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Scientia Plena

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

Alberes Lopes de Lima, I. A. Pedrosa

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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.

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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.
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 discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.
In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.
In this work we present a quantum description of a mesoscopic RLC circuit without source. For this purpose, we model this system for that of a damped harmonic oscillator which is described by the Caldirola-Kanai Hamiltonian. Then, with the aid of the quantum invariant method we solve the Schrödinger equation associated with this Hamiltonian and write the corresponding wave functions in terms of a particular solution of the Milne-Pinney equation. We also construct coherent states for the RLC quantized, and evaluate the quantum fluctuations of the charge and the magnetic flux, as well as the corresponding product of uncertainty.
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)=m0fractt0m(t)=m_0frac{t}{t_0}m(t)=m0​fractt0​, (II) m(t)=m0m(t)=m_0m(t)=m0​ and (III) m(t)=m0ajustfractt02m(t)=m_0ajust{frac{t}{t_0}}^2m(t)=m0​ajustfractt0​2. In all three cases omega(t)=omega0fract0tomega(t)=omega_0frac{t_0}{t}omega(t)=omega0​fract0​t. For class (b) we study the Caldirola-Kanai oscillator (IV)where omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0extexpajustgammatm(t)=m_0 ext{exp}ajust{gamma t}m(t)=m0​extexpajustgammat and the oscillator with omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0ajust1+fractt0alpham(t)=m_0ajust{1+frac{t}{t_0}}^alpham(t)=m0​ajust1+fractt0​alpha, for alpha=2alpha=2alpha=2 (V) and alpha=4alpha=4alpha=4 (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of q(t)q(t)q(t), p(t)p(t)p(t) as well as the phase diagram q(t)q(t)q(t) vs p(t)p(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.
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.