scholarly journals Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

Dynamics ◽  
2021 ◽  
Vol 1 (2) ◽  
pp. 155-170
Author(s):  
Moise Bonilla-Licea ◽  
Dieter Schuch
Keyword(s):  
Coherent States ◽  
Equations Of Motion ◽  
Bohmian Mechanics ◽  
Equation Of Motion ◽  
Time Dependent ◽  
Auxiliary Equation ◽  
Linear Potential ◽  
Quantum Hydrodynamics ◽  
Generalized Coherent States ◽  
Dynamical Invariants

For time dependent Hamiltonians like the parametric oscillator with time-dependent frequency, the energy is no longer a constant of motion. Nevertheless, in 1880, Ermakov found a dynamical invariant for this system using the corresponding Newtonian equation of motion and an auxiliary equation. In this paper it is shown that the same invariant can be obtained from Bohmian mechanics using complex Hamiltonian equations of motion in position and momentum space and corresponding complex Riccati equations. It is pointed out that this invariant is equivalent to the conservation of angular momentum for the motion in the complex plane. Furthermore, the effect of a linear potential on the Ermakov invariant is analysed.

Generalized Coherent States of a Particle in a Time-Dependent Linear Potential

2009 ◽  
Vol 26 (7) ◽  
pp. 070307 ◽  
Author(s):  
L Krache ◽  
M Maamache ◽  
Y Saadi ◽  
A Beniaiche
Keyword(s):  
Coherent States ◽  
Time Dependent ◽  
Linear Potential ◽  
Generalized Coherent States

scholarly journals Electron Trajectories in Molecular Orbitals

2020 ◽  
Author(s):  
Isaiah Sumner ◽  
Hannah Anthony
Keyword(s):  
Lagrange Multiplier ◽  
Equations Of Motion ◽  
Bohmian Mechanics ◽  
Molecular Orbitals ◽  
Quantum Hydrodynamics ◽  
Relativistic Limit ◽  
Quantum Particles ◽  
Quantum Force ◽  
Trajectory Methods ◽  
Structure Calculations

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations, since they predict that the electrons in a ground state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the non-relativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.

scholarly journals DYNAMICS OF GENERALIZED COHERENT STATES

1995 ◽  
Vol 09 (13) ◽  
pp. 823-828 ◽  
Author(s):  
SALVATORE DE MARTINO ◽  
SILVIO DE SIENA ◽  
FABRIZIO ILLUMINATI
Keyword(s):  
Coherent States ◽  
Wave Packets ◽  
Classical Trajectory ◽  
Feedback Mechanism ◽  
The State ◽  
Time Dependent ◽  
Generalized Coherent States ◽  
Classical Evolution

We show that generalized coherent states follow Schrödinger dynamics in time-dependent potentials. The normalized wave-packets follow a classical evolution without spreading; in turn, the Schrödinger potential depends on the state through the classical trajectory. This feedback mechanism with continuous dynamical re-adjustment allows the packets to remain coherent indefinitely.

Coherent states and geometric phases of a generalized damped harmonic oscillator with time-dependent mass and frequency

2014 ◽  
Vol 28 (26) ◽  
pp. 1450177 ◽  
Author(s):  
I. A. Pedrosa ◽  
D. A. P. de Lima
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Classical Equation ◽  
Equation Of Motion ◽  
Time Dependent ◽  
Quantum Invariant ◽  
Dynamical Phase ◽  
Special Cases ◽  
Berry Phases ◽  
Dependent Mass

In this paper, we study the generalized harmonic oscillator with arbitrary time-dependent mass and frequency subjected to a linear velocity-dependent frictional force from classical and quantum points of view. We obtain the solution of the classical equation of motion of this system for some particular cases and derive an equation of motion that describes three different systems. Furthermore, with the help of the quantum invariant method and using quadratic invariants we solve analytically and exactly the time-dependent Schrödinger equation for this system. Afterwards, we construct coherent states for the quantized system and employ them to investigate some of the system's quantum properties such as quantum fluctuations of the coordinate and the momentum as well as the corresponding uncertainty product. In addition, we derive the geometric, dynamical and Berry phases for this nonstationary system. Finally, we evaluate the dynamical and Berry phases for three special cases and surprisingly find identical expressions for the dynamical phase and the same formulae for the Berry's phase.

A COMPARISON OF DIFFERENT SOLUTION ALGORITHMS FOR THE NUMERICAL ANALYSIS OF VEHICLE–BRIDGE INTERACTION

2014 ◽  
Vol 14 (02) ◽  
pp. 1350065 ◽  
Author(s):  
K. LIU ◽  
N. ZHANG ◽  
H. XIA ◽  
G. DE ROECK
Keyword(s):  
Numerical Analysis ◽  
Time Domain ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Good Choice ◽  
Contact Forces ◽  
Time Dependent ◽  
Time Step ◽  
Solution Algorithms ◽  
Coupled Equation

The interaction between a bridge and a train moving on the bridge is a coupled dynamic problem. The equations of motion of the bridge and the vehicle are coupled by the time dependent contact forces. At each time step, the motion of the bridge influences the forces transferred to the vehicle and this, in turn, changes the forces acting on the bridge. In this paper, a comparison of three different time domain solution algorithms for the coupled equation of motion of the train–bridge system is presented. Guidelines are given for a good choice of the time step.

Adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator

2021 ◽  
pp. 2150201
Author(s):  
I. A. Pedrosa
Keyword(s):  
Harmonic Oscillator ◽  
Geometric Phase ◽  
Coherent States ◽  
Time Dependent ◽  
Expectation Values ◽  
Damped Harmonic Oscillator ◽  
Quantum Properties ◽  
Uncertainty Product ◽  
Dynamical Invariants ◽  
Damped Oscillator

In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.

scholarly journals Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Caterina Foti ◽  
Alessandro Coppo ◽  
Giulio Barni ◽  
Alessandro Cuccoli ◽  
Paola Verrucchi
Keyword(s):  
Classical Limit ◽  
Coherent States ◽  
Equations Of Motion ◽  
Quantum Systems ◽  
Physical Systems ◽  
Quantum Time ◽  
First Case ◽  
Hamilton Equations ◽  
Generalized Coherent States ◽  
Evolving System

AbstractWe draw a picture of physical systems that allows us to recognize what “time” is by requiring consistency with the way that time enters the fundamental laws of Physics. Elements of the picture are two non-interacting and yet entangled quantum systems, one of which acting as a clock. The setting is based on the Page and Wootters mechanism, with tools from large-N quantum approaches. Starting from an overall quantum description, we first take the classical limit of the clock only, and then of the clock and the evolving system altogether; we thus derive the Schrödinger equation in the first case, and the Hamilton equations of motion in the second. This work shows that there is not a “quantum time”, possibly opposed to a “classical” one; there is only one time, and it is a manifestation of entanglement.

The Density Matrix: Bloch Equations

2021 ◽  
pp. 314-328
Author(s):  
Duncan G. Steel
Keyword(s):  
Density Matrix ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Time Dependent ◽  
Vacuum Field ◽  
Bloch Equations ◽  
Quantum Device ◽  
Vacuum Radiation ◽  
Reduced Density ◽  
Schrödinger Picture

One of the most powerful tools for calculating quantum device performance is based on the density matrix operator. The operator is unique because it is time dependent in the Schrödinger picture. The approach is quite general, but in the systems of interest here, the Hilbert space of the operator includes both the quantum system such as a nano-vibrator or two-level system and the quantized vacuum radiation field. The equation of motion follows from the time dependent Schrödinger equation. It is possible, as we show, to include the generation of spontaneous emission in this system and then, because observables of interest do not depend on the vacuum field, trace over the vacuum field to create a new density matrix called the reduced density matrix. The resulting equations of motion are the Bloch equations. We use these equations to analyze several problems involving two- and three-level systems.

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