Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia1621710.22363/2312-9735-2017-25-3-283-294Research ArticleOn the Evolution of Converging Wave Packet of an Inverted Quantum Oscillator Driven by Homogeneous Harmonic FieldChistyakovV Vv.chistyakov@corp.ifmo.ruScientific Research University of Informational Technology, Mechanics and Optics1512201725328329406062017Copyright © 2017, Chistyakov V.V.2017<p>The problem investigated refers to periodically driven 1D quantum inverted harmonic oscillator (IHO) with the Hamiltonian of . The model is used widely in huge quantum applications concerned unstable molecular complexes and ions under laser light affection. Non-stationary Schrdinger equation (NSE) was solved analytically and numerically by means of Maple 17 with initial wave function (w.f.) of generalized Gaussian type. This one described the converging 1D probability flux and fitted well the quantum operator of initial conditions (IC). For the IC one can observe, first, the collapse of w.f. packet into extremely narrow 1D space interval of length and, second, its spreading back up to its starting half width, and all that - at dimensionless times. At certain phases j defined by W and s0 the wave packet center displayed nonharmonic oscillating behavior near some slowly drifting space position within this time interval and after that leaved onto infinity while the unlimited packet spreading. And the phases themselves served as bifurcation points separating the NSE solutions with the outgoing to from those with. In resonant case of the values obeyed an inverted Fermi-Dirac formula of; for differing the asymptotic of obeyed well classical law.</p>Mapledriven inverted oscillatornon-stationary Schrödinger equation (NSE)Gaussian wave packetcollapsephase shiftstabilizationbifurcationMapleперевёрнутый квантовый осцилляторпериодическое возмущениенестационарное уравнение Шрёдингера (НУШ)волновой пакетобобщённый гауссовский типколлапсфазовый сдвигдинамическая стабилизациябифуркация[I. Serban, F. Wilhelm, Dynamical Tunneling in Microscopic Systems, Phys. Rev. Lett. 10 (2007) 101–104. URL https://www.researchgate.net/publication/5913852.][S. Baskoutas, A. Jannussistl, R. Mignanig, Dissipative Tunneling of the Inverted Caldirola–Kanai Oscillator, J. Phys. A: Math. Gen. 27 (1994) 2189–2196.][S. Matsumoto, M. Yoshimura, Dynamics of Barrier Penetration in Thermal Medium: Exact Result for Inverted Harmonic Oscillator, Phys. Rev. A 59 (6) (2000) 2201–2238.][L. Pedrosa, A. de Lima, A. de M. Carvalho, Gaussian Wave Packet States of Generalized Inverted Oscillator With Time-Dependent Mass and Frequency, Can. J. of Phys 93 (2015) 3–7.][Y. Nogami, F. Toyama, Nonlinear Schrödinger Soliton in a Time-Dependent Quadratic Potential, Phys. Rev. E 49 (5) (1994) 4497–4501.][N.F. Stepanov, V.I. Pupyshev, Quantum Mechanics of Molecules and Quantum Chemistry: Textbook, MSU Publishers, Moscow, 1991, in Russian.][B.N. Zakhar’ev, Discrete and Continuous Quantum Mechanics Exactly Solvable Models, Physics of Elementary Particles and Atomic Nuclei 23 (5) (1992) 1387–1468, in Russian.][C. A. Muñoz, J. Rueda-Paz, K. B. Wolf, Discrete Repulsive Oscillator Wavefunctions, J. Phys. A: Math. Theor. 42 (2009) 485210.][G. Barton, Quantum Mechanics of the Inverted Oscillator Potential, Annals of Physics 166 (2) (1986) 322–363.][P. Duclosi, E. Soccorsi, P. Stoviček, et al., On the Stability of Periodically Time-Dependent Quantum Systems, Rev. in Math. Phys. 6 (2008) 212–240.][V.G. Bagrov, D.M. Gitman, Coherent States of Inverse Oscillators and Related Problems, J. Phys. A: Math. Theor. 46 (2013) 325305.][I. Zlotnik, Numerical Methods for Solving the Generalized Time-Dependent Schrödinger Equation in Unbounded Domains, Ph.D. thesis, MPEI (2013).]