Use The Theory Of Generalized Quantum Linear Transformation Solving A Class Exactly Time-dependent Quantum Mechanical Problems

For a long time a great attention has been paid to solving exactly time-dependentquantum mechanical problems. In recent years the time-dependent Harmonic Oscillatorcontinues to have widespread applications in various branches of Physics[1-5].such as thefield of experiments with cold atoms, the control of atoms by means of laser beams or otherelectric[6]and magnetic fields[7], the quantum motion of a particle in a Paul trap[8,9]， we usethe theory of generalized Quantum linear transformation solving exactly time-dependentquantum mechanical problems，The results provide a referent value for experimentallystudying time-dependent quantum mechanical problems. For studing the universal,convenient and effective method of solving the time-dependent quantum mechanicalproblems has very important significance.The main works of this thesis are and listed as follows:Firstly, we handle the Hamiltonian of the particles given by two different canonicaltransformations that effected by the linear-damping and time-dependent external force byusing the theory of generalized Quantum linear transformation, we give the rigorous solutionof evolution operator, and the expected value of coordinate and momentum of the particlesquantum fluctuations. Results show that:(1) the two regular translation are equivalent;(2)linear damping has squeezing effect to the momentum of particles, the deviation of themomentum attenuate in the rule of negative index with time, and the bigger the dampingcoefficient, the faster the attenuation;(3) the expected value of coordinate and momentum ofthe particles are separate equal to the classics.Secondly, we use the theory of quantum linear transformation to study the damped andtime-dependent external force harmonic oscillator, give the rigorous solution of evolutionoperator, the expected value of mechanical quantity of the harmonic oscillator，Furthermore，we compared with the classical one and give the law that the expectations of the potentialenergy, kinetic energy and the Hamiltonian close to the classical time-dependent forceddamped harmonic oscillator as tâ†'∞.This thesis is divided into four chapters. The first chapter is introduction, mainlyintroduced meaning which use the theory of generalized Quantum linear transformationsolving exactly time-dependent quantum mechanical problems，and reviews the developmenthistory and the present research status of solving exactly time-dependent quantum mechanical problems. The second chapter describes using generalized linear quantum transformationtheory to solve the problem of time-dependent quantum theoretical basis. Chapter III usinggeneralized linear quantum transformation theory subject to linear damping and containingexternal force particles the quantum behavior, focused on the compression effect of lineardamping on the momentum of a particle, as well as the damping coefficient on themomentum deviation.in this chapter also subject to quantum damped driven harmonicoscillator with time, focusing on the time of the Quantum Damped by the by the time of theharmonic oscillator and the classical damped harmonic oscillator compared. Thefourthchapter gives the main conclusions of this paper and the prospect of other system.