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Planck’s law revisited
Sergei Viznyuk
Abstract
I review a textbook derivation of Planck’s formula for spatial density of radiation
energy. I point out at one inconsistency, and a couple of factitious assumptions used in
derivation. I propose a derivation more aligned with quantum mechanical principles. I
show the de-coherence of oscillator modes is the major factor in Planck’s law.
By nearly universal consent, the day of Dec.14, 1900 when Planck’s formula [1, 2] for spatial
density of radiation energy has been published, is considered [3] the birthday of quantum theory:
(1)
Since then, the formula became a staple piece of every textbook on statistical physics [4, 5].
Its close match to experimental results is touted as one of the greatest achievements of quantum
theory. For instance, the spectral intensity of cosmic microwave background (CMB) deviates from
Planck’s formula less than 0.03% [6, 7]. The Planck’s derivation [1, 2] of (1) follows from:
(2)
, where is the average energy of a resonator of frequency in a thermodynamic ensemble of
resonators, with fixed total energy of the ensemble;
is, supposedly, the number of all
hypothetically possible resonator modes per unit volume of space per unit frequency range. The
formula for the average energy is derived from the energy levels of a quantum harmonic
oscillator:
(3)
, and Boltzmann’s postulate that probability to find a member of thermodynamic ensemble in a
state with energy is proportional to
:
(4)
The reason the so-called zero-point energy term
in (4) is not included into Planck’s formula
(1) is the topic of a hundred-year controversy [8, 9]. A short argument follows here. A radiation
mode manifests itself via electromagnetic interaction with the matter. As a result of this interaction,
the energy quantum is transferred from the matter to the radiation mode, or from radiation mode
to the matter. If radiation mode is in its ground state with
, it cannot lose another
quantum to the matter. That in itself does not prove the zero-point energy does not exist. However,
the gravity which would have been exerted by zero-point energy of all hypothetically possible
radiation modes is over 58 orders of magnitude bigger than the gravity which could be derived
from empirical evidence [8]. I suggest this discrepancy is because, in textbooks, the radiation is
[implicitly] considered a classical object, which exists “out there” in the open space, and possesses
properties independent of the measurement context. Such view is contrary to the base principles
of quantum mechanics [in Copenhagen interpretation]. I argue that the radiation modes which are
present, are the ones which were actually emitted by the matter, not all the hypothetically possible
radiation modes. Since the number of oscillators in the matter, which interact with radiation is
limited, so is the number of radiation modes. A radiation mode and the corresponding matter
oscillator should be viewed as one and the same [entangled] system. Planck may have had similar
view, albeit not clearly stated. The word entanglement was not in physicist’s vocabulary at the
time. Planck’s reasoning was [2]:
Let us consider a large number of monochromatically vibrating resonator – of frequency
(per second), of frequency , of frequency , ..., with all large number – which are
at large distances apart and are enclosed in a diathermic medium with light velocity and
bounded by reflecting walls. Let the system contain a certain amount of energy, the total energy
(erg) which is present partly in the medium as travelling radiation and partly in the resonators
as vibrational energy. The question is how in a stationary state this energy is distributed over
the vibrations of the resonator and over the various of the radiation present in the medium, and
what will be the temperature of the total system…
…we first of all consider the vibrations of the resonators and assign to them arbitrary definite
energies, for instance, an energy to the resonators , to the resonators ,...
…Dividing by , by ,... we obtain the stationary value of the energy , , ... of a
single resonator of each group, and thus also the spatial density of the corresponding radiation
energy…
As it sounds, Planck implied the average energy (4) of a resonator in a given resonator group
is one and the same as the “corresponding radiation energy”. Thus, an actually present radiation
mode should have a corresponding matter resonator. The number of radiation modes is the same
as the number of matter resonators.
On the other hand, (1) still gives the correct result, even as the factor
is considered
as the number of all hypothetically possible radiation modes per unit volume of space per unit
frequency range, not just actually present modes. This contradiction stems from the way the factor
is derived in textbooks. Note, that (4) is obtained from fundamental quantum mechanical
expression (3) in thermodynamic limit of a large number of oscillators present in a given mode.
However, the expression
for the number of modes per unit volume per unit frequency
range is obtained with purely classical approach, by treating each mode as a standing wave
enclosed in a limited volume with ideally conducting walls, so as to nullify the wave amplitude at
the boundary. To combine an expression obtained from fundamentally quantum mechanical
principles, with an expression obtained from purely classical prospective, into a single formula (1)
is the inconsistency I wish to call out. Furthermore, the way the expression
is derived in
textbooks [4, 5] ought to raise eyebrows, since it is based on completely improbable assumptions:
1. that the radiation, e.g. cosmic background, can be considered as enclosed in a cavity
2. that the enclosing cavity has ideal conductor walls so the wave amplitude is a mathematical
zero at the boundary
Both of these assumptions are crucial for considering the available phase space discrete, which is
necessary for the textbook derivation of
factor. Even a miniscule deviation from ideal
conductor walls of the cavity immediately breaks the discreteness of phase space, and effectively
makes the number of hypothetically possible radiation modes of the same frequency infinite. Thus,
the factor
in (2) would have to be derived from a different context, as I do below.
A measurement of radiation intensity with e.g. a bolometer, is effectively the measurement of
the energy contained in matter oscillators inside the sensitive element of the device. Consider an
oscillator immersed into radiation field. If the system oscillator+radiation is in a state at ,
the probability to find it in the same state at , from Schrödinger equation, is:
, where
(5)
, where , are eigenstates and eigenvalues of -matrix. From (5), it follows,
,
i.e. the transition rate is zero. This result is referred to as quantum Zeno effect [10, 11]. It is the
result of a coherent coupling (entanglement) between the oscillator and radiation modes,
manifested by the phase relationship between superposed eigenstates of -matrix in (5):
(6)
From (5), the transition rate is also zero in a more general case, if phase difference
between -states is any analytic function of time, i.e. if phases of -states predictably relate to
each other. In order for the transition to happen, the superposed eigenstates of -matrix must
undergo de-coherence, i.e. the phase relation (6) must be broken. There are various mechanisms
which may cause de-coherence of -states, such as:
1. Rayleigh scattering [12, 13]
2. Brownian motion [14, 15]
3. Dispersive media [16, 17]
4. Recombination of electron-hole pairs in semiconductors [18]
It is not in the scope of this paper to consider de-coherence mechanisms in detail. Rather, I shall
pursue a generic approach. I write (5) in a more general form, given (6) may no longer hold:
, where
(7)
The probability distribution in (7) for an oscillator (e.g. a dipole) over radiation modes
is well known (see e.g. dipole radiation). For this discussion it is only important that the number
of radiation modes within a given spectral width is large, so I can later take the limit .
If the matter is in equilibrium with radiation, a transition changes oscillator energy by
with equal probability in either direction. In between transitions, the phase difference
evolves according to (6). The case of , where , is equivalent to consecutive
transitions in the same direction. In time , the phase of each of the -states in (7) undergoes a
total number
of positive and negative increments with equal probability Here, has a
meaning of mean free time between transitions, i.e. the de-coherence time. The resultant
increments are binomially distributed around mean , with variance of binomial distribution
. The variance in phase
. The
variance in phase difference is
. Here is the angular frequency.
Figure 1 shows numeric simulation of (7), with binomially distributed phases . The
calculation established the following:
(8)
The result (8) is interesting in a couple of ways. First, if , then , i.e. no transition
can occur if oscillator couples into a single radiation mode. This result can be obtained directly
from (5), since means the initial state is also the eigenstate of -matrix. Second, (8)
shows exponential decay over time, which is the characteristic feature of classical behavior, the
result of the de-coherence of radiation modes. In the limit ,
, i.e. the probability
spreads equally among radiation modes. That is the consequence of simplification
.
A more accurate calculation may require a dipole radiation distribution to be used for in (7), a
subject for a separate exercise.
From (8), in the limit of a large number of modes (), the transition rate is:
(9)
To evaluate the de-coherence time in (9), I consider a matter oscillator, formed when a set of
elements (e.g. electron-hole pairs) on the surface of the detector entangle through some medium
(e.g. electromagnetic field). An analog of such entanglement is a Cooper pair in superconductor,
mediated by phonon interaction. The surface area which encloses a set of elements in entangled
Figure 1
Blue line is a calculation of (7), with binomially distributed phases . The
red line is a plot of formula (8). The graphs were calculated using GNU Octave
code http://phystech.com/download/ph2.m with the following parameters:
•
•
•
•
The blue curve smooths out and becomes identical with red curve as .
state is limited by the coherence radius , where is the speed of light, is the
refractive index of the material, and [rad/s] is the spread in internal transition frequencies of
the entangled elements. If is the number of entangled elements per unit surface area of the
detector; the dimensionless scattering rate; the scattering frequency within spectral
width, then, the de-coherence time of the oscillator is:
(10)
, and the transition rate:
(11)
In equilibrium, the loss of a number of oscillators in a particular mode is compensated by the
radiation-stimulated induction into the mode of the same number of oscillators. The energy balance
equation is
(12)
, where is the spectral radiance of incident radiation; is the efficiency of the conversion of
the incident radiation into the oscillator energy; is the number of oscillators per unit surface area
of the detector. I have to subtract zero-point energy term
from the ensemble-average energy
in (12), because an oscillator cannot lose energy in the ground state. Combining (12) with (11):
(13)
The term in square brackets can be considered as pertaining to the incident radiation, and
parameters outside the brackets as properties of the detector. Then, (13) can be split into formula
for the spectral radiance, and formula for the detector efficiency:
(14)
(15)
, where
can be interpreted as the number of entangled elements making up one oscillator.
The Planck’s formula (1) for the spectral energy density readily follows from (14):
(16)
(17)
I have shown the Planck’s law follows from consideration of radiation and matter oscillators as
parts of the same quantum system. I have argued against considering radiation as an entity which
exists and possesses properties independent of the matter it interacts with.
References
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