Abstract
In the preceding chapter we made ourselves familiar with the zeropoint field and studied some of its most important properties, as a preparation for our inquiry into the effects that such a field is bound to have on matter. In this chapter we direct our attention to the fluctuations of the complete radiation field in equilibrium at a given temperature, and under very general assumptions we show that the presence of the zeropoint field affects most dramatically the thermal equilibrium distribution, which becomes described by Planck’s law instead of the classical Rayleigh-Jeans law. The explanation for such astonishing behaviour is found in the extra fluctuations generated by the interference between the thermal and the nonthermal components of the field modes. Other typical quantum properties of this radiation field are studied, although with much less detail and, of course, treating the field everywhere as continuous.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Detailed modern accounts of the Einstein-Stern theory and of the previous attempts by Einstein and Hopf (1910a, b) can be seen in Boyer (1969b) and Milonni (1994). Fully annotated translations into English of these two papers by Einstein and collaborators are given in Bergia et al. (1979, 1980).
See, e.g., Boyer (1969d, 1970a), Theimer (1971), Jimenez et al. (1980, 1982), Marshall (1981), Payen (1984). Other SED papers dealing with the Planck distribution are Park and Epstein (1949) (see §4.1.1), Marshall (1963, 1965a, 1965b), Surdin et al. (1966), Boyer (1980c, 1983, 1984b, d), Theimer and Peterson (1974, 1976), Santos (1975c), Kracklauer (1976), Theimer (1976), Cole (1986, 1990c), França and Maia (1993), de la Peña and Cetto (1993d, 1995b); see also Sachidanandam (1984). A detailed list of references to works up to 1982 may be found in de la Peña (1983).
A way to arrive at g(E) is by considering first the full distribution for all field oscillators of the given frequency, in the corresponding phase space {q n }. The probability for the system to be in a state of energy E(q n ) when the variables qn are within the elementary volume (math) ... can be written in the form (math) ... The integral over all q n corresponding to an energy between E and E + dE at a given moment, gives for the reduced probability density (math)
As discussed in §4.1.2, two basic notions of ‘classical’ can be distinguished in the SED literature. The one formulated in terms of a space-time description does not apply to the present statistical description of the field; we therefore resort to the second notion as the appropriate one in the present context. Accordingly, since the theory contains in an essential way the zeropoint field and differs qualitatively from classical physics, it should not be considered classical. This criterion can be substantiated by recalling that the zeropoint field operates as a special kind of reservoir, a source of extra nonthermal (quantum) fluctuations, which are totally unknown in classical physics.
This equation can be verified by a step-by-step procedure starting from (5.19) and using (5.18), or else applying directly equation (5.17).
We recall that Einstein and Stern (1913) used a similar consideration regarding the correct classical limit, as an argument in favour of retaining the zeropoint term in the Planck distribution.
As is well known, it was Planck who discovered that a quantization of the energy exchanged between atom and field leads to the Planck distribution. That Planck’s law implies quantization of the exchanged energy, was first advanced by Einstein (1905). Since then there have appeared several demonstrations similar to the one given above to show that the quantization rule (5.41) follows from Planck’s law (5.32). Of particular interest for SED are the discussions in Santos (1975c), Theimer (1976) and Landsberg (1981).
Similarly, if the quadratic (classical) term is neglected (which is allowable at very low temperatures), the remaining equation leads upon integration to the (approximate) Wien law (math).
The contribution n2 to the fluctuations can be shown to correspond more generally to any chaotic source of light in the limit of infinite coherence time; a detailed discussion can be seen in Loudon (1973), section 6.7.
That the ‘particle’ (linear) term in the Einstein fluctuations formula can be interpreted as due to the zeropoint fluctuations, as we do here, is explicitly accepted in the quantum literature by some authors. Particularly clear examples are Milonni (1980, 1994) and Milonni and Shih (1991). An interesting reinterpretation of expression (5.45) is offered by Mandel et al. (1964, 1965) within a classical theory of photodetection counts, in which the discrete and fluctuating number n represents the number of absorptions rather than photons.
The experiments are reported in Mizobuchi and Ohtaké (1992) and were realized with light following a suggestion by Ghose et al (1991, 1992); for a discussion of them, see Ghose and Home (1992). The authors recall the fresh enthusiasm of the American physicist H.D. Huffman when he reports having rediscovered on his own the Einsteinian interpretation, in a manuscript of 1989 which apparently he did not get published.
See, e.g., Glauber (1964, 1968), Hillery et al. (1984). Some elements of the theory of the quantum distributions are summarized in §13.3.1.
Recent versions of these calculations can be found in Boyer (1969b), Bergia et al. (1979, 1980), Jiménez et al. (1980), Marshall (1981), Milonni (1981, 1994), and Milonni and Shih (1991).
Einstein and Stern considered a zeropoint energy due to molecular rotational motions and not to vibrations, and argued in detail about the experimental support for their hypothesis from a study of the specific heat of hydrogen. The argument is incorrect according to present quantum knowledge, which attributes no zeropoint energy to molecular rotations. Further, they had to write quite arbitrarily ħω instead of one half this quantity to get the correct result. With the substitution ρ 0 → 2ρ 0 they compensated the absence of the zeropoint energy of the field oscillators, of which they were totally unaware. Detailed discussions can be found in Bergia et al. (1980) and Milonni (1994).
This and related problems have been studied in considerable detail mainly by A. Rueda, who has shown that in nonrelativistic QED the acceleration phenomenon occurs only in the time-symmetric version, but not in the usual form of the theory, expressed in terms of retarded potentials [Rueda 1986b]. Some additional pertinent references are Rueda and Cavalleri (1983), Rueda (1990c), Cavalleri and Spavieri (1986).
Alternatives to the Boyer (1969d) formulation where additional terms are introduced, are proposed in Jimenez et al. (1980, 1983) and Marshall (1981). A more formal attempt is presented in Payen (1984).
In his famous paper on the A-B coefficients, Einstein (1917) demonstrated by means of a statistical study of the atomic recoils that for each quantum of radiation ħω emitted or absorbed in an atomic transition, a linear momentum ħω/c in some well defined direction is exchanged. This considerably reinforced the Einstein notion of light quanta as radiation needles. In earlier works, Einstein (1909) had referred to the quanta of radiation ‘as if radiation is made of independently moving pointlike quanta’, or ‘...(as if) radiation is made of independently moving small complexes with energy ħω’ His latter radiation needles [Einstein 1917] seem to be somewhat closer, though still not equivalent, to the notion used in modern quantum theory.
See, e.g., Milonni 1994. In §§7.4.2 and 11.3.1 these matters are studied in more detail.
Atomic recoil associated with spontaneous emission was reported as apparently observed for the first time by Frisch (1933). More recent observations are discussed in Picque
Theimer (1971) has argued that in SED one should expect the photoelectrons to be released immediately after the incident light beam hits the photocathode, since as a result of the fluctuations produced by the zeropoint field, some electrons may by accident be in the conduction band at the time of illumination and can therefore be more easily released. Thus the incident beam does not by itself excite electrons to the conduction band, which would take some time, but merely affects the statistics of the fluctuations, which is an instantaneous effect, giving opportunity to the electrons that happen to be in the conduction band, to remain definitely there. This instantaneous response of photoelectrons is frequently considered as one of the proofs of the particle-like nature of photons. A similar argument had been given by Lamb and Scully (1969) within a semiclassical approach.
A detailed discussion can be seen, e.g., in Sakurai (1967), pp. 210–240. An entirely different problem are the spontaneous emissions, as they are excluded from a quantum mechanical description that predicts that all excited states are stationary.
Boyer (1969d) also pointed out that the indistinguishability of identical particles, usually considered the essence of Bose-Einstein statistics, should be properly considered a classical concept, since its use in the classical context avoids the Gibbs paradox. [An introductory explanation of the Gibbs paradox can be seen in Reif 1965, pp. 243–246.] There are of course various other known proposals to solve the Gibbs paradox; a fine and illuminating discussion can be seen in Yourgrau et al. (1982).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
de la Peña, L., Cetto, A.M. (1996). The Equilibrium Radiation Field. In: The Quantum Dice. Fundamental Theories of Physics, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8723-5_5
Download citation
DOI: https://doi.org/10.1007/978-94-015-8723-5_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4646-8
Online ISBN: 978-94-015-8723-5
eBook Packages: Springer Book Archive