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).
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 4 shows that, just as Heisenberg’s matrix mechanics represented a break with classical physics and the beginning of a new physics, Schrödinger’s wave mechanics represented another, no less radical, break with classical physics and at the same time opened up another path to the quantum world. Schrödinger’s theory is shown to make substantial use of de Broglie’s waves and related phenomena. The Schrödinger equation, being wave-like, constitutes mathematically an eigenvalue problem for bounded systems, and thus naturally leads to quantization, as is illustrated with a couple of simple examples. The chapter ends with Dirac’s abstract formulation, which has the advantage of being suitable for any description, be it Heisenberg’s, Schrödinger’s or any other.
Abstract Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.
Underpinning the axiomatic formulation of quantum theory presented in this undergraduate textbook is a review of early experiments, a comparison of classical and quantal terminology, a Schroedinger-equation treatment of the one-dimensional quantum box, and a survey of relevant mathematics. Among the many concepts comprehensively discussed are: operators; state vectors and wave functions; experimental observables; classical/quantal connections; and symmetry properties. The theory is applied to a wide variety of systems including the non-relativistic H-atom, external electromagnetic fields, and spin1/2. Collisions are described using wave packets. Various time-dependent and time-independent approximations are discussed; applications include electromagnetic transition rates and corrections to the H-atom energies. The final chapter deals with identical-particle symmetries and their application to the He atom, the Periodic Table and diatomic molecules. There are also brief treatments of advanced subjects such as gauge invariance and hidden variables.
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.
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.
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confined to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic field. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.