The quantities in physics that can be measured are conventionally denoted observables: entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems.

Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield. Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield.

The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.

Quantum mechanics is the theoretical framework that describes physical phenomena on the microscopic scale. In particular, it is used in those situations in which Planck's constant h, approximately equal to 6.62 × 10-34 Js, cannot be assumed to be negligible, despite its apparently tiny magnitude. Macroscopic phenomena, on the other hand, involve magnitudes that are so large that h can be set equal to zero without appreciable error.

As an illustration of this in the macroscopic context, consider the following example: a ladybug (Hippodamia convergens) of mass 1 g moves radially at a speed of 0.01 ms-1 on a turntable rotating at 10 rpm. In the rotating frame of reference, the ladybug has a kinetic energy of about an erg, and, in the laboratory frame, its angular momentum is roughly 10-5 Js when it is at a distance of 0.1 m from the center of the rotating turntable. The latter angular momentum is of the order of 1028h. Thus, relative to this value, h is well approximated by zero and, indeed, only classical mechanics is needed to describe the ladybug's motion.

In contrast to the preceding example, an electron in the “2p” level of hydrogen has an energy of about 10-25 J and an angular momentum whose magnitude is approximately h/(2π) = ħ (pronounced “h-bar”). Here, h cannot be ignored and only a quantum description can accurately account for the properties of this system.

Quantum mechanics was initially formulated as a type of matrix theory by Heisenberg and collaborators. Soon after, it was alternately expressed by Schrödinger as a wave theory through what is now referred to as the Schrödinger wave equation. Schrödinger and Eckart showed that these two approaches were equivalent manifestations of the same underlying formalism, viz., quantum theory.

Quantum theory is formulated in this book (Chapter 5) via six postulates. In order that this postulational approach not be too abstract, it is helpful first to analyze a system that is easily treated by quantum theory. We do so in this chapter, in which the so-called quantal box is studied. The analysis is carried out using the Schrödinger equation, which is introduced in this chapter but without the formal statements of the postulates.

Because the quantal-box problem is similar mathematically to the problem of describing the clamped, vibrating string of classical physics, the string is also analyzed in this chapter. This provides not only a comparison system for the box, but also a brief review of classical waves and wave theory. We also include a section on Hermitian differential operators, Sturm–Liouville systems and the Dirac delta function. All of this material serves as an introduction to Chapters 4 and 5.

We begin with the string. Because many readers will already have seen aspects of the following analysis, we put all of Section 3.1 in small print. This section may be omitted by the knowledgeable reader; the crucial result is Eq. (3.3).

Chapter 1 was concerned with the need for a quantal description of microscopic phenomena. The thrust of Chapter 3 is an analysis and comparison of a classical and a quantum system; some mathematical generalizations of the analysis are presented as well. Chapter 4 provides an introduction to Hilbert-space concepts that are the mathematical framework of quantum theory, while, in Chapter 5, the axioms of quantum mechanics for pure states are stated and illustrated. The present chapter is intended to be a bridge between those chapters and Chapter 1. Here we review many of the concepts of classical mechanics that have relevance in quantum mechanics. Equally important, if not more so, we stress the lack of equivalence between a number of classical concepts and their quantal counterparts.

This inequivalence is emphasized because of the vocabulary common to classical and quantum mechanics. Classical physics is, of course, the source for much of the vocabulary in quantum physics. However, the ways in which the words are used in the two domains often differ substantially, while the concepts are almost necessarily not identical. Some classical concepts have no direct quantal counterparts. For example, velocity and force are not primary quantal entities. This fact means that the quantal definitions of linear and angular momentum cannot be the same as those in classical mechanics. Similarly, the concept of a “state” in classical physics – the collection of the generalized positions and generalized velocities or momenta – does not carry over to quantum physics.

In this chapter the basic concepts of quantum theory are formulated via a self-consistent set of six postulates. These six postulates deal with the following: operator images of physical observables; properties of state vectors and wave functions; the connection with experimental measurements; coordinate-space forms of those operators having classical analogs; the primary dynamical equation of quantum mechanics; and the completeness of the eigenvectors of observables. The postulates are the framework on which the edifice of quantum mechanics is constructed. A separate section is devoted to the statement of each postulate and to a few examples chosen to illustrate or apply it. A comprehensive set of applications is presented in Chapters 6 and 7, where various 1-D systems are used as the vehicles for illustrating the postulates. The basic theory of this chapter is extended and developed in Chapters 8 and 9. Since probability is a key interpretational concept, a brief introduction to this topic is given in an appendix.

We emphasize that our formulation of quantum theory (in this chapter and throughout this book) is restricted to the case of pure states. This is not a major restriction. It means that, when we consider ensembles of individual quantum systems (e.g., single particles, atoms, nuclei, etc.), all of the systems in the ensemble are assumed to be in the same quantum state, not necessarily an eigenstate of a quantal operator. Ensembles for which this assumption does not hold are described by a density operator or density matrix rather than by a pure state; treatments of such cases can be found in references cited later.

Quantum mechanics, the theoretical framework that describes microscopic physical phenomena, is a marvelous intellectual enterprise. Elegant, intriguing, challenging, often counter-intuitive and sprinkled with delightful surprises, it is wholly unlike classical physics. Mastery of quantum theory is an essential ingredient in the education of physicists and chemists. This text, a broad and deep introduction to the theory, is intended not only to help provide that mastery but also to convey to the reader its inherent richness.

The book is designed for use in year-long undergraduate courses, for self-study, and as a supplemental text in graduate courses. Its aims are achieved through a combination of introductory material, including a survey of relevant mathematics; an axiomatic formulation of the theory via six postulates; applications to soluble one-, two-, and three-dimensional systems; theoretical augmentations such as symmetry properties, electro-magnetic interactions, spin ½ and generalized angular momentum; and discussion of approximation methods and their application to electromagnetic transitions and systems of identical fermions (atoms and molecules). The reader is assumed to be familiar with material covered in courses on intermediate mechanics, electromagnetism and elementary modern physics, plus mathematics through vector calculus, elements of partial differential equations (their form and separation-of-variables solutions), and some linear algebra (finite dimensions, matrices).

The writing of this book has been strongly influenced by the author's experiences teaching both the graduate and the undergraduate quantum-theory courses at Brown University. Most of the students who take the undergraduate course are junior-level physics majors, many of whom go on to graduate school in physics.