In this paper we use a canonical and a unitary transformation along with the Lewis and Riesenfeld invariant method to obtain the quantum states of a general system of two time-dependent statically coupled harmonic oscillators. The wave function obtained is written in terms of a c-number quantity, which is solution of the Milne-Pinney equation. We consider three different systems of time-dependent coupled oscillators, and for each one we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.

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In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. 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. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne-Pinney (MP) equation for each system. We obtain the solutions for the system with m 1 = m 2 = m 0 e γt , ω 1 = ω 01 e −γt/2 , ω 2 = ω 02 e −γt/2 and k = k 0 .

Neste trabalho estudamos um sistema clássico de dois osciladores harmônicos acoplados com massas (mi), constantes de mola (ki) e parâmetro de acoplamento (κ) dependentes do tempo. Para encontrar as soluções das equações de movimento de cada oscilador, usamos uma transformação canônica para reescrever a hamiltoniana do sistema acoplado como a soma das hamiltonianas de dois osciladores harmônicos desacoplados com frequências modificadas e massas unitárias. Analisamos o comportamento de xi, vi =ẋi e do diagrama de fase xi vs. vi para o sistema m1 = m2 = moe γt e k1 = k2 = κ = koe γt . Palavras-chave: osciladores acoplados, transformação canônica.In this work we study a coupled system of two classical oscillators with time-dependent masses (mi), spring constants (ki) and coupling parameter (κ). To obtain the solution of the equation of motion for each oscillator, we use a canonical transformation to rewrite the Hamiltonian of the coupled system as the sum of the hamiltonians of two uncoupled harmonic oscillators with modified frequencies and unitary masses. We analyze the behavior of xi, vi =ẋi and the phase diagram xi vs. vi for the system m1 = m2 = moe γt and k1 = k2 = κ = koe γt