Planck's Theorem

Just as the effort to reconcile the mathematical descriptions of Newtonian mechanics and Maxwellian electromagnetism led to Relativity, so the effort to reconcile thermodynamics with electromagnetism led to the discovery of the quantum theory. And just as Einstein introduced postulates that he had inferred intuitively as the basis for his description of the electrodynamics of moving bodies, so Max Planck introduced a postulate the he had inferred intuitively as the basis for his description of the electrodynamics of blackbody radiation. In this essay I want to recapitulate Planck's derivation with reference to the quantum of action, not as a postulate, but as a theorem.

In the Nineteenth Century people estimated the temperature of a stove by looking at a small hole in the stove's side and noting the color of the light emanating from the hole. In 1860 Gustav Robert Kirchhoff presented an insight into that phenomenon that would lead, forty years later, to the beginnings of the quantum theory. He deduced that such light has the same spectral density as does the light that a perfectly black body would emit at the same temperature. He reasoned that the hole effectively absorbs all light falling upon it, just as a perfectly black body would do, and that inside the cavity behind the hole absorption and re-emission of the light would mix it into an equilibrium configuration that would then re-emerge through the hole. Thus, light emerging from a small hole in a large hollow body has a distribution of energy among its frequencies that has no dependence upon any properties of the body except its temperature. Though Kirchhoff was not able to devise the formula himself, he was able to demonstrate that the formula describing the spectral density of light emanating from a black body would contain only the variables describing the frequencies in the radiation and the absolute temperature of the blackbody.

In accordance with that fact, blackbody radiation represents the simplest system of emission of heat-based radiation that Planck could study. Further, because the radiation inside a cavity mimics blackbody radiation perfectly, it gave Planck an easy way to apply Maxwellian electromagnetism to his study, by applying his analysis to a cavity inside a hot thermal reservoir instead of to a black body. So Planck imagined a cavity whose walls have an absolute temperature T and sought to describe the spectral density of the radiation that filled the cavity when the radiation had reached equilibrium with the cavity's walls; that is, when the walls emitted energy as fast as they received it. In what follows I have not followed Planck's derivation exactly, but have laid out a derivation that displays the essence of what Planck did.

Although Planck used an imaginary spherical cavity in his derivation, I will use a cubic cavity for simplicity. I let L represent the length of each side of the cavity and I assert that at equilibrium the radiation filling the cavity comprises only electromagnetic standing waves. In devising a solution of the wave equation that correctly represents the radiation within the cavity I must satisfy two criteria: the reflection path around the cavity must be closed (that is, the wave must come back to its starting point in the same phase it had when it left) and the electric field on the cavity wall must equal zero (lest it generate currents in the wall that would absorb energy from the wave and thereby violate the assumption of equilibrium). Thus we have the wave equation

(Eq'n 1)

which has as its solution for a wave of length λ in this case

(Eq'n 2)

in which equation n_{1}, n_{2},
and n_{3} represent positive integers. If we insert that solution into
Equation 1 and work through the calculations, we obtain the criterion

(Eq'n 3)

Now I want to calculate the number of modes in
the cavity that satisfy that criterion. If we imagine n_{1}, n_{2},
and n_{3} as representing coordinates in some abstract space, then we
can envision Equation 3 as analogous to the Pythagorean theorem, describing the
radius of a sphere in that space. Values of n_{1}, n_{2}, and n_{3}
less than 2L/λ represent legitimate modes of vibration in the radiation field
within our cavity, so we must count all such modes; that is, we must count all
of the possible modes that lie with an abstract spherical volume of radius 2L/λ.
But we can only count modes that we represent with only positive integer values
of n_{1}, n_{2}, and n_{3}, so we must count only
one-eighth the number of modes within the abstract sphere and we must also
account the fact that each mode can hold one of two possible polarizations of
the electromagnetic wave that embodies it. So we end up with the number N of
modes as one fourth of the "volume" of our abstract sphere; that is,

(Eq'n 4)

Referring back to our cubic cavity, we now
calculate the spectral density of modes by calculating the number of modes per
unit volume per unit frequency. Dividing N by L^{3} and then
differentiating with respect to λ gives us that density as

(Eq'n 5)

in which I used the chain rule in the last step to make the substitution

(Eq'n 6)

We now need only multiply Equation 5 by the average energy per mode at any given frequency to obtain the spectral density of energy in blackbody radiation.

In his Nobel Lecture (1920 Jun 02) Planck described the problem and his approach to it: "For many years, such an aim for me was to find the solution to the problem of the distribution of energy in the normal spectrum of radiating heat. Since Gustav Kirchhoff has shown that the state of the heat radiation which takes place in a cavity bounded by any emitting and absorbing substances of uniform temperature is entirely independent of the nature of the substances, a universal function was demonstrated which was dependent only upon temperature and wavelength, but in no way upon the properties of any substance. And the discovery of this remarkable function promised deeper insight into the connection between energy and temperature which is, in fact, the major problem in thermodynamics and thus in the whole of molecular physics. To attain this there was no other way but to seek out from all the different substances existing in Nature one of known emissive and absorptive power, and to calculate the properties of the heat radiation in stationary energy exchange with it. According to Kirchhoff's Law, this would have to prove independent of the nature of the body.

"Heinrich Hertz's linear oscillator, whose laws of emission, for a given frequency, Hertz had just previously completely developed, seemed to me to be a particularly suitable device of this purpose. If a number of such Hertzian oscillators are set up within a cavity surrounded by a sphere of reflecting walls, then by analogy with audio oscillators and resonators, energy will be exchanged between them by the output and absorption of electromagnetic waves, and finally stationary radiation corresponding to Kirchhoff's Law, the so-called black-body radiation, should be set up within the cavity."

Planck worked out a description of the entropy within such a system, because from that description he could derive the average energy of the radiation of a given frequency. In that description we have the entropy as

(Eq'n 7)

in which β=1/kT. When the system reaches equilibrium its entropy reaches a maximum, so we have dS/dβ=0 and that leads to a description of the average energy,

(Eq'n 8)

All Planck had to do from that point was devise a description of the partition function Z of the oscillators in his system.

We have reached the point where Planck worked
the magic that made him famous. As Planck himself put it, "In order to find the
correct resonator entropy S it must be assumed that the energy U of a resonator
with frequency ν can only take on *discrete* energy values, to wit,*
integer* multiples of h times ν, in contrast to classical theory where U can
by *any* multiple, integer or not, of ν. We now say that U is quantized."
(note: Planck used U to represent the energy that I usually represent with E,
though I follow Planck's lead in this essay to avoid confusion of energy and the
strength of the electric field). As you can see, Planck introduced his quantum
of action as an a posteriori assumption to obtain a correct description of the
entropy of blackbody radiation, but I want to introduce it as an a priori
derivation from the finite-value theorem.

We know that light carries momentum and energy, so we should expect that the emission of light constitutes an action. Therefore, the oscillators in Planck's system must emit energy in discrete blobs whose minimum action equals Ut = h. That tells us that in the time t the wall emits a blob of energy U = h/t. Given that inverse time corresponds to a frequency, we infer that this leads to the Planck relation, U=nhν, in which n represents an integer. That relation gave Planck the partition function that yielded the correct description of blackbody radiation.

If we have a system with a number of states with
allowed energies U_{r}, then that system has a partition function

(Eq'n 9)

If, on the other hand, the system allows a continuum of values for its energy, then it has the partition function

(Eq'n 10)

We would expect naturally that we should use
Equation 10 in devising the description of blackbody radiation. Were we to do
that, we would calculate the spectral energy density using the classical average
of the energy, U_{ave}=kT. Multiplying that result into Equation 5
yields the Rayleigh-Einstein-Jeans law with its attendant "ultraviolet
catastrophe".

But Planck's hypothesis obliged him to use Equation 9 and that had a wonderful result. Planck calculated his partition function as

(Eq'n 11)

When we substitute that expression of the partition function into Equation 8, we find that the average energy in the oscillations of a given frequency conforms to

(Eq'n 12)

multiplying that result by Equation 5 gives us the spectral density as

(Eq'n 13)

That equation expresses Planck's law of the blackbody radiation, describing the amount of energy per cubic meter per unit of frequency in the radiation field emanating from a blackbody at temperature T.

But that only tells us how much energy density we have within a very narrow band of frequency. We now want to calculate the total energy density in our blackbody radiation, over the whole range of frequencies. We simply multiply Equation 13 by the minuscule element of frequency, dν, and integrate the result. We obtain

(Eq'n 14)

in which I have made the substitution x=hν/kT
for simplicity's sake. Of course, this represents the density of the radiation
at the surface from which it emanates: the radiation thins out as it expands
outward into space. The value of the definite integral on the right side of the
equation comes to π^{4}/15, so the energy density comes out to

(Eq'n 15)

That equation describes the Stefan-Boltzmann
law, which leads to the better known law of blackbody radiation emission
involving the Stefan-Boltzmann constant, σ=5.6697±0.0029 Watts per meter^{2}
per degree Kelvin^{4}.

Although at first Planck did not accept that his "quantum of action" represented a real quantization of energy, Einstein confirmed that the Planck relation does indeed represent a real quantization in regard to the absorption of radiation in his theory of the photoelectric effect. That effect was discovered by Heinrich Hertz in the course of his experiments to generate and detect the electromagnetic emissions that we call radio waves. Sunlight falling on the detector enhanced the brightness of the spark. Somehow something in sunlight freed up extra electrons to participate in the spark.

Once the phenomenon had been discovered, physicists performed experiments aimed at exploring it. They quickly discovered that the photoelectric effect displayed properties that seemed to conflict with Maxwell's electromagnetic theory. As Planck put it, "...the velocity of the emitted electrons is not determined by the intensity of the radiation, but only by the colour of the light incident upon the substance." In other words, we can torch a metal with infrared radiation until it melts and see not one electron emitted, but the slightest caress of cold metal by a thin haze of ultraviolet radiation will send legions of electrons streaming out of the metal.

Planck had hypothesized his quantum of action in reference to an array of Hertzian electromagnetic oscillators that he presumed to exist in the walls of his spherical cavity. Each of those oscillators would have a fundamental frequency and overtones, a perfectly classical concept that comes directly from Maxwellian electromagnetism, and the array would contain oscillators of every possible frequency. In a sense, then, Planck did not really create the quantum theory, but only a semi-classical version of what we have been taught. It was Einstein, who showed that light can only be absorbed in quanta, who made the radical leap and dragged Planck with him.

Subsequent successes in applying the quantum hypothesis, in explaining the structure of the atom, in explaining the specific heats of metals, and so on, the creation of what we call the old quantum theory, then solidified the hypothesis as part of modern physics.

Thus we know that electromagnetic energy can
only be emitted and absorbed in quanta.

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