Tunneling Gaussian Wave Packets for Inverted Harmonic Potential

Zharko Tasev, Gjorgji Ivanovski

Brent Perreault, Luiz A. Manzoni, Jose Tadeu Lunardi

Guangjie Guo, Zhongzhou Ren, Ju Guo-Xing +1 more

An algebraic method to solve the time-dependent inverted harmonic oscillator (TDIHO) is proposed, and the analytical expressions of the wavefunctions obtained by this method are suitable for discussion of the quantum tunneling effect. With the Gaussian wave packet as the initial state, the general expressions of transmission and reflection probabilities, and dwell time of TDIHO are obtained. The quantum tunneling effect is further illustrated through two specific cases.

Tommy A. Byrd, Megan Ivory, Andrew J. Pyle +4 more

Pavel A. Golovinski

New exact and asymptotic results for a quantum inverted oscillator, driven by the variable external force, are presented. To illustrate the advantages of our approach, we applied the obtained propagator to the descriptions of evolution the initial Gaussian wave packet under arbitrary time-dependent force or a δ-pulse action. The alternative problem of tunneling a particle through a parabolic barrier under the influence of a low-frequency harmonic force we solve in the quasistatic approximation. Going to the Heisenberg picture in the Caldeira-Leggett model, we describe the influence of a time-dependent force on an open inverted oscillator. The resulting dynamics is presented in the form of combination the evolution of the average value of a coordinate operator and the stochastic diffusion, being induced the wave packet spreading and the Brownian motion.

W. van Dijk, F. Masafumi Toyama, Sjirk Jan Prins +1 more

We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.