญหาสมการการเคลอนทของไฮเซนเบรกของอนภาคทถกบงคบดวยแรงภายนอกทขนอยกบเวลาโดยใชตวดำเนนการสรวงและทำลายสถานะ (Solving the Heisenberg Equation of Motion of the Time-Dependent Forced Harmonic Oscillator Using Time-Dependent Annihilation and Creati...

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For this study, the researchers purposed to evaluate the Heisenberg equation of motion for the creation and annihilation operator under the force vibration simple harmonic oscillation. Instead of using the first-order ordinary differential equation to solve the Heisenberg equation of motion, the researchers got the time-dependent creation and annihilation operator under the time-dependent force vibration simple harmonic oscillation. The behavior of wave group in the creation and annihilation operator for image part is destroyed by increasing of the parameter value of .
In this research paper, we developed a theory and model of the forced simple harmonic oscillators in quantum mechanics system in a framework of a general approach to the Heisenberg picture. We can be used the time-dependent Hamiltonian for the forced simple harmonic oscillators. We use first-order ordinary linear differential equation to solve the time-dependent lowering operator and raising operator. We evaluate the lowering operator and raising operator as the function of time for particle mass depend on the parameter frequency of oscillation , the damping coefficient of particle bound in the simple harmonic oscillator potential, the initial force .
รายละเอยดโครงการวจยเรอง การแกไขปญหาสมการการเคลอนทของไฮเซนเบรกของอนภาคทถกบงคบดวยแรงภายนอกทขนอยกบเวลาโดยใชตวดำเนนการสรวงและทำลายสถานะ (Solving the Heisenberg Equation of Motion of the Time-Dependent Forced Harmonic Oscillator Using Time-Dependent Annihilation and Creation Qperators.)
รายละเอยดโครงการวจยเรอง การแกไขปญหาสมการการเคลอนทของไฮเซนเบรกของอนภาคทถกบงคบดวยแรงภายนอกทขนอยกบเวลาโดยใชตวดำเนนการสรวงและทำลายสถานะ (Solving the Heisenberg Equation of Motion of the Time-Dependent Forced Harmonic Oscillator Using Time-Dependent Annihilation and Creati...
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω 0 that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.
In physics, harmonic motion is among the most representative types of motion. A simple harmonic oscillator is often the source of any vibration with a restoring force proportional to Hooke’s law. Every minimum potential has a solution in the form of the harmonic oscillator potential. Little oscillations at the minimum are characteristic of almost all natural potentials and of many quanta mechanical systems. Harmonic motion is an essential building block for these more complex uses. The Schrödinger equation is a defining feature of the harmonic oscillator. Here, we demonstrate that the time-frequency plane is a useful tool for analyzing their dynamics. We numerically integrate several examples involving different input forces and demonstrate that the oscillations are clearly displayed and easily interpretable in the time-frequency plane.