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Nanosystems Physics Chemistry Mathematics

Particle transport in a network of quantum harmonic oscillators

Jambul Yusupov, Kh. Sh. Matyokubov, K.K. Sabirov

Published Apr 25, 2020• 3 citations
Publisher

Abstract

In this paper, we address the problem of a particle dynamics in a network of quantum harmonic oscillators by solving the stationary Schrödinger equation on metric graphs in the presence of harmonic oscillator potential with bond-dependent frequency.Particle transport is analyzed by considering wave packet dynamics in such a system modeled in terms of quantum graph.

DOI
10.17586/2220-8054-2020-11-2-145-152

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In this paper, we address the problem of a particle dynamics in a network of quantum harmonic oscillators by solving the stationary Schrödinger equation on metric graphs in the presence of harmonic oscillator potential with bond-dependent frequency.Particle transport is analyzed by considering wave packet dynamics in such a system modeled in terms of quantum graph.
Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.
The one-dimensional quantum harmonic oscillator is considered. The wave solution of the Schrödinger equation determines the model of motion of particles in the nnn-quantum state as orbits of a stationary random walk at the moment of jump from the n−1n-1n−1-qu
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
In this paper, the analytic solution of the time-dependent Schrödinger equation is obtained for the wave packet in two-dimensional oscillator potential. The quantum dynamics of the wave packet is investigated based on this analytic solution. To our knowledge, this is the first time we solve, analytically and exactly this kind of time-dependent Schrödinger equation in a two-dimensional system, in which the Gaussian parameters satisfy the coupled nonlinear differential equations. The coherent states and their rotations of the system are discussed in detail. We find also that this analytic solution includes four kinds of modes of the evolutions for the wave packets: rigid, rotational, vibrational states and a combination of the rotation and vibration without spreading.