WAVE FUNCTIONS OF TIME-DEPENDENT TWO COUPLED OSCILLATORS BY LEWIS–RIESENFELD METHOD

2006 ◽  
Vol 20 (09) ◽  
pp. 1087-1096 ◽  
Author(s):  
HONG-YI FAN ◽  
ZHONG-HUA JIANG
Keyword(s):  
Schrödinger Equation ◽  
Operator Theory ◽  
Coupled Oscillators ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Invariant Operator ◽  
Wave Functions ◽  
Time Dependent ◽  
General Solutions ◽  
Eigenvectors And Eigenvalues

For the two time-dependent coupled oscillators model we derive its time-dependent invariant in the context of Lewis–Riesenfeld invariant operator theory. It is based on the general solutions to the Schrödinger equation which is obtained and turns out to be the superposition of the generalized atomic coherent states in the Schwinger bosonic realization. The energy eigenvectors and eigenvalues of the corresponding time-independent Hamiltonian are also obtained as a by-product.

Coherent state theory of large amplitude collective motion

1976 ◽  
Vol 54 (19) ◽  
pp. 1941-1968 ◽  
Author(s):  
D. J. Rowe ◽  
R. Bassermann
Keyword(s):  
Schrödinger Equation ◽  
Large Amplitude ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Collective Motion ◽  
State Theory ◽  
Time Dependent ◽  
Hartree Fock ◽  
Starting Point ◽  
Reduced Representations

A theory of large amplitude collective motion of a many-particle system is presented, which is relevant, for example, to nuclear fission. The theory is a combination of techniques used in many areas of physics and mathematics. The starting point is the application of the time-dependent Schrödinger equation to generate invariant subspaces of the Hamiltonian in the Hartree–Fock approximation. This is a generalization of the group-theoretical device of generating orbits of a group in the construction of reduced representations. It is shown how solutions of the time-dependent Schrödinger equation can be expressed as instantaneous stationary states of a constrained static Hamiltonian. Thus contact is made with the traditional cranking models and constrained Hartree–Fock theories of large amplitude collective motion. The collective motion is quantized using the Hill–Wheeler–Griffin method of generator coordinates in a basis of generalized coherent states. One is thereby able to exploit much of the theory of harmonic oscillator coherent states, which have been so successfully used in the quantum theory of the laser. The resulting Schrödinger equation for the collective dynamics is expressed both in the Bargmann representation and in the more familiar Schrödinger representation. It is shown that solution of the Schrödinger equation in the small amplitude harmonic approximation reproduces the well-known RPA result. A pilot calculation for 28Si shows that application in large amplitude is also feasible.

Creating Coherent States in a Mesoscopic Josephson Junction

1998 ◽  
Vol 12 (25n26) ◽  
pp. 1069-1074
Author(s):  
Shao Bin ◽  
Jian Zou ◽  
Qianshu Li
Keyword(s):  
Schrödinger Equation ◽  
Coherent State ◽  
Josephson Junction ◽  
Ground State ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
Classical Field ◽  
The State ◽  
Time Dependent

We examine a system of mesoscopic Josephson junction driven by classical field and solve the time-dependent Schrödinger equation of the system with the help of the time-dependent invariant of the Hamiltonian operator. We show that the state of the system can evolve a pure coherent state when the junction is prepared initially in its ground state.

GEOMETRIC PHASE IN A MESOSCOPIC LC CIRCUIT WITH ALTERNATING SOURCE

2006 ◽  
Vol 20 (32) ◽  
pp. 5373-5381
Author(s):  
YING-HUA JI ◽  
JU-JU HU ◽  
QING LIU ◽  
DAN ZOU
Keyword(s):  
Schrödinger Equation ◽  
Parameter Space ◽  
Geometric Phase ◽  
Schrodinger Equation ◽  
Invariant Operator ◽  
Time Dependent ◽  
Phase Factor ◽  
Evolution Path ◽  
Lc Circuit

In this paper, we examine a mesoscopic LC circuit with alternating source and solve its time-dependent Schrödinger equation by selecting a proper Hermitian invariant operator. We study LR geometric phase in mesoscopic circuit system. Our results indicate that LR geometric phase is present at all times. In the evolution of the circuit system, it is a geometric phase factor accumulated by the wavefunction of external source's interaction with circuit system after nonadiabatic and noncyclic evolution. Its property lies in that it depends on the evolution path of the system wavefunction in the parameter space.

Introduction

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
State Level ◽  
Boltzmann Distribution ◽  
Theoretical Description ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Molecular Reaction ◽  
Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

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