Abstract
The historical problem of the spectral distributionDistribution of the radiation field in equilibrium at a given temperature is revisited in this chapter. Taking into account the inescapable presence of the fluctuatingFluctuations!zero-point zero-point radiation field, a derivation of Planck’s formula is presented that does not introduce any quantum postulate. The starting point is Wien’sWien!law law, which dictates that the mean energy of a Harmonic oscillator!and spinharmonic oscillator of frequency \(\omega \)—or rather, the mean energy of a monochromatic mode of the radiation field—must be proportional to the frequency. The description obtained through a standard thermodynamic analysis is shown to be incomplete, as it does not provide for the existence of fluctuations at zero temperature. This limitation is lifted by means of a statistical analysis that allows to determine the variance of the Fluctuationsfluctuations of the energy at \(T=0.\) A combination of the outcomes of the two analyses leads to a differential equation, which upon integration gives the Planck distribution Planck distributionat any temperature. The different terms contributing Commutator!and correlationto the energy fluctuations acquire thus a new meaning, different from those assigned to it by Planck and by EinsteinEinstein, A.. The implications of the results regarding the question of continuity or discontinuity of the field energy are discussed, and the chapter concludes with a brief discussion on the reality Realityof the zero-point fluctuations and the origin of quantum fluctuations.
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Notes
- 1.
This first part of the exposition borrows from the work ofBoyer, T.H. Boyer (1969b, 2003), who has contributed substantially to the analysis of the Planck distribution Planck distributionfrom a perspective akin to the one developed here. See alsoBoyer, T.H. Boyer (1969a, 1976, 1983, 1984, 2010a, b, 2012), Marshall (1965), andTheimer O. Theimer (1971).
- 2.
Wien!lawWien’s law is a fundamental law of physics, since only simple and very general principles are required for its derivation. It is valid in classical as well as in quantum physics, and is even consistent with relativity, so it was the appropriate law to herald the 20th century. To get a better feeling of its fundamental nature, a derivation based solely on dimensionality arguments can be found in Sommerfeld’s classical book on Thermodynamics (Sommerfeld A. Sommerfeld 1956). Simple clear discussions of Wien’sWien!law law can be seen in two highly pedagogical papers:Piña, E. De la Selva, S. M. T. Piña and de la Selva (2010), andDel Río-Correa, J. L. del Río-Correa (2010)Mendez, A. F. Ramírez, D..
- 3.
We recall that for the derivation of his law, Wien Wienstudied the Doppler effect of the modes of an adiabatically disturbed radiation field in thermal equilibrium (see e.g. Milonni 1994).
- 4.
- 5.
That the only spectrum consistent with relativity (and hence with electromagnetic theory) corresponds to \(\mathcal {E}_{0}(\omega )\sim \omega \) , has been demonstrated independently by several authors. The earliest of such demonstrations are those in Marshall (1963), Santos (1968), andBoyer, T. H. Boyer (1969b). See alsoCole, D. C. Cole (1990),Milonni, P. W. Milonni (1994), Chap. 2; and The Dice, Chap. 4. The present thermodynamic calculation leads to the same expression, Eq. (3.18). Further, the Schrödinger equation provides a similar prediction for the ground-state energy of a particle in a harmonic oscillator potential. Here we have a vivid example of the intrinsic unity of physics, reinforcing the idea that it refers to different aspects of a single reality.
- 6.
Taking \(A=0\) is equivalent to putting the boundary conditionBoundary condition for the solutions of Maxwell’s equations at infinity in the past equal to zero, i. e. no radiation. The choice \(A\ne 0\) replaces this unnatural boundary condition by a zero-point field at infinity, simultaneously restoring time-reversal symmetry in electrodynamics.
- 7.
This form of writing \(W_{g}(\mathcal {E})\) was used, for example, by Einstein (1907) in his early work on the specific heat Specific heatof solids. He considered the distribution in (3.21a) assuming from the start a form for the function \( g(\mathcal {E})\) equivalent to (3.82) below, as was dictated by the quantization discovered by Planck. Here we proceed in the opposite sense, by allowing the theory to determine \(g(\mathcal {E})\).
- 8.
- 9.
\(C_{\omega }\) coincides with the specific heatSpecific heat at constant volume, so the usual notation in this context is \(C_{V}\). Still, we employ the subindex \(\omega \) since we are considering \(\omega \) to be a fixed parameter.
- 10.
This is a well-known statistical result, established for the first time by Lorentz Lorentz, H. A.for the thermal radiation field. A simple demonstration is given inVedral V. Vedral (2005). Inclusion of the zero-point component does not modify this statistical property, since the argument to establish it remains in force.
- 11.
Planck’sBoyer!and Planck law law without zero-point energy (the first relation derived by Planck) is obtained by fixing the constant of integration precisely as \(- \mathcal {E}_{0}\), so that \(U(\beta )= \mathcal {E}_{0} \coth \mathcal {E}_{0}\beta -\mathcal {E}_{0}\). The existence of the zero-point energy remains hidden with this choice.
An additional comment is in place here. At first sight it would seem plausible to take the constant of integration in the first line of Eq. (3.54) as \(\mathcal {E}_{0},\) so that the resulting function, \(U(\omega ,T)=k_{B}T+\mathcal {E}_{0}\) is apparently consistent with both the existence of a nonthermal energy and Wien!lawWien’s law. However, such choice must be discarded since this \(U\) cannot be obtained as a limit case of \({\mathcal {E}}_{0}\ne 0\).
- 12.
- 13.
The rationale behind Planck’s reading of his formula is the following. If the system composed by the walls of the cavity (represented by a collection of material oscillators) and the enclosed radiation field exchanges energy not continuously but by lumps (which he called quanta) of value \(n\hbar \omega \) (\(n=1,2,3,\ldots \)), then the mean equilibrium energy is
$$\begin{aligned} U=\frac{\sum _{n=0}^{\infty }n\hbar \omega e^{-\beta n\hbar \omega }}{ \sum _{n=0}^{\infty }e^{-\beta n\hbar \omega }}. \end{aligned}$$Performing the summations with the aid of the relation \(\sum _{n=0}^{\infty }x^{n}=1/(1-x),\) one gets
$$\begin{aligned} U=\frac{\hbar \omega }{e^{\hbar \omega \beta }-1}, \end{aligned}$$which is just the \(U_{T}\) in Planck’s theory. If by contrast a continuous exchange of energy is assumed instead of a discrete one, the sum above must be replaced by an integral from \(0\) to \(\infty \). The reader can easily check that in this case the result is the classical formula \(U_{T}=1/\beta =k_{B}T.\)
- 14.
A Poisson distributionDistribution!Poisson refers to the probability of \(n\) independent discrete events taking place simultaneously, and has the form
$$\begin{aligned} P_{a}(n)=e^{-a}\frac{a^{n}}{n!}. \end{aligned}$$It is easy to verify that for this distribution the mean of \(n\) is \( \left\langle n\right\rangle =a\) and its variance is precisely \(\sigma _{n}^{2}=\left\langle n\right\rangle .\)
- 15.
ComplementarityGraded realizations of complementarity relations (wave-like or particle-like behaviorPhoton!particle-like behaviour) have been under close scrutiny during the last decades; see e.g.Jaeger, G. Jaeger et al. (1995), Englert (1996),Bergou, J. A. Engert and Bergou (2000),Li L. Liu, N.-L. Liu et al. (2009), Flores, E. V. Flores and de Tata (2010) (see also Ghose and Home 1996).Ghose, P. The general validity of Einstein’s fluctuation formulaEinstein!fluctuation formula (3.71) had been verified experimentally since earlier times; seeAldemade, C. Th. J. Aldemade et al. (1966)Bolwijn, P. T.,Kattke, G. W. Kattke and van der Ziel (1970). The authors are grateful to M. D. Godfrey for drawing their attention to these references.
- 16.
- 17.
This is a most significant quantum result. In the quantum statistical descriptionNonlocality!and statistical description the finite quantity \(\hbar ^{3}\) plays the role of a minimal element of volume in phase space. This idea was introduced formally for the first time by Planck in his early studies of the blackbody spectrum Planck, M.(Planck 1900a, b). Later, in 1924, BoseBose, S. N. assumed that two or more distributions of microstates that differ only in the permutation of phase points within a subregion of phase space of volume \(\hbar ^{3},\) are to be regarded as identical, which already corresponds to the Bose-Einstein statisticsBose-Einstein statistics. In the classical description the volume of such elementary cells is taken to tend to zero in order to recover the continuity of the phase space. It is remarkable that, already in his classical statistical studies, Detailed balance!and Maxwell-Boltzmann statisticsBoltzmannBoltzmann introduced Maxwell-Boltzmann statistics!and detailed balanceformally the idea of a discrete phase space (see e.g., Jones 2008, Chap. 3).
- 18.
The existence Fluctuations!of energyof energy fluctuations associated with the natural linewidth Linewidth!naturaland other processes (see e.g. Schiff 1955; Louisell, W. H. Louisell, W. H. Louisell 1973), effectively dilutes this discrete distribution of energies into a somewhat smoothened-out distribution acquiring a more continuous shape. Thus \(g( \mathcal {E})\) should be seen as a theoretical limiting distribution.
- 19.
The transformation defined by (3.95a) and (3.95b) is an extended canonical Distribution!canonicaltransformationCanonical transformations Goldstein, H.(Goldstein 1980), which differs from a canonical Distribution!canonicalone—from the action and angle variables \((J,\theta ),\) with \(J=\mathcal {E}/\omega ,\) to the phase space variables \((p,q)\)—only by a constant factor \(\omega \). Of course \(\theta =\omega t.\)
- 20.
- 21.
- 22.
This point is discussed in Chap. 6, in connection with the derivation of the Einstein A-B coefficientsEinstein \(A\) and \(B\) coefficients.
- 23.
There is an extended stance against the assumption of the reality of the fluctuating zpf, based mainly on the argument that it does not activate photon detectors. Another frequent argument refers to the unobserved tremendous gravitational effects that such field should produce. The first objection has been answered by offering models of photon counters compatible with the reality Realityof the zero-point fluctuations Santos E.(Santos 2002a, b). The gravitational puzzle is related to the problem of the cosmological constant, andCosmological constant represents an age-old unsolved fundamental problem that besets a broad parcel of physics Weinberg, S.(see e.g. Weinberg 1989).
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de la Peña, L., Cetto, A.M., Valdés Hernández, A. (2015). The Planck DistributionPlanck distribution, a Necessary Consequence of the Fluctuating Zero-Point Field. In: The Emerging Quantum. Springer, Cham. https://doi.org/10.1007/978-3-319-07893-9_3
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