The pseudo Hermitian invariant operator and time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry

2018 ◽  
Vol 59 (7) ◽  
pp. 072103 ◽  
Author(s):  
Walid Koussa ◽  
Naima Mana ◽  
Oum Kaltoum Djeghiour ◽  
Mustapha Maamache
Keyword(s):  
Dynamical Symmetry ◽  
Invariant Operator ◽  
Time Dependent

GEOMETRIC PHASES, SQUEEZED QUANTUM STATES AND GAUSSIAN WAVE PACKET STATES OF RELIC GRAVITONS

2012 ◽  
Vol 18 ◽  
pp. 13-17
Author(s):  
K. BAKKE ◽  
I. A. PEDROSA ◽  
C. FURTADO
Keyword(s):  
Wave Packet ◽  
Linear Time ◽  
Invariant Operator ◽  
Quantum States ◽  
Time Dependent ◽  
Gaussian Wave Packet ◽  
Geometric Phases ◽  
Uncertainty Product ◽  
Quantized Field ◽  
Generalized Time

In this contribution, we discuss quantum effects on relic gravitons described by the Friedmann-Robertson-Walker (FRW) spacetime background by reducing the problem to that of a generalized time-dependent harmonic oscillator, and find the corresponding Schrödinger states with the help of the dynamical invariant method. Then, by considering a quadratic time-dependent invariant operator, we show that we can obtain the geometric phases and squeezed quantum states for this system. Furthermore, we also show that we can construct Gaussian wave packet states by considering a linear time-dependent invariant operator. In both cases, we also discuss the uncertainty product for each mode of the quantized field.

scholarly journals Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Salim Medjber ◽  
Hacene Bekkar ◽  
Salah Menouar ◽  
Jeong Ryeol Choi
Keyword(s):  
Separation Of Variables ◽  
Type Equation ◽  
Invariant Operator ◽  
Wave Functions ◽  
Time Dependent ◽  
Harmonic Potential ◽  
Uncertainty Relations ◽  
Mathematical Methods ◽  
Potential System ◽  
Uvarov Method

The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

scholarly journals On a direct method for the determination of an exact invariant for the time-dependent harmonic oscillator

1977 ◽  
Vol 20 (1) ◽  
pp. 97-105 ◽  
Author(s):  
P. G. L. Leach
Keyword(s):  
Direct Method ◽  
Dynamical Symmetry ◽  
Time Dependent ◽  
Symmetry Groups ◽  
New Approach ◽  
One Dimensional ◽  
The One ◽  
A Direct Method ◽  
Exact Invariant

AbstractAn exact invariant is found for the one-dimensional oscillator with equation of motion . The method used is that of linear canonical transformations with time-dependent coeffcients. This is a new approach to the problem and has the advantage of simplicity. When f(t) and g(t) are zero, the invariant is related to the well-known Lewis invariant. The significance of extension to higher dimension of these results is indicated, in particular for the existence of non-invariance dynamical symmetry groups.

On the dynamics of a time-dependent mesoscopic LC circuit with a negative inductance

2016 ◽  
Vol 30 (12) ◽  
pp. 1650122 ◽  
Author(s):  
I. A. Pedrosa ◽  
E. Nogueira ◽  
I. Guedes
Keyword(s):  
Magnetic Flux ◽  
Wave Packet ◽  
Operator Method ◽  
Invariant Operator ◽  
Time Dependent ◽  
Expectation Values ◽  
Gaussian States ◽  
Faraday’S Law ◽  
Uncertainty Product ◽  
Lc Circuit

We discuss the problem of a mesoscopic LC circuit with a negative inductance ruled by a time-dependent Hermitian Hamiltonian. Classically, we find unusual expressions for the Faraday’s law and for the inductance of a solenoid. Quantum mechanically, we solve exactly the time-dependent Schrödinger equation through the Lewis and Riesenfeld invariant operator method and construct Gaussian wave packet solutions for this time-dependent LC circuit. We also evaluate the expectation values of the charge and the magnetic flux in these Gaussian states, their quantum fluctuations and the corresponding uncertainty product.

scholarly journals Dynamical Invariant Applied on General Time-Dependent Three Coupled Nano-Optomechanical Oscillators

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Sara Hassoul ◽  
Salah Menouar ◽  
Hamid Benseridi ◽  
Jeong Ryeol Choi
Keyword(s):  
Invariant Operator ◽  
Time Dependent ◽  
Unitary Matrix ◽  
Optomechanical System ◽  
Von Neumann ◽  
Classical Counterpart ◽  
Quadratic Invariant ◽  
Fock State ◽  
The Matrix ◽  
General Time

A quadratic invariant operator for general time-dependent three coupled nano-optomechanical oscillators is investigated. We show that the invariant operator that we have established satisfies the Liouville-von Neumann equation and coincides with its classical counterpart. To diagonalize the invariant, we carry out a unitary transformation of it at first. From such a transformation, the quantal invariant operator reduces to an equal, but a simple one which corresponds to three coupled oscillators with time-dependent frequencies and unit masses. Finally, we diagonalize the matrix representation of the transformed invariant by using a unitary matrix. The diagonalized invariant is just the same as the Hamiltonian of three simple oscillators. Thanks to such a diagonalization, we can analyze various dynamical properties of the nano-optomechanical system. Quantum characteristics of the system are investigated as an example, by utilizing the diagonalized invariant. We derive not only the eigenfunctions of the invariant operator, but also the wave functions in the Fock state.

COMPLETE EXACT QUANTUM STATES OF THE GENERALIZED TIME-DEPENDENT HARMONIC OSCILLATOR

2004 ◽  
Vol 18 (24) ◽  
pp. 1267-1274 ◽  
Author(s):  
I. A. PEDROSA
Keyword(s):  
General Solution ◽  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Continuous Spectrum ◽  
Invariant Operator ◽  
Quantum States ◽  
Time Dependent ◽  
Quadratic Invariants ◽  
Generalized Time ◽  
Discrete And Continuous Spectrum

By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.

Sign in / Sign up

Export Citation Format

Share Document