Some Mathematics Needed for Quantum Mechanics

Daniel Minoli, Benedict Occhiogrosso

This chapter presents some of the basic mathematical background and nomenclature needed to gain an introductory understanding of Quantum Mechanics (QM). The more direct application of the math to QM per se is covered in chapters that follow.

A. O. Pittenger

To begin to describe quantum algorithms, we need an understanding of the physical phenomena that motivate quantum mechanics, a simple mathematical model of a potential quantum storage device and the notation to describe and manipulate such a model. In this chapter we present the context in which we will work, motivate the need for quantum mechanics and provide a quick introduction to the essential mathematical tools and notation to be used throughout the book. As part of the price of brevity, we will work with a static model of quantum mechanics and will defer additional mathematical definitions until they are needed. The reader who is already familiar with this material can easily skip to Chapter 2, giving only a cursory glance at Chapter 1 to identify any idiosyncratic notation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

John David Jackson, Robert C. Whitten

Francis T. S. Yu

Since quantum mechanics is one of the pillars of modern science, this chapter discusses one of the most intriguing principles in quantum mechanics, the uncertainty principle. In this it examines Heisenberg's principle was based on a diffraction limited observation, instead of upon a “nature” of time, developing his principle. There is a profound relationship between science and mathematics, in which people have seen that without mathematics there would be no science. In other words, science needs mathematics but mathematics does not need science. Although science is mathematics, mathematics is not science. Since an empty subspace has “no time”, all particles within the subspace collapse or “superimpose” instantly all together at t = 0, because time is distance and distance is time. This is precisely what the “simultaneous and instantaneous” superposition principle does in quantum mechanics. One of the most intriguing principles in quantum theory must be Heisenberg's uncertainty principle.

A. Mangoni