The Principles of Quantum Mechanics

This article is about the book by Paul Dirac. For the book by Ramamurti Shankar, see Principles of Quantum Mechanics.

The Principles of Quantum Mechanics is an influential monograph written by Paul Dirac and first published by Oxford University Press in 1930.^[1] In this book, Dirac presents quantum mechanics in a formal, logically consistent, and axiomatic fashion,^[2] making the book the first of its kind.^[3]: 366 Its 82 sections contain 785 equations with no diagrams.^[2] Nor does it have an index, a bibliography, or a list of suggestions for further reading.^[4]: 178 The first half of the book lays down the foundations of quantum mechanics while the second half focuses on its applications.^[5] Dirac did not pursue a historical approach to the subject. Nor did he discuss at length the philosophy of quantum mechanics.^[4]: 178

The Principles of Quantum Mechanics

Cover of the 4th edition (1958)
Author	Paul Dirac
Language	English
Subject	Quantum mechanics
Genres	Non-fiction
Publisher	Oxford University Press
Publication date	1930
Publication place	United Kingdom
Media type	Print
Pages	357 (1st ed.) 314 (4th ed.)

History

Paul Dirac played a key role in the development of quantum mechanics at the University of Cambridge and the University of Göttingen during the mid- to late-1920s.^[2] In September 1925, Dirac, then a post-graduate student, received the proofs of a paper forwarded to him by his supervisor, Ralph Fowler, from its original author, Werner Heisenberg. Dirac soon recognized the fundamental importance of the non-commutativity of the quantities he encountered. He realized that there was a connection between Heisenberg's matrices and Poisson brackets from classical mechanics, which he could exploit to create his own version of quantum mechanics.^{[4]: 83–9}

Dirac was one of a handful of physicists at Cambridge who were at the cutting edge of the new quantum theory, and, unusually for an early-career researcher, he agreed to write a new textbook on the subject to be published as part of the International Series of Monographs on Physics, edited by Fowler and Peter Kapitza.^[4]: 145 The first edition of The Principles of Quantum Mechanics appeared in 1930.^[6] The Principles was similar in style to the lectures Dirac had been delivering at Cambridge while he was a doctoral candidate during the late 1920s, but not in the contents, which were restricted to what was known at the time. After the completion of the book, Dirac used it for his courses on quantum mechanics over the next four decades, including the regular one he taught at Cambridge during the 1930s in the Lent terms (Spring). (Alan Wilson taught in the Michaelmas terms (Fall), where he focused on the applications of Schrödinger's equation.)^[7]

Dirac synthesized and improved upon the wave and matrix formulations of quantum mechanics, creating his own version as the theory of linear transformations. In addition, he took inspiration from paper published by Cornelius Lanczos presenting quantum mechanics in terms of the theory of linear integral equations. In the preface, Dirac commented, "A great deal of my work was just playing with equations and see what they give."^{[3]: 300–1} Dirac also popularized the distribution or generalized function now named after him, the Dirac ${\displaystyle \delta }$-function, though it had previously been used in the nineteenth century by scholars such as Gustav Kirchhoff in a paper on the Huygens principle in optics and by Oliver Heaviside in his work on electromagnetism.^[3]: 301

Unlike many other works at the time, The Principles was atypical in many different ways. It did not contain my references to the research literature. Chapter One is a purely qualitative exposition on the principle of superposition, illustrated by the polarization of light. Planck's constant ${\displaystyle h}$ was only mentioned later in the book, in a discussion on the general commutation relations, where Dirac introduced the reduced Planck constant ${\displaystyle \hslash }$ ("h bar" or "Dirac's h") as a shorthand for ${\displaystyle h/2\pi }$. Likewise, the Schrödinger equation was presented late in the book. For Dirac, quantum mechanics was no more than a mathematical framework from which one could calculate quantities that could be compared with experiment. Questions of deeper meaning were, according to him, meaningless. Although Dirac never discussed the philosophy of quantum mechanics, his views were, in effect, close to the Copenhagen interpretation.^[7] (Also see positivism.)

In the second edition (1935), Dirac aimed to make the book more accessible than the first,^{[4]: 254–5} and rewrote most of it.^[3]: 367 Unlike the first edition, where he defined the "state" of a quantum-mechanical system as its condition in four-dimensional spacetime, in this edition, he only described the "state" of a system only for a given moment in time. Being aware of the difficulties of making quantum mechanics fully consistent with special relativity in order to create a coherent quantum theory of electrodynamics, he resorted to using three-dimensional space in the sense of classical physics.^[7] He included a new discussion on the action principle, though with a footnote saying that students uninterested in analytical mechanics could skip it, and acknowledged the discovery of the positron, predicted by the negative-energy solution to his relativistic wave equation for the electron.^[5] In the first edition, Dirac, as he did in the 1928 paper, stated—erroneously—the negative-energy solution was the proton.^[7]

In 1947 the third edition of the book was published, in which the chapter on quantum electrodynamics was rewritten particularly with the inclusion of electron-positron creation.^[6] Beginning with this edition, the mathematical descriptions of quantum states and operators were changed to use the bra–ket notation,^[5] introduced in 1939 by Dirac himself.^[8]

In the fourth edition, 1958, the same chapter was revised, adding new sections on interpretation and applications.^[6] It was later reprinted multiple times.^[7] A revised fourth edition appeared in 1967.^[6] For the last editions, the greatest changes were in the final chapter of the book, a reflection of the rapid pace of development of quantum electrodynamics, both theoretical and experimental.^[5] Although Dirac knew about group theory, he did not think of it as a useful tool for quantum mechanics, and consequently did not include it in his book.^[7]

Translations

It has been translated to multiple languages,^[5] including Russian,^[5] Japanese,^[5] German, and French.^[3]: 300 Dirac supplied an additional preface for the Russian edition, thanks to this good relationship with Russian physicists Igor Tamm and Peter Kapitza.^[5] At the request of Dmitri Ivanenko, Dirac added a chapter on methods of approximations, including the Hartree–Fock method (developed by Douglas Hartree and Vladimir Fock) for the Russian edition. In the Russian edition, Ivanenko added multiple appendices and Matvei Bronstein some footnotes.^[7] He also supplied a preface to the Japanese edition as requested by the translators, two of whom were Yoshio Nishina and Sin-Itiro Tomonaga.^[5]

Contents

• The principle of superposition
• Dynamical variables and observables
• Representations
• The quantum conditions
• The equations of motion
• Elementary applications
• Perturbation theory
• Collision problems
• Systems containing several similar particles
• Theory of radiation
• Relativistic theory of the electron
• Quantum electrodynamics

Reception and legacy

According to Laurie M. Brown, it "set the stage, the tone, and much of the language of the quantum-mechanical revolution."^[5]

The book quickly rose to prominence, surpassing other texts published in the 1930s. German and Russian translations were even more popular. In what was then the Soviet Union, it sold 3,000 copies in just a few months. However, the publishing house, GTTI, felt the need to assert that despite appearances to the contrary, Dirac's personal view of quantum mechanics, which was close to what would later be named the Copenhagen interpretation, was not at odds with the ideology of dialectical materialism.^[7] Felix Bloch praised the book for its originality and self-contained presentation, but worried that Dirac did not refer much to the original research literature.^[7] Matvei Bronstein opined that Dirac's textbook was superior to Gruppentheorie und Quantenmechanik (1928) by Hermann Weyl, which, according to him, was overly formal.^[7] Wolfgang Pauli praised it as "an indispensable standard work" but was concerned that it was too distant from experiments.^[4]: 179 He also had a high opinion of the German translation, describing it and the original book as "highly reliable" and "essential" for physicists studying quantum theory.^[5] J. Robert Oppenheimer called the book an "astonishingly complete" and "coherent" treatise, comparing it to Elementary Principles in Statistical Mechanics (1902) by Josiah Willard Gibbs, but warned that like Gibbs' book, it was not suitable as an introduction to quantum mechanics.^[7] Similarly, John Lennard-Jones recommended the book for everyone who wished to stay abreast with modern physics, but cautioned that it was not easy to read.^[7] Charles Galton Darwin complained about the lack of concrete examples to illustrate the theory.^[7] Paul Epstein and Werner Heisenberg complimented the second edition for being more comprehensible than the first.^[5] Albert Einstein admired the book as "the most logically perfect presentation" of the subject and made it his companion.^[4]: 179 Freeman Dyson compared it to a fine work of art.^[4]: 428 Indeed, while textbooks are seldom cited in research papers, Dirac's Principles proved to be an exception during the 1930s.^[7]

However, while physicists had a high opinion of the book, mathematicians were more critical. Bernard Koopman praised Dirac for his profound physical insight but disliked what he perceived as its lack of logical clarity. Koopman also took issue with Dirac's use of the hybrid terms eigenvalue and eigenfunction—from the German Eigenwert and Eigenfunktion—instead of characteristic value and characteristic function.^[7] Garrett Birkhoff and John von Neumann complained about the way Dirac employed mathematics, including the delta distribution.^[7] Max Jammer observed that Dirac's primary motivation in writing the book was creating an exposition in physics, treating mathematics as a tool. In this regard, von Neumann's Mathematical Foundations of Quantum Mechanics (1932), with its uncompromising emphasis on mathematical rigour, was a supplement to Dirac's book.^[3]: 367 On the other hand, Laurent Schwartz, who put the notion of distributions or generalized functions on a secure mathematical footing in 1945, credited Dirac with thinking of the Dirac delta function as the kernel of an integral transform.^[5]

Among students, opinions were mixed. Some were displeased that the book was basically a transcript of Dirac's lectures at Cambridge.^[4]: 179 Most students looked elsewhere because they wanted a text that could teach them how to carry out computations. But the most gifted of them viewed the book as a "bible" of modern physics.^[4]: 255 One such student was Subrahmanyan Chandrasekhar, who later became an astrophysicist.^[4]: 179 Abdus Salam and Eugene Wigner compared Dirac's The Principles of Quantum Mechanics with the Principia, written by Dirac's predecessor as the Lucasian Professor of Mathematics at Cambridge, Isaac Newton.^[5] In his doctoral dissertation, Richard Feynman cited Dirac's discussion of the action principle in the second edition.^[5] Still in print in the early twenty-first century, it has influenced many generations of young physicists.^[4]: 428 John Gribbin recommended that students at least borrow a library copy to read the first chapter.^[9]: 361

See also

• Physics portal
• Books portal

• The Evolution of Physics (Einstein and Infeld)
• The Feynman Lectures on Physics Vol. III (Feynman)
• The Physical Principles of the Quantum Theory (Heisenberg)
• Mathematical Foundations of Quantum Mechanics (von Neumann)
• Modern Quantum Mechanics (Sakurai)