عنوان مقاله [English]
In this article, a quantum many body system is considered. Then two time-dependent interactions have been added to the system. Changing of the interactions are assumed in general form. After that, by using algebraic method, time evolution of this many body system is investigated. In order to study the time evolution, Lewis-Riesenfeld dynamical invariant and time evolution operator method have been used. Appropriate dynamical invariants are constructed and their Eigen values are derived as well as appropriate time evolution operators are constructed. At the end, a comparison has been done between the derived results in the article considering a special case.
 M. Maamache, A. Bounames, N. Ferkous, Comment on “Wave functions of a time-dependent harmonic oscillator in a static magnetic field”, Physical Review A 73 (2006) 016101.
 C.M.A. Dantas, I.A. Pedrosa, B. Baseia, Harmonic oscillator with time-dependent mass and frequency and a perturbative potential, Physical Review A 45 (1992) 1320.
 I.A. Pedrosa, Exact wave functions of a harmonic oscillator with time-dependent mass and frequency, Physical Review A 55 (1997) 3219.
 L.H. Yu, C.P. Sun, Evolution of the wave function in a dissipative system, Physical Review A 49 (1994) 592.
 J.Y. Ji, J.K. Kim, S.P. Kim, Heisenberg-picture approach to the exact quantum motion of a time-dependent harmonic oscillatorm, Physical Review A 51 (1995) 4268.
 B. Remaud, E.S. Hernandez, Damping of wave packet motion in a general time-dependent quadratic field Journal of Physics A: Mathematical and General 13 (1980) 2013.
 I.A. Pedrosa, G.P. Serra, I. Guedes, Wave functions of a time-dependent harmonic oscillator with and without a singular perturbation, Physical Review A 56 (1997) 4300.
 J.R. Choi, Exact quantum theory of noninteracting electrons with time-dependent effective mass in a time-dependent magnetic field, Journal of Physics: Condensed Matter 15 (2003) 823.
 G. Harari, Y. Ben-Aryeh, A. Mann, Propagator for the general time-dependent harmonic oscillator with application to an ion trap, Physical Review A 84 (2011) 062104.
 R.J. Glauber, Photon Correlations, Physical Review Letters 10 (1963) 84.
 G.S. Agarwal, S.A. Kumar, Exact quantum-statistical dynamics of an oscillator with time-dependent frequency and generation of nonclassical states, Physical Review Letters 67 (1991) 3665.
 H.P. Yuen, Two-photon coherent states of the radiation field, Physical Review A 13 (1976) 2226.
 W. Paul, Electromagnetic traps for charged and neutral particles, Review of Modern Physics, 62 (1990) 531.
 L.S. Brown, Quantum motion in a Paul trap, Physical Review Letters 66 (1991) 527
 S.A. Fulling, Aspects of Quantum Fields in Curved Space, Cambridge University Press, Cambridge (1982).
 D.G. Vergel, E.J. Villasenor, The time-dependent quantum harmonic oscillator revisited: Applications to quantum field theory, Annals of Physics 324 (2009) 1360.
 Y.Q. Li, X.Y. Pan, V. Sahni, Wave function for time-dependent harmonically confined electrons in a time-dependent electric field, The Journal of Chemical Physics 139 (2013) 114301.
 H.R. Lewisand, W.B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, Journal of Mathematical Physics 10 (1969) 1458.
 H. Sobhani, H. Hassanabadi, Two-dimensional linear dependencies on the coordinate time-dependent interaction in relativistic non-commutative phase space, Communications in Theoretical Physics 64 (2015) 263-268.
 J. Wei, E. Norman, Lie Algebraic Solution of Linear Differential Equations, Journal of Mathematical Physics 4 (1963) 575.
 H. Sobhani, H. Hassanabadi, Rashba Effect in Presence of Time-Dependent Interaction Communications in Theoretical Physics, 65 (2015) 543.
 H. Sobhani, H. Hassanabadi, Study of Time Evolution for Approximation of Two-Body Spinless Salpeter Equation in Presence of Time-Dependent Interaction, 2016 (2016) 3647392.
 L. Naderi, H. Hassanabadi, H. Sobhani, Bohr Hamiltonian with time-dependent potential, International Journal of Modern Physics E 25 (2016) 1650029.
 H. Sobhani, H. Hassanabadi, Davydov–Chaban Hamiltonian in presence of time-dependent potential, Physics Letters B 760 (2016) 1–5.