With many physical applications already on the table, in this chapter, we return to some of the simplified ones and re-complexify them.These problems require more sophisticated, and incomplete, solutions.Instead of finding the position of the bob for the simple pendulum, we find the period of motion for the ``real" pendulum.Instead of the classical harmonic oscillator, with its familiar solution, we study the period of the relativistic harmonic oscillator, and find that in the high energy limit, a mass attached to a spring behaves very differently from its non-relativistic counterpart.

An exploration of the wave equation and its solutions in three dimensions.The chapter's mathematical focus is on vector calculus, enough to understand and appreciate the harmonic functions that make up the static solutions to the wave equation.

A review of the problem of motion for masses attached to springs.That's the physics of the chapter, a familiar problem from high school and introductory college classes, meant to orient and refresh the reader.The mathematical lesson is about series solutions (the method of Frobenius) for ordinary differential equations (ODEs), and the definition of trigonometric special functions in terms of the ODEs that they solve.This is the chapter that reviews complex numbers and the basic properties of ODEs and their solutions (superposition, continuity, separation of variables).

There's a great deal of interesting physics in the Schrödinger equation and its solutions, and the mathematical underpinnings of that equation can be expressed in several ways. It's been my experience that students find it helpful to see a combination of Erwin Schrödinger's wave mechanics approach and the matrix mechanics approach of Werner Heisenberg, as well as Paul Dirac’s bra and ket notation. So these first two chapters provide the mathematical foundations that will help you understand these different perspectives and “languages” of quantum mechanics, beginning with the basics of vectors in Section 1.1. With that basis in place, you can move on to Dirac notation in Section 1.2 and abstract vectors and functions in Section 1.3. The rules pertaining to complex numbers, vectors, and functions are reviewed in Section 1.4, followed by an explanation of orthogonal functions in Section 1.5, and using the inner product to find components in Section 1.6. The final section of this chapter (as in all later chapters) is a set of problems that will allow you to exercise your understanding of the concepts and mathematical techniques presented in this chapter. Remember that you can find full, interactive solutions to every problem on the book's website.

And since it's easy to lose sight of the architectural plan of an elaborate structure when you’re laying the foundation, as mentioned in the Preface you’ll find in each section a plain-language statement of the main ideas of that section as well as a short paragraph explaining the relevance of that development to the Schrödinger equation and quantum mechanics.

As you look through this chapter, don't forget that this book is modular, so if you have a good understanding of the included topics and their relevance to quantum mechanics, you should feel free to skip over this chapter and jump into the discussions of operators and eigenfunctions in Chapter 2. And if you’re already up to speed on those topics, the Schrödinger equation and quantum wavefunctions await your attention in later chapters.

Vector Basics

If you pick up any book about quantum mechanics, you’re sure to find lots of discussion about wavefunctions and the solutions to the Schrödinger equation. But the language used to describe those functions, and the mathematical techniques used to analyze them, are rooted in the world of vectors.

The concepts and techniques discussed in the previous chapter are intended to prepare you to cross the bridge between the mathematics of vectors and functions and the expected results of measurements of quantum observables such as position, momentum, and energy. In quantum mechanics, every physical observable is associated with a linear “operator” that can be used to determine possible measurement outcomes and their probabilities for a given quantum state.

This chapter begins with an introduction to operators, eigenvectors, and eigenfunctions in Section 2.1, followed by an explanation of the use of Dirac notation with operators in Section 2.2. Hermitian operators and their importance are discussed in Section 2.3, and projection operators are introduced in Section 2.4. The calculation of expectation values is the subject of Section 2.5, and as in every chapter, you’ll find a series of problems to test your understanding in the final section.

Operators, Eigenvectors, and Eigenfunctions

If you’ve heard the phrase “quantum operator” and you’re wondering “What exactly is an operator?,” you’ll be happy to learn that an operator is simply an instruction to perform a certain process on a number, vector, or function. You’ve undoubtedly seen operators before, although you may not have called them that. But you know that the symbol “ √ ” is an instruction to take the square root of whatever appears under the roof of the symbol, and “d( )/dx” tells you to take the first derivative with respect to x of whatever appears inside the parentheses.

The operators you’ll encounter in quantum mechanics are called “linear” because applying them to a sum of vectors or functions gives the same result as applying them to the individual vectors or functions and then summing the results.

If you’ve worked through Chapters 1 and 2, you’ve already seen several references to the Schrödinger equation and its solutions. As you’ll learn in this chapter, the Schrödinger equation describes how a quantum state evolves over time, and understanding the physical meaning of the terms of this powerful equation will prepare you to understand the behavior of quantum wavefunctions. So this chapter is all about the Schrödinger equation, and you can read about the solutions to the Schrödinger equation in Chapters 4 and 5.

In the first section of this chapter, you’ll see a “derivation” of several forms of the Schrödinger equation, and you’ll learn why the word “derivation” is in quotes. Then, in Section 3.2, you’ll find a description of the meaning of each term in the Schrödinger equation as well as an explanation of exactly what the Schrödinger equation tells you about the behavior of quantum wavefunctions. The subject of Section 3.3 is a time-independent version of the Schrödinger equation that you’re sure to encounter if you read more advanced quantum books or take a course in quantum mechanics.

To help you focus on the physics of the situation without getting too bogged down in mathematical notation, the Schrödinger equation discussed in most of this chapter is a function of only one spatial variable (x). As you’ll see in later chapters, even this one-dimensional treatment will let you solve several interesting problems in quantum mechanics, but for certain situations you’re going to need the three-dimensional version of the Schrödinger equation. So that's the subject of the final section of this chapter (Section 3.4).

Origin of the Schrödinger Equation

If you look at the introduction of the Schrödinger equation in popular quantum texts, you’ll find that there are several ways to “derive” the Schrödinger equation. But as the authors of those texts invariably point out, none of those methods are rigorous derivations from first principles (hence the quotation marks). As the brilliant and always-entertaining physicist Richard Feynman said, “It's not possible to derive it from anything you know. It came out of the mind of Schrödinger.”

So if Erwin Schrödinger didn't arrive at this equation from first principles, how exactly did he get there? The answer is that although his approach evolved over several papers, from the start Schrödinger clearly recognized the need for a wave equation from the work of French physicist Louis de Broglie.

The conclusions reached in the previous chapter concerning quantum wavefunctions and their general behavior are based on the form of the Schrödinger equation, which relates the changes in a particle or system's wavefunction over space and time to the energy of that particle or system. Those conclusions tell you a great deal about how matter and energy behave at the quantum level, but if you want to make specific predictions about the outcome of measurements of observables such as position, momentum, and energy, you need to know the exact form of the potential energy in the region of interest. In this chapter, you’ll see how to apply the concepts and mathematical formalism described in earlier chapters to quantum systems with three specific potentials: the infinite rectangular well, the finite rectangular well, and the harmonic oscillator.

Of course, you can find a great deal more information about each of these topics in comprehensive quantum texts and online. So the purpose of this chapter is not to provide one more telling of the same story; instead, these example potentials are meant to show why techniques such as taking the inner product between functions, finding eigenfunctions and eigenvalues of an operator, and using the Fourier transform between position and momentum space are so important in solving problems in quantum mechanics. As in previous chapters, the focus will be on the relationship between the mathematics of the solutions to the Schrödinger equation and the physical meaning of those solutions. And although we live in a universe with (at least) three spatial dimensions in which the potential energy may vary over time as well as space, most of the essential physics of quantum potential wells can be understood by examining the one-dimensional case with time-independent potential energy. So in this chapter, the Schrödinger equation is written with position represented by x and potential energy by V(x).

Infinite Rectangular Potential Well

The infinite rectangular well is a potential configuration in which a quantum particle is confined to a specified region of space (called the “potential well”) by infinitely strong forces at the edges of that region. Within the well, no force acts on the particle. Of course, this configuration is not physically realizable, since infinite forces do not occur in nature. But as you’ll see in this section, the infinite rectangular potential well has several features that make this a highly instructive configuration.

This book has one purpose: to help you understand the Schrödinger equation and its solutions. Like my other Student's Guides, this book contains explanations written in plain language and supported by a variety of freely available online materials. Those materials include complete solutions to every problem in the text, in-depth discussions of supplemental topics, and a series of video podcasts in which I explain the most important concepts, equations, graphs, and mathematical techniques of every chapter.

This Student's Guide is intended to serve as a supplement to the many comprehensive texts dealing with the Schrödinger equation and quantum mechanics. That means that it's designed to provide the conceptual and mathematical foundation on which your understanding of quantum mechanics will be built. So if you’re enrolled in a course in quantum mechanics, or you’re studying modern physics on your own, and you’re not clear on the relationship between wave functions and vectors, or you want to know the physical meaning of the inner product, or you’re wondering exactly what eigenfunctions are and why they’re so important, then this may be the book for you.

I’ve made this book as modular as possible to allow you to get right to the material in which you’re interested. Chapters 1 and 2 provide an overview of the mathematical foundation on which the Schrödinger equation and the science of quantum mechanics is built. That includes generalized vector spaces, orthogonal functions, operators, eigenfunctions, and the Dirac notation of bras, kets, and inner products. That's quite a load of mathematics to work through, so in each section of those two chapters you’ll find a “Main Ideas” statement that concisely summarizes the most important concepts and techniques of that section, as well as a “Relevance to Quantum Mechanics” paragraph that explains how that bit of mathematics relates to the physics of quantum mechanics.

So I recommend that you take a look at the “Main Ideas” statements in each section of Chapters 1 and 2, and if your understanding of those topics is solid, you can skip past that material and move right into a term-byterm dissection of the Schrödinger equation in both time-dependent and timeindependent form in Chapter 3. And if you’re confident in your understanding of the meaning of the Schrödinger equation, you can dive into Chapter 4, in which you’ll find a discussion of the quantum wavefunctions that are solutions to that equation.

If you’re wondering how the abstract vector spaces, orthogonal functions, operators, and eigenvalues discussed in Chapters 1 and 2 relate to the wavefunction solutions to the Schrödinger equation developed in Chapter 3, you should find this chapter helpful. One reason that relationship may not be obvious is that quantum mechanics was developed along two parallel paths, which have come to be called the “matrix mechanics” of Werner Heisenberg and the “wave mechanics” of Erwin Schrödinger. And although those two approaches are known to yield equivalent results, each offers benefits in elucidating certain aspects of quantum theory. That's why Chapters 1 and 2 focused on matrix algebra and Dirac notation while Chapter 3 dealt with plane waves and differential operators.

To help you understand the connections between matrix mechanics and wave mechanics, the first section of this chapter explains the meaning of the solutions to the Schrödinger equation using the Born rule, which is the basis for the Copenhagen interpretation of quantum mechanics. In Section 4.2, you’ll find a discussion of quantum states, wavefunctions, and operators, along with an explanation of several dangerous misconceptions that are commonly held by students attempting to apply quantum theory to practical problems.

The requirements and general characteristics of quantum wavefunctions are discussed in Section 4.3, after which you can see how Fourier theory applies to quantum wavefunctions in Section 4.4. The final section of this chapter presents and explains the form of the position and momentum operators in both position and momentum space.

The Born Rule and Copenhagen Interpretation

When Schrödinger published his equation in early 1926, no one (including Schrödinger himself) knew with certainty what the wavefunction ψ represented. Schrödinger thought that the wavefunction of a charged particle might be related to the spatial distribution of electric charge density, suggesting a literal interpretation of the wavefunction as a real disturbance – a “matter wave.” Others speculated that the wavefunction might represent some type of “guiding wave” that accompanies every physical particle and controls certain aspects of its behavior. Each of these ideas has some merit, but the question of what is actually “waving” in the quantum wavefunction solutions to the Schrödinger equation was very much open to debate.

LEARNING FROM NUMBERS

The equations of motion of classical mechanics, whether Newton's, Lagrange's, or Hamilton's, have us believe that given the state of the system at a particular time, one can always predict what its state would be any time later, or, for that matter, what it was any time earlier. This is because of the fact that the equations of motion are symmetric with respect to time-reversal: (t)→(–t). Classical mechanics relies on the assumption that position q and momentum p of a mechanical system are simultaneously knowable. Together, the pair (q, p) provides a signature of the state of the system. Their time-dependence, i.e., provided by the equations of motion, accurately describes their temporal evolution. For macroscopic objects, this is an excellent approximation, and the classical laws of mechanics are stringently deterministic. This is stringently correct, but there is an important caveat, expressed succinctly by Stephen Hawking: “Our ability to predict the future is severely limited by the complexity of the equations, and the fact that they often have a property called chaos… a tiny disturbance in one place, can cause a major change in another.” The difficulty Hawking alludes to has nothing to do with the quantum principle of uncertainty, but to a challenge within the framework of the fully deterministic classical theory. The solution to the temporal evolution may become chaotic, even as they remain deterministic, due to extreme sensitivity to the initial conditions that are necessary to obtain the solution to the equation of motion. Careful admission of the previous remark would prepare you to embark your journey on the exciting field of chaos. Along the way, you will also meet objects having weird fractional dimensions. The field covered by chaos theory is vast, though relatively young. It is a very rich field and can be introduced from a variety of perspectives.

The general field of chaos theory is often regarded as a study of a ‘dynamical system’ which is just about any quantity which changes, and one has reasons to track these changes and the sequence of values it may take, for example, over a time interval.

DISSIPATIVE SYSTEMS

Energy dissipative systems prompt us to think of physical phenomena in which energy is not conserved, it is lost. We attribute these losses to ‘friction’, which is the common term used to describe energy dissipation. Now, in our everyday experience, we are primarily involved with the gravitational and electromagnetic interactions, and both of these are essentially conservative. What is it, then, that makes friction non-conservative? What does non-conservation of energy, or ‘energy-loss’, really mean? Fundamental interactions in nature allow energy to be changed from one form to another, but not created or destroyed. It therefore seems that the term ‘energy loss’ is used somewhat loosely. We must be really careful when we talk about dissipative phenomena.

Losing money is often a matter of concern, as also other things we sometimes ‘lose’ from time to time, including time itself. Time wasted does not ever come back, of course; but nor does the time well-spent. The difference between the two is that the latter is accounted for by the gains made, and for the former there is simply no account. Isn't it merely a matter of book-keeping? It is not at all uncommon that we plan to do something during the day, and end up not doing it. We then sit and wonder where lost track of time. Did the day just skip over the afternoon and ring in the evening? If that did not happen, where was the time ‘lost’? You possibly remember everything that you did since morning, and you may be able to account for every hour you spent, except perhaps for what happened between 2:30 pm and 3 pm. That was the time you ransacked your house to find your mathematics text book, and did not realize that it took you half an hour to find it. The book had not really vanished, even if you suspected that it was lost. You had to look for it all over, from your Dad's room to your Sister's. As such, neither the book was lost, nor was half an hour scooped out of the day.