Abstract This chapter gives a summary, as non-technical as possible, of the rest of the book. It thus gives a brief synopsis of the development of matrix mechanics, wave mechanics, and two formalisms, statistical transformation theory and the Hilbert space formalism, tying the two together. It describes how the book covers the transition from the old to two distinct forms of the new quantum theory, including Bohr’s (unsuccessful) attempts in the early 1920s to account for the periodicity of the table of elements, and the introduction of two ingredients the first of which was indispensable for this task: Pauli’s (1925b) exclusion principle and Uhlenbeck and Goudsmit’s (1925) electron spin. It revisits the successes and failures of the old quantum theory and shows how the new theory took care of the failures and put the successes on a more solid basis.
Abstract This chapter gives a summary, as non-technical as possible, of the rest of the book. It thus gives a brief synopsis of the development of matrix mechanics, wave mechanics, and two formalisms, statistical transformation theory and the Hilbert space formalism, tying the two together. It describes how the book covers the transition from the old to two distinct forms of the new quantum theory, including Bohr’s (unsuccessful) attempts in the early 1920s to account for the periodicity of the table of elements, and the introduction of two ingredients the first of which was indispensable for this task: Pauli’s (1925b) exclusion principle and Uhlenbeck and Goudsmit’s (1925) electron spin. It revisits the successes and failures of the old quantum theory and shows how the new theory took care of the failures and put the successes on a more solid basis.
Abstract We provide an overview, as non‐technical as possible, of the contents of Vol. 1 of the book. Reflecting the structure of the volume, this overview consists of two parts. In the first part, we summarize the most important early contributions to quantum theory (covered in detail in Chs. 2–4). This part starts with Planck’s work on black‐body radiation culminating in the introduction of Planck’s constant in 1900. It then moves on to Einstein’s 1905 light‐quantum hypothesis, his theory of specific heats, and his formulas for energy and momentum fluctuations in black‐body radiation. After summarizing Bohr’s path to his quantum model of the atom, it concludes with Einstein’s 1916–17 radiation theory combining elements of Bohr’s model with his own light‐quantum hypothesis. In the second part we summarize our analysis of the old quantum theory (given in detail in Chs. 5–7). After a brief overview of the career of Sommerfeld, who together with Bohr took the lead in developing the old quantum theory, we review the three principles we have identified as the cornerstones of the theory (the quantization conditions, the adiabatic principle, and the correspondence principle). We then discuss three of the theory’s most notable successes (fine structure, Stark effect, X‐ray spectra) and, finally, three of its most notorious failures (multiplets, Zeeman effect, helium).
Abstract This is the second of two volumes on the genesis of quantum mechanics in the first quarter of the twentieth century. It covers the period 1923–1927. It traces how the old quantum theory of Bohr and Sommerfeld gave way to the matrix mechanics of Heisenberg and others, and to the wave mechanics of Schrödinger. It then turns to the emergence of statistical transformation theory and the Hilbert space formalism unifying these two forms of the new quantum theory. It uses the metaphor of an arch built on a scaffold to capture these developments. The volume provides detailed analyses of the classic papers of Heisenberg, Born, Jordan, Pauli, Dirac, De Broglie, Einstein, Schrödinger, von Neumann, and others, which helped to lay the foundations of modern quantum mechanics. Drawing on correspondence and later reminiscences of these authors and synthesizing much of the extensive secondary literature on the breakthrough to modern quantum mechanics in the mid-1920s, this volume presents an authoritative new account of the genesis of quantum mechanics.
Abstract This chapter covers the emergence of the statistical interpretation of the new quantum mechanics. After covering the earliest suggestions in this direction by Born and Pauli, it covers the statistical transformation theory of Dirac and Jordan, focusing on Jordan’s version. This version was an important part of the inspiration of Heisenberg’s uncertainty paper, which is covered next. The chapter concludes with the Como and Solvay conferences of late 1927 where Bohr revealed his complementarity principle and where the Copenhagen-Göttingen matrix-mechanics camp of Bohr, Born, Heisenberg, and Pauli squared off with Schrödinger, Einstein, De Broglie, and others in the wave-mechanics camp over the interpretation of the new quantum formalism.
Our modern understanding of atoms, molecules, solids, atomic nuclei, and elementary particles is largely based on quantum mechanics. Quantum mechanics grew in the mid-1920s out of two independent developments: the matrix mechanics of Werner and the wave mechanics of Erwin Schrödinger. For the most part this chapter follows the path of wave mechanics, which is more convenient for all but the simplest calculations. The general principles of the wave mechanical formulation of quantum mechanics are laid out and provide a basis for the discussion of spin, identical particles. and scattering processes. The general principles are supplemented with the canonical formalism to work out the Schrödinger equation for charged particles in a general electromagnetic field. The chapter ends with the unification of the approaches of wave and matrix mechanics by Paul Dirac, and a modern approach, known as Hilbert space, is briefly described.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
Abstract This chapter covers the completion of the formalism of quantum mechanics and its statistical interpretation in the first two installments of a remarkable trilogy of work by von Neumann in 1927. In the first, von Neumann introduces the modern Hilbert space formalism and demonstrates the equivalence of matrix and wave mechanics by showing that they are just two different instantiations of Hilbert space, one being the space of square-summable sequences, the other being the space of square integrable functions. In the second, drawing on work by von Mises on the foundations of probability theory, he provides a careful formulation of the statistical interpretation of this formalism. In doing so, he introduces such staples of modern quantum mechanics as density operators characterizing quantum statistical ensembles and the distinction between pure and mixed states.