In a recent thermodynamic analysis of the harmonic oscillator Boyer has shown, using an interpolation procedure, that the existence of a zero-point energy leads to Planck’s law. We avoid the interpolation procedure by adding a statistical argument to arrive at Planck’s law as a consequence of the existence of the zero-point energy. As in Boyer’s argument, no explicit assumption of quantum mechanics is introduced. We discuss the relation of our results to the analysis of Planck and Einstein which led to the notion of the quantized radiation field. We then inquire into the discrete or continuous behavior of the energy and pinpoint the origin and meaning of the discontinuities. To include zero-point fluctuations (which are neglected in the thermodynamic analysis), we discuss the statistical (in contrast to the purely thermodynamic) description of the oscillator, which accounts for both the thermal and temperature-independent contributions to the dispersion of the energy.

1.
T. H.
Boyer
, “
Thermodynamics of the harmonic oscillator: Wien’s displacement law and the Planck spectrum
,”
Am. J. Phys.
71
,
866
870
(
2003
).
2.
M.
Planck
, “
Über eine Verbesserung der Wienschen Spektralgleichung
,”
Verh. Dtsch. Phys. Ges.
2
,
202
204
(
1900
).
English translation in Ref. 4. See also
M.
Planck
, “
Über das Gesetz der Energieverteilung im Normalspektrum
,”
Ann. Phys.
4
,
553
563
(
1901
).
3.
A.
Einstein
, “
Zum gegenwärtigen Stand des Strahlungsproblems
,”
Phys. Z.
10
,
185
193
(
1909
). English translation in Ref. 4.
4.
D.
ter Haar
,
The Old Quantum Theory
(
Pergamon Press
,
Oxford
,
1967
).
English translations of the Einstein’s papers cited here also appear in
The Collected Papers of Albert Einstein
, Vol.
2
: The Swiss Years: Writings, 1900–1909 (English translation supplement, translated by Anna Beck (
Princeton U.P.
,
1989
).
5.
As demonstrated in Ref. 6, the single dependence of the zero-point energy on frequency allowed by special relativity is E0(ω)ω. The previous classical thermodynamic calculation leads to the same expression, Eq. (5). The Schrödinger equation provides a similar prediction for the ground state energy of a particle in a harmonic oscillator potential.
6.
That the single spectrum of the zero-point radiation field which is consistent with relativity (and hence with electromagnetic theory) corresponds to E0(ω)ω has been demonstrated independently by several authors. The earliest such demonstrations are by
T. W.
Marshall
, “
Random electrodynamics
,”
Proc. R. Soc. London, Ser. A
276
,
475
491
(
1963
);
E.
Santos
, “
Is there an electromagnetic background radiation underlying the quantum phenomena?
,”
An. R. Soc. Esp. Fis. Quim.
LXIV
,
317
320
(
1968
);
T. H.
Boyer
, “
Derivation of the blackbody radiation spectrum without quantum assumptions
,”
Phys. Rev.
182
,
1374
1383
(
1969
).
See also
P. W.
Milonni
,
The Quantum Vacuum
(
Academic
,
New York
,
1994
), Chap. 2.
7.
E. W.
Montroll
and
M. F.
Shlesinger
, “
Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: A tale of tails
,”
J. Stat. Phys.
32
,
209
230
(
1983
).
8.
See, for example,
K.
Huang
,
Statistical Mechanics
(
Wiley
,
New York
,
1963
).
9.
A.
Einstein
, “
Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme
,”
Ann. Phys.
22
,
180
190
(
1907
).
English translation in the
Collected Papers
, Ref. 13.
10.
Einstein considered the probability distribution in Eq. (7) assuming from the start a form for the function g(E) equivalent to Eq. (65), as was dictated by the quantization discovered by Planck. Here we let the theory determine g(E), moving in the opposite sense.
11.
A short early account of the material in this section was given in
L.
de la Peña
and
A. M.
Cetto
, “
Planck’s Law as a consequence of the zeropoint radiation field
,”
Rev. Mex. Fis.
48
Suppl. 1,
1
8
(
2002
).
12.
For consistency in notation, we continue to write the specific heat as Cω; it coincides with the specific heat at constant volume, so the usual notation in this context is CV.
13.
Letter from Einstein to
Conrad
Habicht
; May 18 or 25 1905, in
The Collected Papers of Albert Einstein
, Vol.
5
, The Swiss Years, Correspondence, 1902–1914, edited by
Martin J.
Klein
,
A. J.
Kox
, and
Robert
Schulmann
(
Princeton University Press
,
Princeton
,
1993
), pp.
31
32
. Translated into English by Anna Beck in the accompanying translation, pp.
19
20
. A discussion of this point is given by
J. S.
Ridgen
, “
Einstein’s revolutionary paper
,” ⟨physicsworld.com/cws/article/print/21818⟩.
14.
V.
Vedral
,
Modern Foundations of Quantum Optics
(
Imperial College Press
,
London
,
2005
), Chap. 3.
15.
Related discussions are given in
E.
Santos
, “
Comment on ‘Presenting the Planck’s relationE=nhν
,”
Am. J. Phys.
43
743
744
(
1975
);
O.
Theimer
, “
Blackbody spectrum and the interpretation of the quantum theory
,”
Am. J. Phys.
44
,
183
185
(
1976
);
P. T.
Landsberg
, “
Einstein and statistical thermodynamics. II. Oscillator quantisation
,”
Eur. J. Phys.
2
,
208
212
(
1981
).
16.
We should take into account at least the fluctuations that lead to the natural linewidth. See, for example,
W. H.
Louisell
,
Quantum Statistical Properties of Radiation
(
Wiley
,
New York
,
1973
), Chap. 5.
17.
T. H.
Boyer
, “
Classical statistical thermodynamics and electromagnetic zero-point radiation
,”
Phys. Rev.
186
,
1304
1318
(
1969
). Boyer uses the terms caloric and probabilistic entropies instead of thermal and statistical, respectively, used here.
18.
The factor ω is the absolute value of the Jacobian of the canonical transformation from the space E,θ to the space q,p for the harmonic oscillator. See, for example,
G. R.
Grimmett
and
D. R.
Stirzaker
,
Probability and Random Processes
(
Clarendon
,
Oxford
,
1983
), Chap. 4;
A.
Papoulis
,
Probability, Random Variables, and Stochastic Processes
(
McGraw–Hill
,
Boston
,
1991
), Chap. 6.
19.
M.
Hillery
,
R. F.
O’Connell
,
M. O.
Scully
, and
E. P.
Wigner
, “
Distribution functions in physics: Fundamentals
,”
Phys. Rep.
106
,
121
167
(
1984
).
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