Mastering Quantum Mechanics: Essentials, Theory, and Applications

Book Preface

Mastering Quantum Mechanics offers students a complete overview of quantum mechanics, beginning with an introduction to the essential concepts and results, followed by the theoretical foundations that provide the conceptual framework of the subject, and closing with the tools and applications they will need for advanced studies and research. The book emerged naturally from my many years of teaching quantum mechanics to both Massachusetts Institute of Technology (MIT) undergraduates and students of all ages and backgrounds worldwide. In fact, while teaching MIT’s cycle of quantum mechanics courses, I began working with the university’s online initiatives, edX and MITx, to offer the courses online at the same high academic level that my colleagues and I teach on campus, a feat possible thanks to the sophisticated software of the online platforms. As I developed and taught these courses over the last ten years, I collected and refined a large amount of lecture notes and received comments and input from hundreds of learners. These materials form the backbone of this book. It is my hope that students will use this book in the manner best suited for them, whether to gain a solid understanding of the key components of quantum mechanics or to prepare themselves for graduate work in physics. Most importantly, I hope that my book will guide students at all levels through the complex and fascinating world of quantum mechanics in a clear, precise, accessible, and useful manner. I hope to convey the beauty and elegance of the subject and my enthusiasm for it.
Mastering Quantum Mechanics aims to present quantum mechanics in a modern and approachable way while still including traditional material that remains essential to a well-rounded understanding of the subject. The book is accessible to anyone with a working knowledge of the standard topics typically covered during the first three semesters of undergraduate physics (mechanics, electromagnetism, and waves). It assumes some fluency in mathematics, specifically with multivariable calculus and ordinary differential equations. Familiarity with linear algebra is helpful but not strictly required. I do not assume any previous knowledge of quantum mechanics. By working through this textbook, students will master the essential tools and internalize the main concepts of quantum mechanics. They will gain the strong foundation in quantum theory that is required for graduate work in physics, whether in quantum field theory, string theory, condensed matter physics, atomic physics, quantum computation, or any other subject that uses quantum mechanics. Put differently, Mastering Quantum Mechanics need not be a student’s final stop in the study of quantum mechanics. On the contrary, the knowledge gained here can be a springboard for future studies, whether a graduate-level course in quantum mechanics or specialized courses in the above-mentioned fields.
When writing this book, I tried to find the most efficient way to make quantum mechanics comprehensible and help the student internalize it. Below are a few ways in which this book attempts to smooth the learning process, in terms of pedagogy.

1. There is a gradual increase in difficulty as the book progresses, closely matching students’ increasingly sophisticated understanding of the material. I aim to keep pushing learners at the right pace throughout, challenging them at a constant level.
2. I give a fairly in-depth discussion of the mathematical background required to understand the ideas of quantum mechanics clearly and precisely. This material, dealing with complex vector spaces and linear operators, is developed in two chapters at the beginning of part II. The concepts are illustrated at every step with examples from the quantum systems covered in part I. This approach helps students remember and internalize what they have already learned, as well as understand the math. With this mathematical background in place, students can focus on comprehending the physics at hand.
3. The exposition style is explicit and deliberate, aiming to help students’ thought processes by anticipating their potential questions. I also attempt, when possible, to look at facts from various angles and perspectives so that students pause to appreciate the nuances.
4. I have addressed a number of topics at various points at different levels of detail. Readers will encounter Hermitian operators, uncertainty, the harmonic oscillator, angular momentum, and central potentials both in parts I and II. The hydrogen atom appears in all three parts! It would take too long to do full justice to any of these subjects in part I, and students of part I are not yet prepared to absorb all of their important implications. Returning to these topics in parts II and III, with more experience, is in fact helpful to the learning process.
5. When useful, I quote and use important results that will be proven later in the book. For example, while perturbation theory is seen in full detail in part III, some of its results are cited and used in parts I and II. This is a win-win strategy: the student who will not get to part III will gain some exposure to the subject, and the student who gets to part III approaches the subject with some familiarity.
6. I discuss multiparticle states and tensor products ahead of turning to the subject of addition of angular momentum. This sequencing greatly facilitates understanding the latter, which is often a stumbling block in the learning of quantum mechanics.

7. Undergraduate textbooks generally devote limited resources to the advanced material in part III. I have endeavored, at various points, to present the more extended analysis needed to properly grasp this intrinsically subtle material. I do that for density matrices, degenerate perturbation theory, semiclassical approximation, and the adiabatic theorem.
8. Readers can support their studies by viewing videos, recorded in a classroom setting, in which I explain about 90 percent of the material in this book. These videos are available for free on MIT OpenCourseWare (OCW). There are also three online courses (8.04x, 8.05x, and 8.06x) that run at various times and include computer-graded exercises and problems.

Mastering Quantum Mechanics assumes no previous knowledge of quantum mechanics. It is an undergraduate textbook suitable for physics sophomores or juniors. While three semesters are needed to cover most of the material, the book can be used in a number of different ways: for self-learning, for a one-semester or a two-semester course, or to supplement a graduate-level course. All chapters contain exercises and problems. The exercises, inserted at various points throughout the text, are relatively straightforward and can be used by learners to assess their comprehension. The problems at the ends of the chapters are more challenging and sometimes develop new ideas. Mastery of the material requires solving all the exercises and a good fraction of the problems. I have starred the sections that one can safely skip in a first reading of this book.
The book is organized into three parts: part I, “Essentials,” presents the basic knowledge of the subject. Part II, “Theory,” provides the conceptual framework of quantum mechanics, solidifying the understanding of part I and extending it in various new directions. Finally, part III, “Applications,” allows students to internalize all they have learned as valuable techniques are introduced. For the benefit of interested readers and instructors, I sketch below the main features and contents of each part.
I: Essentials This material aims to give the student taking just one semester of quantum mechanics a sound introduction to the subject. Such an introduction, I believe, must include exposure to the following subjects: states and probability amplitudes, the Schrödinger equation, energy eigenstates of particles in potentials, the harmonic oscillator, angular momentum, the hydrogen atom, and spin one-half.
Chapter 1 gives a preview of many of the key ideas that will be developed in the book. Chapter 2 begins with the very surprising possibility of interaction-free measurements in the context of photons on a Mach-Zehnder interferometer, a kind of two-state system illustrating the concept of probability amplitudes. We then trace the ideas that led to the development of quantum theory, beginning with the photoelectric effect and continuing with Compton scattering and de Broglie waves. I explain the stationary phase principle, a tool used throughout this book. Then comes the Schrödinger equation in chapter 3, at which point I introduce position and momentum operators. While deriving the probability current, Hermitian operators appear for the first time. In chapter 4 we use wave packets to motivate Heisenberg’s uncertainty principle, and we examine how they evolve in time. Through Fourier transformation and Plancherel’s theorem, I introduce the idea of momentum space. Expectation values of operators, and their time dependence, are explored in chapter 5. With the introduction of an inner product, the notation is streamlined to discuss Hermitian operators and their spectrum. We take a preliminary look at the axioms of quantum mechanics. We learn about the uncertainty of an operator in a state, showing that this uncertainty vanishes if and only if the state is an eigenstate of the operator.
With chapter 6 we begin a three-chapter sequence on energy eigenstates. We begin with instructive examples: the particle on a circle, the infinite square well, the finite square well, the delta function potential, and the linear potential, a system naturally solved in momentum space. Chapter 7 focuses on general features. We explore the properties of bound states in one dimension and develop the semiclassical approximation to understand qualitatively the behavior of energy eigenstates. We continue with the node theorem and explore the numerical calculation of energy eigenstates with the shooting method. Here, Students learn how to “remove” the units from the Schrödinger equation, making numerical work efficient. We continue with the virial theorem, the variational principle, and the Hellman-Feynman lemma. The third chapter in the sequence, chapter 8, deals with energy eigenstates that are part of the continuum–scattering states. We use the stationary phase principle to analyze the time-dependent process in which a packet hitting a barrier gives rise to a reflected packet and a transmitted packet.
In chapter 9 we study the harmonic oscillator, finding the energy eigenstates by solving the Schrödinger differential equation, as well as using algebraic methods, with creation and annihilation operators. Our first look into angular momentum and central potentials comes in chapter 10, where we do just enough to be able to understand the hydrogen atom properly. We find the angular momentum commutator algebra and show that we can find simultaneous eigenstates of 2 and z—these are spherical harmonics. For central potentials we discuss the nature of the spectrum of bound states and the boundary conditions at r = 0. In chapter 11 we study the hydrogen atom, going first through the reduction of the two-body problem into a trivial center-of-mass motion and a relative motion in a Coulomb potential. We derive the energy spectrum from an analysis of the differential equation, discuss the degeneracies, and consider Rydberg atoms. Part I concludes with chapter 12, where we consider the “simplest” quantum system and discover that the natural operators in this system describe intrinsic angular momentum. We discuss the Stern-Gerlach experiment and see how to construct general states of spin one-half particles.
Part I does not use Dirac’s bra-ket notation except for a preview in chapter 12. A proper discussion of bra-kets is lengthy, as one needs to think of bras as elements of dual vector spaces. This subject is taken up in part II, chapter 14. Instead of bra-kets, part I uses inner products. This mathematical structure, well worth learning about, makes the definition of Hermitian operators very simple.
II: Theory Part II aims to develop the theory required to understand the foundations of quantum mechanics, to better understand the quantum systems of part I, and to extend the scope of quantum systems under study. It begins with mathematical tools and then turns to the pictures of quantum mechanics and the axioms of quantum mechanics. We learn about entanglement while introducing tensor products, and we deepen our understanding of angular momentum by studying the addition of angular momentum. Identical particles complete part II.
Chapters 13 and 14 give the reader the requisite foundations in complex vector spaces and linear operators. At every step, physical examples illustrate the mathematical concepts. These examples are from systems already encountered in part I, thus serving both as a review of concepts as well as a way to deepen the understanding of these systems. These chapters also present important computational tools: index manipulation, Pauli matrix identities, matrix representations, matrix exponentials, commutator identities, including Hadamard’s lemma, and simple cases of the Campbell-Baker-Hausdorff formula. We discuss orthogonal projectors and the construction of rotation operators for spin states. Finally, after touching upon it in part I, I present the bra-ket notation of Dirac, showing how it relates to inner products and to dual spaces.
In chapter 15 we review the concept of uncertainty, giving it a geometric interpretation and then proving the uncertainty inequality. We prove the spectral theorem for finite dimensional vector spaces, leading to the concept of a complete orthonormal set of projectors. I demonstrate how to build a complete set of commuting observables. Chapter 16 discusses unitary time evolution, showing how it implies the Schrödinger equation. The chapter also presents the Heisenberg picture of quantum mechanics. Finally, having all the mathematical and physics background ready, we discuss the axioms of quantum mechanics. Chapter 17 studies the dynamics of quantum systems. We explore two different, important systems: coherent states of the harmonic oscillator and nuclear magnetic resonance. We also develop the factorization, or “supersymmetric” method, useful for finding algebraically the spectrum of one-dimensional potentials. Multiparticle states and tensor products of vector spaces appear in chapter 18. We consider entangled states, the questions raised by Albert Einstein, Boris Podolsky, and Nathan Rosen, and the remarkable resolution provided by the work of John Bell. Teleportation and no-cloning are also presented. Chapter 19 offers our second look at angular momentum and central potentials. This time we use the algebra of angular momentum to work out the finite-dimensional representations. We study spherical free-particle solutions and derive Rayleigh’s formula, useful for scattering. In chapter 20 we focus on addition of angular momentum, giving a preview of perturbation theory that allows us to study this subject in the context of the hydrogen atom. We present a derivation of the spectrum of the hydrogen atom using the simultaneous conservation of angular momentum and of the Runge-Lenz vector. The last chapter of part II, chapter 21, deals with identical particles. We emphasize the action of the permutation group on tensor products, giving a clear motivation for the symmetrization postulate and the existence of bosons and fermions. We briefly explain why exotic statistics are allowed when space is two-dimensional.
III: Applications The material in part III helps the student master the key theoretical concepts of part II while developing a wider perspective and a host of tools valuable for research and advanced work. Apart from the important subjects of density matrices and particles on electromagnetic fields, most of the work deals with approximation methods.
Chapter 22 introduces mixed states and density matrices. I discuss bipartite systems and give a brief introduction to open systems, decoherence, and the Lindblad equation. We also discuss measurements in quantum mechanics, touching on the possible relevance of decoherence. We turn to quantum computation in chapter 23, showing how quantum superpositions and interference allow for a surprising speedup of certain computations. In chapter 24 we study the coupling of charged particles to electromagnetic fields, whose gauge transformations are supplemented by transformations of the wave function. We derive the Landau levels of a charged particle inside a uniform constant magnetic field.
We then begin a sequence of chapters on approximation methods: for small, time-independent perturbations; for potentials that vary slowly with position; for small, time-dependent perturbations; and, finally, for perturbations that vary slowly in time. Chapter 25 deals with time-independent perturbation theory. We pay particular attention to the perturbation of degenerate states and to the possibility that degeneracies are not lifted to first order in the perturbation. The fine structure of the hydrogen atom and the Zeeman effect are worked out to illustrate the theory. In chapter 26 the students learn about the WKB approximation, useful for slowly varying potentials. We introduce the so-called connection formulae, show how to use them, and derive them using complex-analytic methods. We apply the WKB method to tunneling and to the delicate calculation of level splitting in double-well potentials.
Chapter 27 is devoted to time-dependent perturbation theory. We obtain a number of results for first-order transitions and give a derivation of Fermi’s golden rule, both for time-independent and time-dependent transitions. As a classic application of this theory, we discuss the interaction of light and atoms. We also model and solve for the time dependence of a system where a discrete state is coupled to a continuum. The subject of chapter 28 is the adiabatic approximation, where we consider perturbations that vary slowly in time. We state and prove the quantum adiabatic theorem and go on to discuss Landau-Zener transitions, Berry’s phase, and the Born-Oppenheimer approximation.
The last two chapters of part III, chapters 29 and 30, deal with scattering. The first of these examines scattering in one dimension, or, more precisely, scattering on the half line. This is a good setup to learn many of the ideas relevant to three-dimensional scattering. We discuss Levinson’s theorem as well as resonances. Turning to three-dimensional scattering in the last chapter, we discuss cross sections, phase shifts, and partial waves. We conclude by setting up integral methods that lead to the Born approximation.
A note on units Unless explicitly noted, I use Gaussian-cgs units. This is particularly convenient when dealing with the hydrogen atom, magnetic moments, and particles in electromagnetic fields. With these units, the Coulomb potential energy of hydrogen, for example, is V = −e2/r. Numerical estimates are easily done by using the following values for the fine-structure constant α and for the product ℏc:

When dealing with magnetic fields and magnetic moments, I often also give values in SI units (from the French Système International). Thus, magnetic fields are often expressed in gauss as well as in tesla. Recall that tesla = (104/c) gauss, with the factor of c needed because magnetic fields have different units in the Gaussian and SI systems. This issue first comes up in section 12.2.
How to Use This Textbook
This book emerged from a three-semester structure but is easily adaptable to other settings. Instructors can pick and choose, as there is lots of material for extra learning. Here are some ideas on how to deal with one-semester or two-semester courses and how to use the book to supplement a graduate-level course.
A one-semester course Such a course would be naturally built from part I, which contains most of what could be considered basic material in quantum mechanics. Some instructors wishing to supplement this content could do it as follows. Discuss the basics of time-independent perturbation theory, as summarized in section 20.3, after the study of the Hellmann-Feynman lemma in section 7.9. Give the algebraic derivation of angular momentum multiplets in section 19.3, after studying chapter 10. Supplement the discussion of bra-ket notation in chapter 12 by including the material in section 14.10.
A two-semester course Parts I and II are the natural foundation for a one-year undergraduate course. By skipping some of the part II material (such as the factorization method, Rayleigh’s formula, Runge-Lenz vector, and hidden symmetry in the hydrogen atom, among others) the instructor can make some room to include part III material. Prime candidates for inclusion are chapter 22 on density matrices and decoherence, and chapter 24 on charged particles in electromagnetic fields.
Supplementing a graduate-level course Some of the part II and much of the part III material could be used to supplement a graduate course suitable for students who have had just two semesters of undergraduate quantum mechanics. Such a course, using part II material, could start with the axioms of quantum mechanics, as discussed in section 16.6, including background content from the spectral theorem. This would be followed by chapter 18 on multiparticle states and by chapter 21 on identical particles. From part III, many choices are possible, as the chapters are largely independent. Density matrices, charged particles on electromagnetic fields, semiclassical approximation, and adiabatic approximation would be natural choices for students who have already learned time-independent and time-dependent perturbation theory.