The nature of time: Explaining quantum mechanical phenomena

Sayed Kazem Mousavi

A. Ruschhaupt, Reinhard F. Werner

João Philipe Macedo Braga

Quantum mechanics plays a fundamental role in the description and understanding of the natural phenomena. Actually, the phenomena that take place in atomic and subatomic scale can not be well explained without the quantum mechanics approach. Furthermore, there are a lot of phenomena in macroscopic scale that reveals the quantum behavior of nature. In this sense, we can say that quantum mechanics is fundamental for the understanding of all natural phenomena. In Quantum Mechanics the state of a particle is mathematically described by the wave function Ψ(r,t) and its time evolution is governed by time-dependent Schrödinger equation. Thus, we can state that the fundamental problem of quantum mechanics is to solve the Schrödinger Equation in an arbitrary situation. In this work, we study a numerical technique to solve the time-dependent and time-independent Schrödinger Equation known as Split Operator technique. This aproach uses approximations for the exponencial of sum of operators that do not commute in order to implement the time-evolution operator. It makes possible to reduce the solution of the Schrödinger equation to a successive processes of multiplication and solution of tridiagonal system of linear equations. It can be easily performed using a computer. The technique was studied in detail using cartesian coordinates, and we also explained how to use the technique with periodic or finite boundary conditions. We make use this technique to study the behavior of an electron subjected to a random potential. In this situation we face the Anderson Localization phenomena. Furthermore, we developed the Split Operator technique using generalized coordinates, and studied the problem of an electron confined to a cylinder surface. It was verified that the numerical results agree with the analytical ones. So we can conclude that the Split Operator technique using generalized coordinates produce reliable results.

Eduardo R. Mucciolo

In this chapter we review the ideas and discoveries that lead to the development of quantum mechanics. Like other theories in physics, it was motivated by a series of experimental results that were at odds with the prevailing theories of the time. New concepts were introduced to explain those experiments, which in turn led to new experiments and more discoveries. Quantum mechanics took its current shape about 100 years ago and continues to provide a formidable framework for describing not only natural phenomena but also for leading us to new technologies, such as quantum information processing.

Avshalom C. Elitzur, Shahar Dolev

Simon W. Saunders