Numerical studies of quantum systems

Leonardi Hernández-Sánchez, I. Ramos-Prieto, Ariel Flores-Rosas +2 more

Obtaining exact solutions to the Schrödinger equation in complex quantum systems poses significant challenges. In this context, numerical methods emerge as valuable tools for analyzing such systems. This article proposes a numerical approach using the fourth-order Runge-Kutta method, implemented in Python, to tackle radiation-matter interaction systems. This methodology is applicable to various Hamiltonians, including that of the Jaynes-Cummings model. The accuracy of the numerical results is validated by comparing them with analytical solutions in simplified cases, demonstrating its effectiveness in studying quantum systems where exact solutions cannot be derived.

Tôru Sakai

Marco Patriarca

We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schrödinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.

S. S. Kancharla

G. Ball, Kendra vant, Nelson L Christensen

We present a detailed numerical study of a chaotic classical system and its quantum counterpart. The system is a special case of a kicked rotor and for certain parameter values possesses cantori dividing chaotic regions of the classical phase space. We investigate the diffusion of particles through a cantorus. A quantum analysis confirms that the cantori act as barriers. We numerically estimate the classical phase space flux through the cantorus per kick and relate this quantity to the behavior of the quantum system. We introduce decoherence via environmental interactions with the quantum system and observe the subsequent increase in the transport of quantum particles through the boundary.

Wolfgang Schweizer