Pascual Jordans resolution of the conundrum of the wave-particle duality of light
📝 Abstract
In 1909, Einstein derived a formula for the mean square energy fluctuation in black-body radiation. This formula is the sum of a wave term and a particle term. In a key contribution to the 1925 Dreimaennerarbeit with Born and Heisenberg, Jordan showed that one recovers both terms in a simple model of quantized waves. So the two terms do not require separate mechanisms but arise from a single consistent dynamical framework. Several authors have argued that various infinities invalidate Jordan’s conclusions. In this paper, we defend Jordan’s argument against such criticism. In particular, we note that the fluctuation in a narrow frequency range, which is what Jordan calculated, is perfectly finite. We also note, however, that Jordan’s argument is incomplete. In modern terms, Jordan calculated the quantum uncertainty in the energy of a subsystem in an energy eigenstate of the whole system, whereas the thermal fluctuation is the average of this quantity over an ensemble of such states. Still, our overall conclusion is that Jordan’s argument is basically sound and that he deserves credit for resolving a major conundrum in the development of quantum physics.
💡 Analysis
In 1909, Einstein derived a formula for the mean square energy fluctuation in black-body radiation. This formula is the sum of a wave term and a particle term. In a key contribution to the 1925 Dreimaennerarbeit with Born and Heisenberg, Jordan showed that one recovers both terms in a simple model of quantized waves. So the two terms do not require separate mechanisms but arise from a single consistent dynamical framework. Several authors have argued that various infinities invalidate Jordan’s conclusions. In this paper, we defend Jordan’s argument against such criticism. In particular, we note that the fluctuation in a narrow frequency range, which is what Jordan calculated, is perfectly finite. We also note, however, that Jordan’s argument is incomplete. In modern terms, Jordan calculated the quantum uncertainty in the energy of a subsystem in an energy eigenstate of the whole system, whereas the thermal fluctuation is the average of this quantity over an ensemble of such states. Still, our overall conclusion is that Jordan’s argument is basically sound and that he deserves credit for resolving a major conundrum in the development of quantum physics.
📄 Content
1 The recovery of Einstein’s fluctuation formula in the Dreimännerarbeit
In the final section of the famous Dreimännerarbeit of Max Born, Werner Heisenberg, and Pascual Jordan (1926)-cited hereafter as ‘3M’-the Umdeutung [= reinterpretation] procedure of (Heisenberg, 1925) is applied to a simple system with infinitely many degrees of freedom, a continuous string fixed at both ends. In a lecture in Göttingen in the summer of 1925 (3M, p. 380, note 2)-attended, it seems, by all three authors of the Dreimännerarbeit-Paul Ehrenfest (1925) had used this system as a one-dimensional model for a box filled with black-body radiation and had calculated the mean square energy fluctuation in a small segment of it. The string can be replaced by a denumerably infinite set of uncoupled harmonic oscillators, one for every possible mode of the string. The harmonic oscillator is the simplest application of Heisenberg’s new quantum-theoretical scheme. The basic idea behind this scheme was to retain the classical equations of motion but to reinterpret them-hence the term Umdeutung-as expressing relations between arrays of numbers, soon to be recognized as matrices (Born and Jordan, 1925), assigned not to individual states but to transitions between states and subject to a noncommutative multiplication law. 1 When this Umdeutung procedure is applied to the harmonic oscillators representing the modes of a string and the mean square energy fluctuation in a small segment of the string and a narrow frequency interval is calculated, one arrives at a surprising result. In addition to the classical wave term, proportional to the square of the mean energy, one finds a term proportional to the mean energy itself. This term is just what one would expect for the mean square energy fluctuation in a system of particles.
For this simple model, one thus recovers both terms of the well-known formula for the mean square energy fluctuation in a subvolume of a box with black-body radiation that Albert Einstein showed in 1909 is required by the Planck formula for black-body radiation and some general results in statistical mechanics. After deriving a similar formula for the mean square fluctuation ⋆ This paper was written as part of a joint project in the history of quantum physics of the Max Planck Institut für Wissenschaftsgeschichte and the Fritz-Haber-Institut in Berlin.
of momentum, Einstein remarked that in both cases “the effects of the two causes of fluctuation mentioned [waves and particles] act like fluctuations (errors) arising from mutually independent causes (additivity of the terms of which the square of the fluctuation is composed)” (Einstein, 1909a, p. 190).
He reiterated this point in his lecture later that year at a meeting of the Gesellschaft Deutscher Naturforscher und Ärzte in Salzburg (Einstein, 1909b, p. 498). 2 To paraphrase a modern commentator, black-body radiation seems to behave as a combination of waves and particles acting independently of one another (Bach, 1989, p. 178). These fluctuation formulae famously led Einstein to predict at the beginning of his Salzburg lecture “that the next phase of the development of theoretical physics will bring us a theory of light that can be interpreted as a kind of fusion of the wave and emission theories” (Einstein, 1909b, pp. 482-483).
Reluctant to abandon the classical theory of electromagnetic radiation and to embrace Einstein’s light-quantum hypothesis, several physicists in the following decade tried to find either holes in Einstein’s derivation of the (energy) fluctuation formula so that they could avoid the formula or an alternative derivation of it with which they could avoid light quanta. 4 In the wake of the Compton effect (Compton, 1923) and Satyendra Nath Bose’s new derivation of Planck’s black-body radiation law (Bose, 1924), however, both Einstein’s fluctuation formula and his light quanta began to look more and more unavoidable. As a result, the problem of reconciling the wave and the particle aspects of light took on greater urgency. In the paper that provided the simple model used in the Dreimännerarbeit, for instance, Ehrenfest (1925) emphasized the paradoxical situation that quantizing the modes of a classical wave according to a method proposed by Peter Debye (1910) gives the correct Planck formula for the energy distribution in black-body radiation but the wrong formula for the mean square fluctuation of the energy (Stachel, 1986, p. 379). This problem is also highlighted in the Dreimännerarbeit (3M, p. 376). As Einstein characterized the situation in an article on the Compton effect in the Berliner Tageblatt of April 20, 1924: “There are . . . now two theories of light, both indispensable and-as one must admit today despite twenty years of tremendous effort on the part of theoretical physicists-without any logical connection.” 5
One possibility that was seriously considered at the time, especially after the decisive refutation in April 1925 of the theor
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