Osciladores harmônicos acoplados dependentes do tempo

For N -coupled time-dependent harmonic oscillators with any couplings coordinates and momentum, we have derived the primary invariant and the generalized (Lewis and Riesenfeld) invariant represented in terms of the classical solution matrix of Hamilton's oscillator equation. Using the Lewis and Riesenfeld invariant method and exploiting the properties of the symplectic matrix we have obtained the exact Schrödinger wavefunctions for the oscillators.

Journal of Physics A Mathematical and GeneralOct 31, 20014 citations

Quantum dynamics of a particle trapped by oscillating fields

Alberes Lopes de Lima, Alexandre Rosas, I. A. Pedrosa

Journal of Modern OpticsDec 17, 20084 citations

Osciladores harmônicos acoplados dependentes do tempo

Diego Ximenes Macedo

Published Jan 1, 2012• 0 citations

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Abstract

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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.