On the theory of quantum mechanics. Offprint from Proceedings of the Royal Society A, vol. 112, 1926

[London: Harrison & Sons for the Royal Society, 1926. First edition. FERMI-DIRAC STATISTICS AND THE FIRST STEPS TOWARDS QUANTUM ELECTRODYNAMICS.

First edition, extremely rare offprint, of Dirac's paper, which "is justly seen as a major contribution to quantum theory" (Kragh, p. 36). It introduced his quantum mechanical derivation of what is now called Fermi-Dirac statistics, which describes a distribution of particles (now known as fermions, a name coined by Dirac in 1945) in certain systems containing many identical particles that obey the Pauli exclusion principle-meaning that no two of the particles can occupy the same quantum state simultaneously. It also contains Dirac's first steps towards quantum electrodynamics. The paper "will be remembered as the first in which quantum mechanics is brought to bear on statistical mechanics. Recall that the earliest work on quantum statistics, by Bose and by Einstein, predates quantum mechanics. Also, Fermi's introduction of the exclusion principle in statistical problems, though published after the arrival of quantum mechanics, is still executed in the context of the 'old' quantum theory. All these contributions were given their quantum mechanical underpinnings by Dirac, who was, in fact, the first to give the correct justification of Planck's law, which started it all: 'Symmetrical eigenfunctions ... give just the Einstein-Bose statistical mechanics . . . (which) leads to Planck's law of black-body radiation'" (Pais, p. 6). Dirac's paper is credited "for having laid the foundations of the integration of quantum mechanics and quantum statistics because they introduced the quantum-mechanical expression of the symmetry of a system under exchanges of equal particles. The quantum formalism of exchange symmetry is regarded as having solved at once long-standing difficulties regarding the statistical properties of both equal particles and light quanta by clarifying and legitimizing the previously foggy notion of indistinguishable particles" (Monaldi, p. 125). The second part of the present paper contained the seed of Dirac's invention of quantum electrodynamics, which was brought to fruition a few months later in 'The Quantum Theory of the Emission and Absorption of Radiation'. In the present paper, "Dirac considered a system of atoms subjected to an external perturbation that could vary arbitrarily with the time ... [Dirac obtained results] 'in agreement with the ordinary Einstein theory,' that is, with the quantum mechanical derivation of the B coefficients that occurred in Einstein's theory of 1917 [that gave the probability of absorption and stimulated emission of radiation]. Since he made use of a classical description of the electromagnetic field, Dirac was not at the time able to proceed further, and he noted, 'One cannot take spontaneous emission [i.e. the A coefficients] into account without a more elaborate theory.' This more elaborate theory was ready less than half a year later" (Kragh, pp. 120-121). OCLC lists University of Florida only (where Dirac spent his last years). No copy in auction records.

Provenance: Bertha Swirles (1903-99) (signature on front wrapper, extensive annotations to lower margins of last two pages of text, including several equations). As an undergraduate at Cambridge Swirles attended lectures by J. J. Thomson and Rutherford. She remained at Cambridge in 1925 to undertake research in mathematical astronomy under the supervision of Ralph Fowler; another of Fowler's research students, a couple of years ahead of Swirles, was Paul Dirac. After periods at Bristol, Imperial College, London, and Manchester, Swirles took up a lectureship in mathematics at Girton College, Cambridge in 1938, where she remained for the rest of her career.

The present paper was "Dirac's first published response to Schrödinger's theory [i.e., wave mechanics]. He had corresponded with Heisenberg while completing his PhD thesis in Cambridge in the spring of 1926. Many years later, he wrote in his recollections that he did the [present] work on many-particle systems after Heisenberg convinced him of the usefulness of wave mechanics. Dirac felt 'at first a bit hostile' to this theory because it seemed to him that it represented a regress to 'the pre-Heisenberg stage.' In a non-extant letter to Heisenberg, he criticized Schrödinger because 'the wave theory of matter must be inconsistent just like the wave theory of light'. Heisenberg agreed with this criticism but nonetheless saw Schrödinger's theory as progress. Thanks to Heisenberg's detailed explanation of the relation between the two formal schemes, Dirac could see that wave mechanics 'would not require us to unlearn anything that we had learned from matrix mechanics' but rather 'supplemented the matrix mechanics and provided very powerful mathematical developments which fitted perfectly with the ideas of matrix mechanics'.

"In Dirac's retrospective account, it was the study of Schrödinger's formalism that suggested to him the possibility of symmetric and antisymmetric wave functions for a system of similar particles. These 'symmetry questions,' in turn, 'brought in the possibility of new laws of Nature' ...

"Instead of confronting Schrödinger's undulatory interpretation, Dirac set out to reformulate Schrödinger's formal apparatus in general terms according to his own mathematical approach. He deduced the expression of the general solution of a quantum-mechanical problem as a linear expansion with arbitrary constants in 'a set of independent solutions,' which he called eigenfunctions (p. 664). This formal milestone enabled him to develop a quantum-mechanical treatment of multiparticle systems and to reach three lasting results. He arrived at the symmetry and antisymmetry of the wave functions, formulated the statistics that we now know as Fermi-Dirac statistics, and derived a calculation of Einstein's coefficients of absorption and stimulated emission ...

"Dirac adopted 'an atom with two electrons' as the simplest multiparticle system. In his atom, however, all interactions between electrons could be neglected. He did not resort to the analogy with the classical phenomenon of resonance as a theoretical tool, but used only the symmetry of the two-electron system supplemented by the methodological principle for which he credited Heisenberg:

'[Heisenberg's matrix mechanics] enables one to calculate just those quantities that are of physical importance, and gives no information about quantities such as orbital frequencies that one can never hope to measure experimentally. We should expect this very satisfactory characteristic to persist in all future developments of the theory' (p. 667).

"Dirac indicated with (𝑚𝑛) 'the state of the atom in which one electron is in an orbit labelled 𝑚 and the other in the orbit 𝑛.' He then asked: Were the 'physically indistinguishable' states (𝑚𝑛) and (𝑛𝑚) to be counted as distinct or as identical? This question was inconsequential in classical mechanics, but in the matrix formalism, it implied a choice between two different matrix representations. In one, the matrix elements corresponding to the transitions (𝑚𝑛) → (m'n') and (𝑚𝑛) → (n'm') would be represented by two separate matrix elements, in the other they would be represented by the same element ... [Dirac] asserted that the two transitions, (𝑚𝑛) → (m'n') and (𝑚𝑛) → (n'm'), were 'physically indistinguishable' and that 'only the sum of the intensities for the two together could be determined experimentally' (p. 667). From this proposition he drew the answer:

'Hence, in order to keep the essential characteristic of the theory that it shall enable one to calculate only observable quantities, one must adopt the second alternative that (𝑚𝑛) and (𝑛𝑚) count as only one state' (p. 667).

"Having so fixed the matrix formalism, Dirac applied his formula for the general solution of the two-particle model. He formed the eigenfunctions of the whole system as linear combinations of products of the eigenfunctions of the single electrons; then, he imposed the condition that they correspond to the matrices. This condition could be satisfied only by combinations that were symmetrical or antisymmetrical under exchange of the electrons. Either one of these two possibilities gave 'a complete solution of the problem' and quantum mechanics did not dictate which was the correct one (p. 669). The choice, Dirac stated, was to be made by appealing to Pauli's exclusion principle:

'An antisymmetrical eigenfunction vanishes identically when two of the electrons are in the same orbit. This means that in the solution of the problem with antisymmetrical eigenfunctions there can be no stationary states with two or more electrons in the same orbit, which is just Pauli's exclusion principle' (pp. 669-670).

"The symmetrical solution, however, could not be correct for 'the problem of electrons in an atom' because it allowed any number of electrons in the same orbit (p. 670). These results could be straightforwardly extended to any system composed of similar particles, in particular, to an assembly of molecules. Dirac thus applied them to the ideal gas. He obtained the eigenfunction of the assembly by multiplying the single-molecule eigenfunctions and choosing either the symmetrical or the antisymmetrical linear combinations. At this point, he turned to statistical considerations. He implicitly made the assumption that the new states, represented by symmetrical and antisymmetrical wavefunctions, represented the energy distributions, or macrostates, of statistics. Then, he explicitly adopted as a 'new assumption' the simplest extension of Bohr's rule, namely, that 'all the stationary states of the assembly (each represented by one eigenfunction) have the same a priori probability' (p. 671). In the case of symmetrical eigenfunctions, this rule corresponded to the Bose-Einstein statistics. In the case of the antisymmetrical eigenfunctions, whereby the number of molecules associated with each single-particle eigenfunction could only be 0 or 1, it led to the new statistics that is now known as the Fermi-Dirac statistics. Dirac concluded:

'The solution with symmetrical eigenfunctions must be the correct one when applied to light quanta, since it is known that the Einstein- Bose statistical mechanics leads to Planck's law of black-body radiation. The solution with antisymmetrical eigenfunctions, though, is probably the correct one for gas molecules, since it is known to be the correct one for electrons in an atom, and one would expect molecules to resemble electrons more closely than light quanta' (p. 672)

"Despite having just derived the two quantum statistics from the same set of assumptions (with the difference of the Pauli principle), Dirac separated them starkly in their applicability. His integration of quantum statistics and quantum mechanics was thus sealed with an uncompromising rejection of Einstein's analogy between light quanta and material corpuscles ...

"The fact that Dirac disregarded the undulatory interpretation of the wave function and the consequences of antisymmetry for the independence of material particles does not mean that he refrained completely from any interpretation of the general solution of the wave equation. He did put forward an interpretation in the last section of the paper, in which he outlined a perturbation theory and fruitfully put it to use. He wrote the wave equation of 'an atomic system subjected to a perturbation from outside (e.g., an incident electromagnetic field),' and showed that the general solution could be written as Ψ = Σn anΨn, where the Ψn were the wave functions associated with the stationary states of the unperturbed atom, and the an coefficients depending on time. He thus deftly switched interpretive models. He proceeded to consider the general solution as no longer representing an atom but an assembly of atoms, and to assume that the square modulus of the coefficient an represented 'the number of atoms in the 𝑛th state' (pp. 646-647). The general solution now was a new theoretical representation of a multiparticle system that avoided any representation of individual particles and therefore bypassed, for the time being, the question of whether two states differing only by particle exchange should be counted as a distinct or identical. Determining the time evolution of the an under the effect of the perturbation, Dirac was then able to derive the coefficients of absorption and stimulated emission of Einstein's theory of radiation ...

"Dirac returned to the difference between electrons and light quanta and the emission and absorption of radiation half a year later, after having spent several months at Bohr's institute in Copenhagen formulating the general transformation theory and a general statistical interpretation of it. As a result of that work, he was able to forge a link between his two representations of multiparticle systems for the case of light quanta, and thereby launched quantum electrodynamics" (Monaldi, pp. 137-143).

"'On the Theory of Quantum Mechanics' became the most cited of Dirac's early papers and was studied with interest by both matrix and wave theorists. Although the paper was recognized as an important work, many physicists felt that it was difficult to understand and even cryptic. Schrödinger may be representative in this respect. In October [1927], when Dirac was in Copenhagen, Schrödinger told Bohr about his troubles in reading Dirac:

'I found Dirac's work extremely valuable, because it translates his interesting set of ideas at least partly into a language one can understand. To be sure, there is still a lot in this paper which I find obscure ... Dirac has a completely original and unique method of thinking, which - precisely for this reason - will yield the most valuable results, hidden to the rest of us. But he has no idea how difficult his papers are for the normal human being'" (Kragh, p. 37).

"Following the publication of Dirac's paper, the new statistics was eagerly taken up and applied to a variety of problems. The first application was made by Dirac's former teacher, Fowler; as an expert in statistical physics, he was greatly interested in the Fermi-Dirac result. Fowler studied a Fermi-Dirac gas under very high pressure, thus beginning a chapter in astrophysics that, a few years later, would be developed into the celebrated theory of white dwarfs by his student Chandrasekhar. In Germany, Pauli and Sommerfeld made other important applications of the new quantum statistics, with which they laid the foundation for the quantum theory of metals in 1927" (ibid., p. 36).

Kragh, Dirac: A Scientific Biography, 1990; Monaldi, 'Early interactions of quantum statistics and quantum mechanics,' pp. 125-147 in: Traditions and Transformations in the History of Quantum Physics. HQ-3: Third International Conference on the History of Quantum Physics, Berlin, June 28 - July 2, 2010 (Katzir, Lehner, & Renn, eds.), 2017. Pais, 'Paul Dirac: Aspects of his life and work,' in Paul Dirac: The Man and his Work, ed. P. Goddard, 1998, pp. 1-45.

8vo (254 x 179 mm), pp. [661], 662-677, [3, blank]. Original printed wrappers (lightly soiled and foxed).