Applied mathematics and physics

Analytical Computation of the Caldirola-Kanai Oscillator Parameters by the Dynamic Invariant Method

Chadia Qotni, Az-Eddine L. Marrakchi, Salaheddine Sayouri, Y. Achkar

Published Oct 18, 2019• 0 citations

Abstract

The method of the invariant dynamical linear operator is very simple and may be useful to solve the Schrodinger equation, in particular in the case of the problem of a harmonic oscillator with a time dependent mass and frequency. Indeed, we have successfully used this approach to the Caldirola-Kanai oscillator. In particular, we have obtained explicit expressions of the uncertainty product and the quantum correlation coefficient. The results obtained are in good agreement with those of the literature.

DOI: 10.12691/amp-7-1-2

Related papers

Teoria quântica de invariantes aplicada ao oscilador harmônico forçado dependente do tempo de Caldirola-Kanai

Alberes Lopes de Lima, I. A. Pedrosa

Scientia PlenaJan 1, 20080 citations

Osciladores log-periódicos e tipo Caldirola-Kanai

Vagner Henrique Loiola Bessa

Jan 1, 20120 citations

Gaussian wave packet states of a generalized inverted harmonic oscillator with time-dependent mass and frequency

I. A. Pedrosa, Alberes Lopes de Lima, Alexandre M. de M. Carvalho

Canadian Journal of PhysicsJan 12, 201512 citations

Analytical Study of the Forced Harmonic Oscillator with Time Dependent Parameters by Means of the Invariant Dynamical Linear Operator

Chadia Qotni, Az-Eddine L. Marrakchi, Salaheddine Sayouri +1 more

A general treatment of the quantum forced harmonic oscillator with time dependent mass and frequency is presented. We use the invariant dynamical linear operator method. We analyse the uncertainty principle and obtain the quantum correlation of the forced quantum harmonic oscillator with parameters depending upon time. We prove that these parameters do not depend on the external force. We evaluate the time dependent probability density; we prove that this parameter depends on the external force.

International review of physicsApr 30, 20130 citations

Wave functions of a time-dependent harmonic oscillator with and without a singular perturbation

I. A. Pedrosa, G. P. Serra, I. Guedes

We use the Lewis and Riesenfeld invariant method to obtain the exact Schr\"odinger wave functions for a time-dependent harmonic oscillator with and without an inverse quadratic potential. As a particular case we also obtain the wave functions for the Caldirola-Kanai oscillator.

Physical Review ANov 1, 1997120 citations

Quantum Dynamics of a Particle in a Linear Potential: Invariant Operator Approach and Discrete Spectrum Solutions

Mustapha Maamache, Aymen Bendjoudl

quant-pharXiv:2606.02112Jun 2, 20260 citations

Applied mathematics and physics

Analytical Computation of the Caldirola-Kanai Oscillator Parameters by the Dynamic Invariant Method

Chadia Qotni, Az-Eddine L. Marrakchi, Salaheddine Sayouri, Y. Achkar

Published Oct 18, 2019• 0 citations

Publisher

Abstract

DOI: 10.12691/amp-7-1-2

Related papers

Teoria quântica de invariantes aplicada ao oscilador harmônico forçado dependente do tempo de Caldirola-Kanai

Alberes Lopes de Lima, I. A. Pedrosa

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrodinger equation for the Caldirola-Kanai time-dependent forced harmonic oscillator in terms of the solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the quantum fluctuations in coordinate and momentum as well as the uncertainty product. It is also shown that the quantum fluctuations and the width of the Gaussian packet do not depend on the external force.In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrodinger equation for the Caldirola-Kanai time-dependent forced harmonic oscillator in terms of the solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the quantum fluctuations in coordinate and momentum as well as the uncertainty product. It is also shown that the quantum fluctuations and the width of the Gaussian packet do not depend on the external force.

Scientia PlenaJan 1, 20080 citations

Osciladores log-periódicos e tipo Caldirola-Kanai

Vagner Henrique Loiola Bessa

In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) $m(t)=m_0frac{t}{t_0}$ , (II) $m(t)=m_0$ and (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$ . In all three cases $omega(t)=omega_0frac{t_0}{t}$ . For class (b) we study the Caldirola-Kanai oscillator (IV)where $omega(t)=omega_0$ and $m(t)=m_0 ext{exp}ajust{gamma t}$ and the oscillator with $omega(t)=omega_0$ and $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$ , for $alpha=2$ (V) and $alpha=4$ (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of $q(t)$ , $p(t)$ as well as the phase diagram $q(t)$ vs $p(t)$ . To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) $m(t)=m_0frac{t}{t_0}$, (II) $m(t)=m_0$ and (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$. In all three cases $omega(t)=omega_0frac{t_0}{t}$. For class (b) we study the Caldirola-Kanai oscillator (IV)where $omega(t)=omega_0$ and $m(t)=m_0 ext{exp}ajust{gamma t}$ and the oscillator with $omega(t)=omega_0$ and $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$, for $alpha=2$ (V) and $alpha=4$ (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of $q(t)$, $p(t)$ as well as the phase diagram $q(t)$ vs $p(t)$. To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.

Jan 1, 20120 citations

Gaussian wave packet states of a generalized inverted harmonic oscillator with time-dependent mass and frequency

I. A. Pedrosa, Alberes Lopes de Lima, Alexandre M. de M. Carvalho

We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.

Canadian Journal of PhysicsJan 12, 201512 citations

Analytical Study of the Forced Harmonic Oscillator with Time Dependent Parameters by Means of the Invariant Dynamical Linear Operator

Chadia Qotni, Az-Eddine L. Marrakchi, Salaheddine Sayouri +1 more

International review of physicsApr 30, 20130 citations

Wave functions of a time-dependent harmonic oscillator with and without a singular perturbation

I. A. Pedrosa, G. P. Serra, I. Guedes

Physical Review ANov 1, 1997120 citations

Quantum Dynamics of a Particle in a Linear Potential: Invariant Operator Approach and Discrete Spectrum Solutions

Mustapha Maamache, Aymen Bendjoudl

We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.

quant-pharXiv:2606.02112Jun 2, 20260 citations

m(t)=m_0frac{t}{t_0}

, (II)

m(t)=m_0

and (III)

m(t)=m_0ajust{frac{t}{t_0}}^2

. In all three cases

omega(t)=omega_0frac{t_0}{t}

. For class (b) we study the Caldirola-Kanai oscillator (IV)where

omega(t)=omega_0

and

m(t)=m_0 ext{exp}ajust{gamma t}

and the oscillator with

omega(t)=omega_0

and

m(t)=m_0ajust{1+frac{t}{t_0}}^alpha

, for

alpha=2

(V) and

alpha=4

(VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of

q(t)

p(t)

as well as the phase diagram

q(t)

p(t)

. To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.