BKS theory

The Bohr-Kramers-Slater (BKS) theory [N. Bohr, Collected Works, J. Kalckar, ed. North-Holland, Amsterdam, etc., 1985, Vol. 5, pp. 3-216.] [J. Mehra and H. Rechenberg, The historical development of quantum theory, Springer-Verlag, New York, etc., 1982, Vol. 1, Part 2, pp. 532-554.] [N. Bohr, H.A. Kramers, and J.C. Slater, Phil. Mag. 47, 785-802 (1924) (German version: Zeitschr. f. Physik 24, 69-87 (1924)).] (1924) is perhaps more a program than a genuine physical theory, the ideas that are developed not being worked out in a quantitative way. The theory might be seen as a final attempt at understanding the interaction of matter and electromagnetic radiation on the basis of the so-called Old quantum theory, in which quantum phenomena are treated by imposing quantum restrictions on classically describable behaviour. In particular does the BKS theory stick to a "classical" wave description of the electromagnetic field.

The initial idea of the BKS theory originates with
Slater [Letters from J.C. Slater, November, December 1923, reprinted in Ref. 1, pp. 8, 9.] , who proposed to Bohr and
Kramers the following elements of a theory of emission and absorption of radiation by atoms, to be developed during his stay in Copenhagen:
# Emission and absorption of electromagnetic radiation by matter is realized in agreement with Einstein's photon concept;
# A photon emitted by an atom is guided by a classical electromagnetic field (compare de Broglie's ideas published September 1923 [L. de Broglie, Comptes Rendues 177, 507-510 (1923).] ) consisting of spherical waves, thus enabling to explain interference;
# Even when there are no transitions there exists a classical field to which all atoms contribute; this field contains all frequencies at which an atom can emit or absorb a photon, the probability of such an emission being determined by the amplitude of the corresponding Fourier component of the field; the probabilistic aspect is provisional, to be eliminated when the dynamics of the inside of atoms is better known;
# The classical field is not produced by the actual motions of the electrons but by `motions with the frequencies of possible emission and absorption lines' (to be called `"virtual" oscillators', creating a field to be referred to as `virtual' as well).

Slater's main intention seems to be to reconcile the two conflicting models of radiation, viz. the wave and particle models. He may have had good hopes that his idea with respect to oscillators vibrating at the "differences" of the frequencies of electron rotations (rather than at the rotation frequencies themselves) might be attractive to Bohr because it solved a problem of the latter's atomic model, even though the physical meaning of these oscillators was far from clear. Nevertheless, Bohr and Kramers had two objections to Slater's proposal:
# The assumption that photons exist. Even though Einstein's photon hypothesis could explain in a simple way the photoelectric effect, as well as conservation of energy in processes of de-excitation of an atom followed by excitation of a neighboring one, has Bohr always been reluctant to accept the reality of photons, his main argument being the problem of reconciling the existence of photons with the phenomenon of interference. Bohr and Kramers hoped to be able to evade the photon hypothesis on the basis of results obtained by Kramers on describing dispersion of light by means of a classical theory of interaction of radiation and matter;
# The impossibility to account for conservation of energy in a process of de-excitation of an atom followed by excitation of a neighboring one. This impossibility followed from Slater's probabilistic assumption, which did not imply any correlation between processes going on in different atoms.

In the BKS paper the Compton effect is discussed as an application of the idea of `"statistical" conservation of energy and momentum' in a continuous process of scattering of radiation by a sample of free electrons, where "each of the electrons contributes through the emission of coherent secondary wavelets". Although Compton already had given an attractive account of his experiment on the basis of the photon picture (including conservation of energy and momentum in "individual" scattering processes), is it stated in the BKS paper that "it seems at the present state of science hardly justifiable to reject a formal interpretation as that under consideration [i.e. the weaker assumption of "statistical" conservation] as inadequate". This statement may have prompted experimental physicists to improve `the present state of science' by testing the hypothesis of `statistical energy and momentum conservation'. Anyway, already after one year the BKS theory was falsified by experiments studying correlations between the directions into which the emitted radiation and the recoil electron are emitted in individual scattering processes. Such experiments were independently performed by Bothe and Geiger [W. Bothe and H. Geiger, Zeitschr. f. Phys.26, 44 (1924); Naturwiss. 13, 440-441 (1925).] , as well as by Compton and Simon [A.H. Compton, Proc. Natl. Acad. Sci. U.S.A. 11, 303-306 (1925); A.H. Compton and A.W. Simon, Phys. Rev. 26, 289-299 (1925).] . They provided experimental evidence pointing into the direction of energy and momentum conservation in individual scattering processes (at least, it was shown that the BKS theory was not able to explain the experimental results). More accurate experiments were performed much later [R. Hofstadter and J.A. McIntyre, Phys. Rev. 78, 24-28 (1950).] [W.G. Cross and N.F. Ramsey, Phys. Rev. 80, 929-936 (1950).] .

As is suggested by his letter to Born [Letter of April 29, 1924 in: The Born-Einstein Letters, Correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955 with commentaries by Max Born, Walker and Company, New York, 1971.] was for Einstein the corroboration of energy and momentum conservation probably even more important than his photon hypothesis: "Bohr's opinion of radiation interests me very much. But I don't want to let myself be driven to a renunciation of strict causality before there has been a much stronger resistance against it than up to now. I cannot bear the thought that an electron exposed to a ray should by its own free decision choose the moment and the direction in which it wants to jump away. If so, I'd rather be a cobbler or even an employee in a gambling house than a physicist. It is true, my attempts to give the quanta palpable shape have failed again and again, but I'm not going to give up hope for a long time yet."

Bohr's reaction, too, was not primarily related to the photon hypothesis. According to Heisenberg [Interview with Mehra, quoted in Ref. 2, p. 554] Bohr would have remarked: "Even if Einstein sends me a cable that now an irrevocable proof of the physical existence of light-quanta had been found, the message cannot reach me, because it has to be transmitted by electromagnetic waves.". For Bohr the lesson to be learned from the falsification of the BKS theory was not that photons do exist, but rather the limited applicability of classical space-time pictures in understanding phenomena within the quantum domain. This theme would become particularly important a few years later in developing the notion of `complementarity'. According to Heisenberg also Born's statistical interpretation had its ultimate roots in the BKS theory. Hence by its failure the BKS theory may yet have provided an important contribution to the revolutionary transition from classical mechanics to quantum mechanics.

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