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CRC Press eBooks

The Language of Quantum Mechanics

Eduardo R. Mucciolo

Published Aug 6, 2025• 0 citations
Publisher

Abstract

In this chapter we go through the essential mathematical tools needed to develop a quantitative description of quantum systems. The content will seem a bit dry and abstract, but gaining a good grasp of the concepts and the math of vectors and operators will make understanding quantum information processing a lot easier. So, take your time when studying this chapter. It will pay off.

DOI
10.1201/9781003485124-3

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In this chapter we go through the essential mathematical tools needed to develop a quantitative description of quantum systems. The content will seem a bit dry and abstract, but gaining a good grasp of the concepts and the math of vectors and operators will make understanding quantum information processing a lot easier. So, take your time when studying this chapter. It will pay off.
The ideas involved in quantum mechanics can be expressed quite concisely, yet a detailed study of individual quantum mechanical systems seems to rely heavily upon mathematics. In some ways this is unfortunate, as there is a danger that the underlying concepts may be lost in this detail. However, it is essential to be able to apply these ideas if we are to obtain results that are of use, and which may be compared with experimental data. In this chapter, we will deal with the mathematics needed for some of the simpler concepts before looking at more complicated situations that need more specialized tools.
Abstract Quantum mechanics is conceptually challenging and requires mathematics, but it is an essential basis for understanding nanoscience. Though simplified, the material in this chapter covers what might be learned in an entire semester, so if you have not been exposed to the subject before, panic would be a reasonable reaction! The chapter is long because it (almost) serves as a stand-alone guide to the subject (taken in conjunction with Appendix C). This outline lists the essential “take-home” points needed to follow the rest of the book. If you grasp these, you can always come back to this chapter later for more background.
Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.
Quantum mechanics is based on linear algebra. The general theory uses infinite dimensional vector spaces. Fortunately for us, to describe spin or polarization we need only finite dimensions, which makes things much easier. In fact, we need only a few tools. At the end of this chapter I have given a list. The rest of the chapter explains how to use these tools and what the calculations mean. There are many examples. It is important to work carefully through all of them. The mathematics introduced here is essential to everything that follows. Like much mathematics, it can seem complicated when it is first introduced, but it becomes almost second nature with practice. The actual computations only involve addition and multiplication of numbers, along with an occasional square root and trigonometric function.
Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information