Download Planck`s law as a consequence of the zeropoint radiation field

REVISTA MEXICANA DE FÍSICA 48 SUPLEMENTO 1, 1 - 8
SEPTIEMBRE 2002
Planck’s law as a consequence of the zeropoint radiation field
L. de la Peña and A.M. Cetto
Instituto de Fı́sica, Universidad Nacional Autónoma de México
Apdo. Post. 20-364, 01000 México, D.F., Mexico
Recibido el 19 de marzo de 2001; aceptado el 3 de julio de 2001
In this paper we show that a strong link exists between Planck’s blackbody radiation formula and the zeropoint field. Specifically, the
hypothesis that the equilibrium field (including the zeropoint term) follows a canonical distribution, in combination with the requirements
that its specific heat remain finite as the temperature T → 0 and that it behave classically as T → ∞, implies that the energy of the zeropoint
field has a fixed value and leads to Planck’s law for the equilibrium distribution at arbitrary T ; this in turn implies quantization of the energy
of the field oscillators. There is no need to introduce any quantum property or ad hoc assumption for the purpose of the present derivation.
Keywords: Zeropoint field; Planck’s law; blackbody radiation; energy quantization.
En este trabajo se exhibe la existencia de una estrecha relación entre la fórmula de Planck para la radiación de cuerpo negro y el campo
de punto cero. Especı́ficamente, la hipótesis de que el campo de equilibrio (incluido el término de punto cero) sigue una distribución
canónica, junto con las demandas de que su calor especı́fico se mantenga finito conforme T → 0 y de que se comporte clsicamente para
T → ∞, implica que la energı́a del campo de punto cero tiene un valor fijo y conduce a la ley de Planck para la distribución de equilibrio
a temperatura arbitraria; esto a su vez implica la cuantización de la energı́a de los osciladores del campo. No hay necesidad de introducir
propiedades cuánticas o supuestos ad hoc para el propósito de la presente derivación.
Descriptores: Campo de punto cero; ley de Planck; radiación de cuerpo negro; cuantización de la energı́a.
PACS: 02.50.Ey; 03.65.Ta; 05.40.+j
To our colleague Leopoldo Garcı́a Colı́n, with whom we have the fortune to share a long and warm friendship —and a
common interest in the fundamentals of physics.
1. Introduction
As has been repeatedly recalled on occasion of its centennial, the first problem to find an answer in terms of quantum
physics was the determination of the spectrum of blackbody
radiation. Planck noted right from the start [1, 2] —and Einstein elaborated on it briefly afterwards [3]— that the correct
formula for the spectral distribution of the blackbody radiation field in thermal equilibrium was obtained by introducing
a discrete element; this led to a picture that is now a fundamental part of quantum theory.
What has not been so clearly established, however, is the
link between the Planck distribution (with the ensuing energy quantization) and the zeropoint field. Planck discovered
the latter only in 1911, while formulating his so-called second theory [4], in which the total mean energy of the matter
oscillators contains a zeropoint term ~ω/2. The possibility
of a direct relationship between quantization as revealed by
Planck’s law, and the zeropoint energy of matter oscillators,
was envisaged for the first time, still in a classical context,
by Einstein and Stern in 1913 [5]. However, it was Nernst [7]
who a few years later suggested to read Planck’s complete
blackbody radiation formula as saying that also the electromagnetic field possesses such zeropoint fluctuations, and to
consider these fluctuations as the source of quantization. A
nice discussion of some of these and related points can be
seen in Milonni’s book [8], where also the case is made for
the need to recognize the reality of the zeropoint radiation
field.
The proposal of Planck and Nernst, appealing as it
sounds, remained idle for decades in an obscure corner of
physics until it was finally revived by Boyer [9], after some
previous attempts, notably those by Park and Epstein [10] and
Surdin et al. [11] In a first attempt to put to the proof the feasibility of the idea, Boyer showed that a classical treatment of
the radiation field including a random zeropoint component
of average energy ~ω/2 per normal mode, along with some
assumptions on the role of the container walls in restoring
and maintaining equilibrium, leads to Planck’s distribution
law. This work aroused interest in the subject and triggered
a whole series of efforts (see e.g. Refs. 12–22), several of
which contain significant positive results; however, what is
still wanting is a definitive derivation of the blackbody spectrum as a direct result of the presence of the zeropoint field
free of ad hoc assumptions.
In the present paper, which is a revision and updating of
an earlier work [23], we turn back to this problem, carefully
avoiding the use of any ad hoc assumption but maintaining
at the same time the simplicity of the reasoning. A couple of
elementary and customary statistical and thermodynamic arguments are here used to demonstrate that the quantization
of the radiation field as implied by Planck’s law, and this law
itself, follow indeed from the hypothesis of the existence of
a real fluctuating zeropoint field with energy ~ω/2. There is
no need to introduce any explicit quantum property or ad hoc
assumption for the purpose of the present derivation.
2
ENGLISHL. DE LA PEÑA AND A.M. CETTO
2. Inconsistency between the zeropoint field
and equipartition of energy
Let us consider the equilibrium radiation field inside a cavity at temperature T . The field is made of independent modes
characterized by a given frequency ω, so that we may focus on one frequency in particular. Irrespective of whether
the system is classical or quantum, the energy is distributed
among the modes of a given frequency within the energy interval between E and E + dE following a canonical distribution,
W (E) dE =
1
g(E)e−βE dE,
Z
r
(4)
As we see, g(E) = 1 leads to equipartition of energy E = kT
among the radiation oscillators and thus it is in contradiction
with experiment, as is well known after a century of quantum physics. Thus, results such as Eq. (4) obviously require
revision.
For the general case with arbitray g(E), the partition
function is given by
Z
Z = g(E)e−βE dE,
(5)
so that its derivative with respect to β is
Z 0 = −Z E.
(6)
Further, for an arbitrary function f (E) one has
Z
1
f (E) =
f (E)g(E)e−βE dE,
Z
so that with the help of Eq. (6) one gets
0
f (E) ≡
df¯
= E f¯ − Ef .
dβ
dE
2
= E − E2.
dβ
(7)
(9)
Another useful formula derived from Eq. (5) is
1 dr Z
,
Z dβ r
(10)
of which Eq. (6) is a particular case and Eq. (8) is an immediate consequence. We recall that Eqs. (5)–(10) hold for any
function g(E), including of course the classical value g = 1.
From Einstein’s formula, rewritten as
(2)
and the higher-order moments are given by
(8)
This result is a generalization of the well-known Einstein formula for the thermal energy fluctuations, obtained for r = 1,
(1)
while in quantum physics g(E) takes the form of a discrete
distribution. The main task of the present investigation is to
derive the function g(E) from the assumption of the existence
of the zeropoint radiation field [see Eq. (48)].
Given their relevance for the present discussion, we recall
a couple of (classical) results that are obtained for g(E) = 1.
Firstly, the average energy of every field oscillator is given by
Z ∞
1
E=
EW (E) dE = ,
(3)
β
0
E r = r!E .
0
E r = EE r − E r+1 .
E r = (−)r
where β = 1/kT and g(E) represents the intrinsic probability for the states with energy around E. This is our central
assumption. In classical physics one has
g(E) = 1,
For f = E r , r = 0, 1, 2, . . ., one obtains in particular
kT 2
∂E
2
= E 2 − E ≡ σE2 ,
∂T
(11)
a simple relation between the energy fluctuations σE2 and the
specific heat of the field cV = ∂E/∂T follows, namely,
kT 2 cV = σE2 .
(12)
This result holds in principle for any temperature, including T = 0 [21, 24]. Hence, since according to the third law
of thermodynamics cV approaches zero as T → 0, also the
energy variance of the field must go to zero as T → 0. More
specifically, it must go to zero as rapidly as T 2 cV (below we
see that this means exponentially rapid).
2
In classical physics, with E 2 = 2E according to Eq. (4),
this demand on the energy fluctuations implies that E → 0 as
T → 0, which is automatically satisfied by the equipartition
formula E = kT and leads to the Rayleigh-Jeans distribution
when the result is inserted into the Planck formula for the
equilibrium spectral energy density, ρ(ω) = (ω 2 /π 2 c3 )E.
Things are different in the presence of the zeropoint field,
however. Let us denote by E0 the average value of the energy
of this field. Equation (12) tells us that even if E0 > 0, the
variance of the zeropoint energy must vanish, which means
that
E2 = E
2
at T = 0,
(13)
contrary to the (classical) equation [Eq. 4]. This result will
play an important role in what follows. In fact, the present
derivation of Planck’s law differs from all previous attempts
(except for that of Ref. 23) by this crucial point. Going back
to Eq. (1), it means that the function g(E) must indeed be a
non trivial function of the energy. What is surprising is that
this function turns out to be uniquely determined by the mere
demand that E0 6= 0, in combination with the vanishing of
the variance at T = 0, as will be shown below.
We have noted that the mere inclusion of a zeropoint
term implies a necessary modification of the classical laws,
and in particular, the invalidity of the equipartition of energy,
Rev. Mex. Fı́s. 48 S1 (2002) 1 - 8
3
ENGLISHPLANCK’S LAW AS A CONSEQUENCE OF THE ZEROPOINT RADIATION FIELD
(see the related discussions in Refs. 22, 24, and 25). But another important consequence of our considerations is that a
description of the zeropoint field consistent with the distribution W (E) in Eq. (1), demands the energy of this field to be
¡ ¢1/2
a non-fluctuating quantity, E 2 T =0 = E T =0 = E0 .
relation. In particular, with θ2 = 0 we would get θr = 0
for r > 2, which would carry us back to the classical solution, Eq. (4).
To find θ2 (E) in Eq. (15) for r = 2,
E 2 = 2E + E02 θ2 ,
(21)
3. Higher moments of the energy distribution
we write it as a power series in x
X
θ2 (x) =
Ck xk .
(22)
Equation (8) gives a recurrence relation for the energy moments,
2
2
0
E r+1 = E E r − E r .
(14)
To solve this equation we recall once more that in the clasr
sical limit, i.e., for high temperatures (β → 0), E r = r!E
must hold, so that for arbitrary temperatures we propose to
write
r
E r = r!E + E0r θr ,
(15)
where the θr are functions of β to be determined, and the
coefficient E0r has been introduced so as to make each θr
dimensionless. It is possible to specify certain properties that
the θr must possess. In the first place, by integrating Eq. (9)
Z
dE
β=−
(16)
2,
E2 − E
we observe that E 2 can be expressed indeed as a function
of E(β); hence according to Eq. (14), every E r can in its
turn be expressed as a function of E, and therefore also
θr = θr (E).
Secondly, for r = 0, 1 Eq. (15) gives
θ0 = 0,
We know from Eq. (4) that at high temperatures E 2 = 2E ,
so there can be no term in Eq. (22) with k ≥ 2. Negative values for k must be excluded because they would describe an
unphysical behavior for the fluctuations. We are thus left with
θ2 (x) = C0 + C1 x = C0 + C1
where according to Eq. (18),
C0 + C1 = −1.
(24)
4. Planck’s law
Let us use the above results to find E as a function of β.
Firstly we combine Eqs. (21) and (22) to write
2
E 2 = 2E + C1 E0 E + C0 E02 ,
(25)
and get rid of C1 with the help of Eq. (24),
2
E 2 = 2E − (1 + C0 )E0 E + C0 E02 .
(17)
E0 β =
2
For r = 2 we get from Eq. (15) E 2 = 2E + E02 θ2 ; taking
T = 0 and using Eq. (13), it follows that
2
σE2 T =0 = E02 − E 0 = E 0 + E02 θ2T =0
E = E0 +
whence
θ2T =0 = −1.
(18)
Introducing Eq. (15) into (14) we get the recurrence relation
µ ¶r−1
1 dθr
E
E
=−
+ θr + rr!
θ2 ,
E0 dβ
E0
E0
or in terms of x ≡ E/E0 ,
¡
¢
θr+1 = x2 + θ2 θr0 + xθr + rr!xr−1 θ2 ,
1
x − C0
ln
,
1 − C0
x−1
where the integration constant has been chosen so that
E → ∞ as β → 0. Inverting,
= E02 + E02 θ2T =0 = 0,
θr+1
(23)
By introducing this into Eq. (16) we get upon integration
θ1 = 0.
2
E
,
E0
(19)
(20)
(1 − C0 )E0
.
e(1−C0 )E0 β − 1
(26)
We have thus obtained a continuous family of solutions in
terms of the parameter C0 . As shown below, for all values of
C0 6= 1 there is energy quantization, C0 = 1 leading to the
classical (continuous) limit. It is important to observe that
in Eq. (26) the functional dependence of the mean energy
in both the frequency and the temperature is already determined, since we know that E0 ∼ ω (as follows from Wien’s
law) [21, 24]. The value to be assigned to C0 can be determined in different ways, two direct ones of which are the
following.
Planck’s constant was introduced by Planck by writing
(in modern notation)
where θr0 = dθr /dx. Note that the function θ2 determines
all θr for r > 2, but θ2 itself is left undetermined by this
Rev. Mex. Fı́s. 48 S1 (2002) 1 - 8
(1 − C0 )E0 = ~ω,
(27)
4
ENGLISHL. DE LA PEÑA AND A.M. CETTO
as follows from the requirement E0 ∼ ω. By inserting this relation into Eq. (26) we recast the latter in the form
~ω
~ω
E=
+ ~ωβ
,
1 − C0
e
−1
(28)
where obviously C0 < 1 for E0 to have a positive value. This
equation is in fact a bit more general than Planck’s law, as it
allows for the zeropoint field to be in a state with an energy
Er = ~ω/(1 − C0 ) > ~ω/2 for C0 > −1, without affecting at all the thermal part of the distribution; below we relate
such states with the squeezed states of the vacuum. As follows from Eq. (28), one must set C0 = −1 to assign the
correct value to the energy of the zeropoint field.
In a better but still simple procedure we consider the hightemperature behaviour of Eq. (26) by expanding the exponential function in the denominator, to get
(1 − C0 )E0
E = E0 +
1
1 + (1 − C0 )E0 β + (−C0 )2 E02 β 2 + . . .−1
2
1
1
→ + (1 + C0 )E0 ,
(29)
β
2
for β → 0. Since at high temperatures we must have
E = 1/β, we get, using also Eq. (24),
C0 = −1,
C1 = 0,
(30)
θ2 = −1.
(31)
whence
For any other C0 the vacuum field gives a constant contribution to be added to 1/β in Eq. (29). With this value of the
parameter Eq. (26) gives exactly Planck’s law for the average
energy,
E = E0 +
2E0
1
~ω
= ~ω + ~ω/kT
,
2
e2E0 β − 1
e
−1
2
The remaining functions θr (r > 2) are obtained from
Eq. (30) with C0 = −1,
(34)
Since with the values given in Eq. (30) θ2 becomes an even
function of x, all remaining θr acquire a well-defined parity
as a function of x (of E), being an even or odd function for r
even or odd, respectively. For the first few functions θr one
obtains
θ3 = −5x,
θ4 = 5 − 28x2 ,
Z = e−
E dβ
=
e−C0 E0 β
,
−1
e(1−C0 )E0 β
(37)
or
Z=
e−E0 β
1 − e−(1−C0 )E0 β
.
(38)
A power series expansion of the denominator in terms
of e−(1−C0 )E0 β gives
Z(β) = e−E0 β
∞
X
e−(1−C0 )E0 βn
n=0
≡
∞
X
e−En β ,
(39)
n=0
where we have introduced the abbreviation
£
¤
En = 1 + (1 − C0 )n E0
= E0 + n~ω,
n = 0, 1, 2, . . .
(40)
using Eq. (27). The intrinsic probability function g(E) is
readily obtained now by taking the inverse
R Laplace transform
of Z(β), as follows from Eq. (5), Z = g(E)e−βE dE. The
inverse transform of Eq. (39) leads to
" ∞
#
X
−1
−En β
g(E) = LE
,
e
n=0
or
∞
X
δ(E − En ).
(41)
n=0
Hence for all values of C0 but C0 = 1 there is energy quantization of the field oscillators for the equilibrium state. The
quantization of the field is essentially independent of the
value of the parameters, so in this particular sense the theory is essentially insensitive to them.
Equation (1) takes now the form usual in quantum theory
for the canonical ensemble
W (E) =
∞
1 X −En β
e
δ(E − En ),
Z n=0
(42)
and the average value of f (E) is given as with the corresponding density matrix in quantum theory,
(35)
a.s.o.
R
(32)
(33)
θr+1 = (x2 − 1)θr0 + xθr − rr!xr−1 .
By using Eq. (26) for the average energy and integrating
Eq. (7) one obtains for the partition function
g(E) =
whilst Eq. (25) reduces to
E 2 = 2E − E02 .
5. Energy quantization
(36)
Any other selection C0 6= −1 destroys this symmetry.
Rev. Mex. Fı́s. 48 S1 (2002) 1 - 8
f (E) =
∞
X
pn f (En ),
n=0
pn = Z −1 exp(−En β).
(43)
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ENGLISHPLANCK’S LAW AS A CONSEQUENCE OF THE ZEROPOINT RADIATION FIELD
6. Some comments on quantization and final
remarks
The Planck law has been derived without introducing any discrete property or quantum rule for the field or matter oscillators. This seems to be the kind of derivation that Planck was
looking for when he tried to avoid the need to introduce the
discrete interchange of energy among field and matter oscillators. In the present derivation, the Planck distribution appears as a direct consequence of the zeropoint energy being
different from zero. It should be stressed that the hypotheses used in the derivation, namely that the equilibrium field
is canonically distributed, that it has the correct classical behaviour when T → ∞, and that cV remains finite at T = 0 (in
agreement with the third law of thermodynamics), are widely
acknowledged as reasonable assumptions both in classical
and quantum physics, and they clearly do not contain any
quantum assumption. Thus the quantization of the energy of
the oscillators as given by Eq. (40) and the Planck law should
be seen here to be a consequence of the reality of the zeropoint field.
To the above list of postulates one should apparently add
Wien’s law in the form E(ω, T ) = aωf (ω/T ), with a a constant and f an unknown function, to obtain E0 ∼ ω. However,
it has been demonstrated by Cole in a long series of careful
studies (see Ref. 24 and references therein) from first principles that the energy of the thermal equilibrium field at T = 0
is necessarily of the form const·ω, which is of course equivalent to demonstrating that Wien’s law can be extended down
to T = 0, as we have done above.
To consider the zeropoint field as the source of quantization is not only in agreement with the ideas explored initially by Einstein and Stern in 1913 [5] and by Nernst around
1916 [7], but has also been the Leitmotiv of stochastic electrodynamics. For this theory, having a neat derivation of the
Planck distribution based on first principles is of fundamental importance, which explains the sustained efforts devoted
to this task along the years [9-23].
It must be stressed that we have obtained only the correct Planck law for the average energy E(ω, T ) of the
(radiation field) oscillators, and not the spectral density
ρ(ω, T ) = (ω 2 /π 2 c3 )E(ω, T ), to which the term “Planck’s
distribution” normally applies, the difference between the
two quantities lying in the factor that gives the density of
modes, ω 2 /π 2 c3 . This simplification is acceptable here, since
we are interested in the quantization of the energy, an information that is fully contained in E(ω, T ). Although Planck
derived and used the correct factor (ω 2 /π 2 c3 ) right from the
begining of his investigations, its proper calculation for quantum particles (and not only for classical waves) requires a
knowledge of quantum statistics, as we now know, so it was
only many years after that this problem was settled.
Although the existence of a zeropoint field does not in
principle fall in conflict with classical electromagnetic theory, and may even be regarded more natural than its nonexistence [9, 12, 23] it is inconsistent with classical thermo-
2
dynamics, where the combined requisites E 2 = 2E and
σE2 = 0 at T = 0, imply E = 0 at T = 0. From this point
of view classical physics (including thermodynamics) is incompatible with the notion of a zeropoint energy. This wellknown fact reappears here because the quantized energy of
the field, a direct consequence of the existence of the zeropoint energy, is contrary to the law of equipartition of energy
among the field oscillators.
In his 1907 paper on the specific heat of solids [3] Einstein derived Planck’s law starting from Eq. (42) as an alternative to Planck’s proposal. This procedure is similar to the
modern density matrix formulation, which is equivalent to the
use of Eq. (43). Here we have followed the inverse procedure,
in going from Planck’s law to the quantized energy spectrum.
The demonstration that Planck’s law [Eq. (32)] implies the
quantization rule [Eq. (40)] has been given by various authors at different times, as, e.g., in Refs. 26–28 Of course the
description in terms of the canonical distribution can refer to
any set of oscillators, whether those of the radiation field, or
oscillating dipoles attached to gas molecules enclosed in the
container, or those that model the walls of the container itself,
as long as they are in thermodynamic equilibrium with the
radiation field. Hence the conclusions of the above analysis
can be applied equally well to the field and to the oscillators
of material systems in thermal equilibrium.
It is particularly striking that a theory with no quantum assumptions leads to the conclusion that at zero temperature the
value of the field energy is strictly fixed at a value different
from zero. This observation may be reinforced by noting that
from the derivative of order r of the partition function Z(β)
one gets
Z (r) = (−)r E0r Z
for
β → ∞,
for
β → ∞.
whence, using Eq. (10),
r
E r → E = E0r
(44)
Thus the energy of the field at T = 0 is absolutely fixed at
the value E0 . More generally, for any system described by
the distribution W (E) of Eq. (1) the zeropoint energy is dispersionless, and the same comment applies to the excitation
energies of the field oscillators.
From the point of view of quantum theory this result is
not surprising, since in equilibrium at T = 0 only the ground
state of the system is realized, and this is an eigenstate of
the hamiltonian. At higher temperatures the rest of the energy eigenvalues are activated. However, when the field is assumed to be stochastic, the fixed (quantized) energies come
as a quite unexpected and strange result. A partial explanation
is that we are not dealing with just one independent mode of
the field, but with the averaged energy of the infinity of modes
of a given frequency contained in the cavity. This quantity
is much less fluctuating than the energy of a single mode,
and although it is still fluctuating, in the present description
these relatively small fluctuations are neglected. It should be
clear, however, that the notion that (field or particle) energy
Rev. Mex. Fı́s. 48 S1 (2002) 1 - 8
6
ENGLISHL. DE LA PEÑA AND A.M. CETTO
fluctuations completely disappear at T = 0 (or for an eigenstate) cannot be but an approximation, as follows from several arguments. For example and as remarked by Cole [21],
the mere existence of thermal equilibrium at T = 0 between
charged particles and the radiation field becomes impossible under static conditions according to Earnshaw’s theorem.
One is thus led to conclude that quantum mechanics is an approximate theory of nature, valid as long as one assumes that
energy eigenvalues are realized in isolation. Of course, this
approximation is alleviated at least in part in quantum electrodynamics, where radiative corrections to the motions and
states are allowed. These crucial points are discussed with
some detail elsewhere [29] and in the earlier work given in
Refs. 23 and 30.
A further remark should be added about the fixed energy values. The vanishing of the energy fluctuations does
not mean that all fluctuations disappear. For instance, both
quadratures of an oscillator are still fluctuating quantities (although highly correlated), even at T = 0. As an illustration,
considering q and p to be the quadratures of the field oscillators of a certain frequency ω, we have q̄ = 0, p̄ = 0, and from
the virial theorem it follows that ω 2 σq2 = σp2 = E, whence
2
σq2 σp2 =
2
E
E02
E + 2E0 E T
1
=
+ T
≥ ~2 ,
2
2
2
ω
ω
ω
4
(45)
where E T ≥ 0 is the thermal equilibrium energy, E −E0 . The
contribution E02 /ω 2 = ~2 /4 represents the minimum value
for this expression, and is an irreducible limit attained just
at T = 0; the second contribution being extrinsic (temperature dependent), it can acquire any value from zero to infinity.
We also verify that the minimun value of σq2 σp2 , just as that
of E ≥ ~ω/2, is fixed by the zeropoint field.
We can now get a better insight of the mechanism by
which the introduction of the zeropoint field leads to Planck’s
law and energy quantization instead of the classical RayleighJeans distribution and its continuous energy spectrum. To this
end we first recast Eq. (32) in the form
E = E0 + E T ,
~ω
.
(46)
e~ω/kT − 1
Now using Eq. (33) we write for the energy fluctuations of
the field oscillators of frequency ω
vacuum field. These extra interferences are here identified as
being responsible for the transition from the classical to the
quantum result. It is well known that the linear term ~ωE T is
just the one that Einstein interpreted as due to the corpuscular
(photon) structure of the radiation field at low densities. Here
we have two contrasting interpretations for the same term,
and thus for the origin of quantization.
The quantized field described by Planck’s law is a radiated field, which means that the atoms radiate bunches of
energy, endowing this field with a quantum structure. This
language applies whether one considers the walls of the container as made of atoms, or one models them by means of
harmonic oscillators. However, with the latter model a new
possibility appears, since field and matter oscillators should
be thermodynamically indistinguishable, namely, that a given
matter oscillator of frequency ω can absorb succesively n
photons of energy ~ω from the field, and thus acquire a free
(i.e., interchangeable) energy ~ωn, n = 1, 2, 3, . . . With such
a picture the quantum levels appear not as internal, but as free
energy.
As stated before, the continuous family of field states described by the different values of the constant C0 can be seen
to correspond to vacuum states that have been somehow manipulated (engineered). For example we observe that Eq. (27)
with C 0 > −1 can be used to describe a squeezed vacuum
field by writing the dispersions of q and p for an oscillator in
terms of the squeeze parameter r as usual [33]
σq2 =
E0 2r
e ,
ω2
σp2 = E0 e−2r ,
whence
Er =
1 2 2 1 2
e2r + e−2r
1
ω σq + σp = E0
= ~ω cosh 2r. (48)
2
2
2
2
For a squeezed state, |r| 6= 0 and the energy of the vacuum
field is larger than E0 = ~ω/2.
ET =
2
2
σE2 = E − E02 = E T + 2E0 E T ,
(47)
which is the famous result that Einstein used several times to
introduce the photon structure of the radiation field [31]. In
the Appendix we come back to this formula. With E0 = 0
(which would correspond to the limit of high temperatures)
2
one gets the maxwellian result σE2 = E T . In this limit all
fluctuations are due to the random interferences between the
thermal field modes of frequency ω. In the limit of very
low temperatures, however, E T → 0 and Eq. (47) reduces
to σE2 = 2E0 E T . Now the fluctuations are due to the random interferences between the modes of the thermal and the
Appendix
In this Appendix we make a brief detour to discuss the relationship between the first quantum paper by Planck in 1900,
in which he made his famous interpolation that led to the
Planck distribution and the quantization of the energy of the
oscillators, and the one by Einstein in 1909 in which he used
the complete formula Eq. (47), intimately related with his famous heuristic hypothesis about the photons in his “photoelectric” paper of 1905 [32]. We recall that from 1905 until 1909 Einstein considered the linear term alone, which he
deduced from Wien’s phenomenological approximation to
the correct (Planck) law, E = Aωe−Bω/T , an expression
valid for low temperatures or high frequencies.
Rev. Mex. Fı́s. 48 S1 (2002) 1 - 8
ENGLISHPLANCK’S LAW AS A CONSEQUENCE OF THE ZEROPOINT RADIATION FIELD
7
In his 1900 paper [1] Planck first observed that according
to classical physics one can write for the entropy S of oscillators of a fixed frequency in thermal equilibrium, with U the
mean energy (which we have been calling E)
the Planck constant being still concealed in the (by then unknown) expression for the constant that appears in U0 [34].
In his turn, Einstein realized that Planck’s distribution requires that the energy fluctuations be given by Eq. (47),
1
k
∂S
=
= .
∂U
T
U
σE2 = E T + 2E0 E T ,
2
(49)
(55)
This is immediate from the equipartition of energy, but at the
end of the 19th century matters related to this theorem were
still poorly understood, so the result was less obvious than
it appears now. If instead of the classical result one uses the
Wien approximation, written in the form (A and B are parameters, independent of the frequency)
(neither Planck nor Einstein knew about the zeropoint field,
so here 2E0 stands for ~ω) and interpreted the linear term
on the right-hand side of this expression as the result of a
discrete structure of the radiation field (in equilibrium, we
should add). Combining with Eq. (13) for the energy of the
thermal fluctuations,
U ≡ E = Aωe−Bω/T ,
σE2 T = kT 2 cV ,
(50)
(56)
one obtains once more
one gets
∂S
1
1
=
=
ln
∂U
T
Bω
µ
¶
Aω
.
U
2
(51)
kT 2 cV = E T + 2E0 E T = U 2 + 2E0 U.
(57)
∂2S
∂ 1
1 1
=
=− 2 ,
∂U 2
∂U T
T cV
(58)
Since
To simplify this result we perform a second derivative, and
obtain
∂2S
1 k
=−
.
2
∂U
Bkω U
(52)
combining both equations one gets
The equivalent classical result that follows from Eq. (49) is
∂2S
k
= − 2.
∂U 2
U
(53)
This pair of equations suggested to Planck what appears to
be the most productive of all interpolations in the history of
theoretical physics, namely to write
∂2S
k
=−
∂U 2
U (U + Bkω)
≡−
k
,
U (U + U0 )
(54)
where U0 = Bkω = const ω. As is well known, Planck
used this expression to deduce the Planck distribution and
several months afterwards he demonstrated that his quantum
hypothesis was required to arrive at Eq. (54). Thus, strictly
speaking, Eq. (54) was the first quantum law to be discovered,
1.
2.
3.
4.
∂2S
k
=−
,
2
∂U
U (U + 2E0 )
M. Planck, Annalen der Physik 1 (1900) 69.
M. Planck, Annalen der Physik 4 (1901) 553.
A. Einstein, Annalen der Physik 22 (1907) 180.
M. Planck, Verh. Dtsch. Phys. Ges 13 (1911) 138, see also Annalen der Physik 37 (1912) 642.
5. A. Einstein and O. Stern, Annalen der Physik 40 (1913) 551,
see Ref. 6 for an English translation.
6. S. Bergia, P. Lugli, and N. Zambrini, Ann. Fond. L. de Broglie
5 (1980) 39.
(59)
which is just the Planck interpolation formula [Eq. (54)
with U0 = 2E0 .
Thus we see that Planck’s and Einstein’s observations are
intimately related; one could even say that they are equivalent
in their physical meaning. However, the conclusions drawn
by each of these authors from their respective observations
were radically different. So different indeed that Planck never
accepted Einstein’s point of view, from which the latter never
retracted. For Planck concluded that Eq. (54) means quantization of the energy, in the sense of Eq. (40) (with C0 = −1, of
course), whereas Einstein interpreted Eq. (55) as exhibiting
the photon structure of the (equilibrium) field at low densities, whence he concluded that the description of the radiation
field as given by Maxwell’s equations is incomplete. This is
just the conclusion that Planck rejected.
7. W. Nernst, Verh. Dtsch. Phys. Ges. 18 (1916) 83.
8. P.W. Milonni, The Quantum Vacuum. An Introduction to Quantum Electrodynamics, (Academic Press, Boston, 1993).
9. T.H. Boyer, Phys. Rev. 182 (1969) 174.
10. D. Park and H.T. Epstein, Am. J. Phys. 17 (1949) 301.
11. M. Surdin, P. Braffort, and A. Taroni, Nature 210 (1966) 405,
see also M. Surdin Ann. Inst. Henri Poincaré 15A (1971) 203.
Rev. Mex. Fı́s. 48 S1 (2002) 1 - 8
8
ENGLISHL. DE LA PEÑA AND A.M. CETTO
12. T.H. Boyer, Phys. Rev. 186 (1969) 1304. See also Phys. Rev. D
1 (1970) 1526.
25. D.C. Cole, in Essays on the Formal Aspects of Electromagnetic
Theory, edited by A. Lakhakia, (Word Scientific, Singapore,
1993).
13. O. Theimer, Phys. Rev. D 4 (1971) 1597. See also O. Theimer
and P.R. Peterson, Phys. Rev. D 10 (1974) 3962.
26. E. Santos, Am. J. Phys. 43 (1975) 1743.
14. A.F. Kracklauer, Phys. Rev. D 14 (1976) 654.
27. O. Theimer, Am. J. Phys. 44 (1976) 183.
15. J.L. Jiménez, L. de la Peña, and T.A. Brody, Am. J. Phys. 48
(1980) 840.
28. P.T. Landsberg, Eur. J. Phys. 2 (1981) 208.
16. R. Payen, J. Phys. 45 (1981) 805.
30. L. de la Peña and A.M. Cetto, in New Perspectives on Quantum
Mechanics, Proceedings of the XXXI Latin-American School
of Physics (ELAF), edited by S. Hacyan, R. Jáuregui, and R.
López-Peña, AIP Conference Proceedings (American Institute
of Physics, New York, 1999) pp. 464.
17. T.W. Marshall, Phys. Rev. D 24 (1981) 1509.
18. J.L. Jiménez and G. del Valle, Rev. Mex. Fı́s. 28 (1982) 627, see
also Rev. Mex. Fı́s. 29 (1983) 259.
19. T.H. Boyer, Phys. Rev. D 27 (1983) 2906; 29 (1984) 2408.
20. A.M. Cetto and L. de la Peña, Found. Phys. 19 (1989) 419.
21. D.C. Cole, Phys. Rev. A 42 (1990) 1847.
31. A. Einstein, Phys. Z. 10 (1909) 185.
32. A. Einstein, Annalen der Physik 17 (1905) 132.
33. M. Scully and M. Zubairy, Quantum Optics, (Cambridge University Press, United Kingdom, 1997).
22. D.C. Cole, Found. Phys. 29 (1999) 1819.
23. L. de la Peña and A.M. Cetto, The Quantum Dice. An Introduction to Stochastic Electrodynamics, (Kluwer Academic Press,
Dordrecht, 1996).
24. D.C. Cole, Phys. Rev. A 45 (1992) 8471.
29. L. de la Peña, and A.M. Cetto, in preparation (January 2001).
34. These matters are discussed in detail in the nice introductory book by L. Garcı́a-Colı́n, La naturaleza estadı́stica de la
teorı́a de los cuantos, (Universidad Autnónoma Metropolitana,
México, 1987).
Rev. Mex. Fı́s. 48 S1 (2002) 1 - 8