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MAX PLANCK AND THE THEORY OF QUANTA1
G. A. Lorentz.
It rarely happens that, within a span of time scarcely encompassing a single decade, science should, as the result of the discovery of new phenomena and the growth of new theories, undergo so profound a transformation as occurred in physics 25 years ago. The expiring century brought, in rapid succession, the discovery of X-rays, radioactivity, and the Zeeman effect; the first conclusions concerning the nature of the electron were obtained, and in 1905 there already appeared Einstein’s first work on the theory of relativity.
In the middle of this remarkable period there arose the theory of quanta, which played an entirely exceptional role in the transformation of physics, since it led to the atomistics of energy and deepened our views on the significance of discontinuity in the phenomena of nature. Gradually it conquered ever broader domains. It was precisely this theory that disclosed the mystery of the structure of the atom and gave the key to understanding the language of spectra. Thus, for the physicists of our day, it has become the most necessary and reliable guide, whose indications they willingly follow. And although its propositions sometimes recall the incomprehensible utterances of an oracle, we are convinced that behind them truth always stands.
It is fitting now, with gratitude and astonishment, to recall how Planck, at a meeting of the German Physical Society on December 14, 1900, in developing his hypothesis of energy elements, laid the foundation of the theory of quanta, and to traverse once more the path that he then indicated with the hand of a master.
The task was to clarify theoretically the character of the dependence of the intensity of thermal radiation on temperature and wavelength; the basis on which the theory was built was, on the one hand, thermodynamics and statistical mechanics, and, on the other hand, Maxwell’s theory of the electromagnetic field. Since the time
Kirchhoff knew that the energy density of black radiation is a function of temperature and wavelength, independent of the special properties of the body. Boltzmann, by his theoretical derivation of Stefan’s law, and W. Wien, by his derivation of the “displacement law,” elucidated important properties of this universal function. It still remained, however, to reveal its form completely.
This was precisely the task that Planck set himself in several earlier works. The investigations of irreversible processes of radiation, published in the Sitzungsberichte der Berliner Akademie at the beginning of 1900 and combined in an article printed in the Annalen der Physik, were devoted exactly to the detailed study of the exchange of energy between matter and the ether. Matter was here schematized to a high degree; as its representatives there appeared the well-known linear vibrators, or Planck resonators. Each of these small formations had a definite number of oscillations and carried known electric charges, by means of which the interaction with the field of radiant energy took place. It was not necessary to make any definite assumptions about their structure; it was assumed only that a vibrator could lose its energy simply as a result of radiation, and not through resistances of some other origin. On the other hand, already existing rays could set the resonators into oscillation or strengthen or weaken their motion, so that, generally speaking, the energy \(U\) of a resonator could increase or decrease with time.
Planck, however, repeatedly emphasized as an essential feature of the thermodynamic treatment the following circumstance: not all rapidly and disorderly changing details of processes are subject to investigation, but only what can be detected by “macroscopic” observation. Consequently, by \(U\) one must understand the mean value computed over a time interval that embraces many periods of oscillation but is nevertheless sufficiently small that the changes undergone during this interval by the observed quantities may be neglected. Precisely the same is true for the radiation field. The subject of investigation is not individual oscillations, but the intensities of rays intersecting in various directions; moreover, for each direction and for each interval of wavelengths the intensity is measured by the quantity of energy passing through a surface element placed perpendicular to the direction of the rays, and rays are considered whose directions lie within a small solid angle. The energy density is connected in a simple way with the quantities introduced in the expression of these amounts of energy.
As a first result from purely electromagnetic considerations, a formula was derived which makes it possible to compute
the change of the vibrator’s energy with time, i.e. the quantity \(\frac{dU}{dt}\), if, for the chosen moment of time, the value of \(U\) and the intensity of the radiation are known. Hence the condition of equilibrium is also obtained, if \(\frac{dU}{dt}\) is set equal to zero.
If the energy density corresponding to the interval of wavelengths \(d\lambda\) is denoted by \(u\,d\lambda\), then the result—assuming isotropy of the field of rays—will have the form:
\[ u=\frac{8\pi}{\lambda^{4}}\,U . \tag{1} \]
Thus the problem would be solved as soon as, for each resonator, i.e. for each frequency of oscillation, the energy \(U\) were known as a function of the temperature. Then the function of radiant energy \(u\) would also be known as a function of \(\lambda\) and \(T\). At the same time it should also be noted that \(u\) determines not only the density of the radiation energy, but also the emissive power of an absolutely black body. Knowing the latter, one can then, on the basis of Kirchhoff’s law, obtain the radiation of any other body whose absorptive power is known.
Here precisely was the point from which it became clear that, using the laws which had previously been considered valid in thermodynamics and kinetic theory, it was impossible to attain the goal, and that, consequently, a new path had to be laid out. Indeed, according to the well-known law of “equipartition of energy,” in thermal equilibrium one and the same kinetic energy \(\frac{1}{2}kT\) must correspond to each degree of freedom of a particle. Planck’s linear vibrator, for which the mean values of the potential and kinetic energies are equal to one another, should therefore have the total energy
\[ U=kT, \]
and the formula for the radiation function would be
\[ u=\frac{8\pi}{\lambda^{4}}\,kT . \tag{2} \]
No delicate observations are needed in order to be convinced that experiment contradicts this result, for the formula shows no signs of the maximum which the radiation function at constant temperature actually has for a certain definite wavelength. It is likewise easy to be convinced that in many cases, especially when small wavelengths are involved,
the formula leads to excessively large energy densities and to excessively strong radiations. Consider, for example, a polished silver plate at a temperature of \(15^\circ\text{C}\), and for yellow light. Since, at normal incidence, the plate reflects approximately \(90\%\) of the incident energy, its absorptivity is equal to \(0.1\). Consequently, its emissive power in the direction of the normal must amount to \(0.1\) of the emissive power of an absolutely black body. For the latter, if equation (2) were valid, the radiation would be proportional to the absolute temperature; at a temperature of \(15^\circ\) it would therefore be approximately 50 times smaller than at a temperature of \(1200^\circ\). The radiating power of a cold silver plate would thus amount to \(1/50\) of the emission of a black body heated to \(1200^\circ\). The radiation of the latter, however, is so intense that \(1/50\) of it could not escape observation. The silver plate would have to be visible in the dark. The circumstance that this is not observed shows that the vibrators contained in the plate and corresponding to yellow light by no means possess the thermal motion that was ascribed to them in the derivation of formula (2).
Similar considerations show that the law of the uniform distribution of energy, as applied to radiation phenomena, must necessarily be abandoned. This conclusion was reached in various ways, and Rayleigh, for example, in an article published in the summer of 1900, in which he derived a formula corresponding to equation (2), remarked that this formula, for constant \(\lambda\), is valid in the limiting case of high temperatures. He tried, just as W. Wien had already successfully done earlier (1896), to replace this formula by a better one.
Planck, however, still had to provide a theoretical justification for the inapplicability of the principle of the uniform distribution of energy. In the work that has been discussed up to now, he confined himself to attempts analogous to those of Wien and Rayleigh. But a fortunate idea came to him, which subsequently led him to the definitive solution: to consider the entropy of the vibrators and of the radiation. He came to the conclusion that, in order to solve the problem, it would be sufficient to know the entropy as a function of the energy.
Planck first introduced the hypothesis according to which the entropy of a resonator with energy \(U\) has the value
\[ S=-\frac{U}{a\nu}\log\frac{U}{eb\nu}, \tag{3} \]
where \(a\) and \(b\) are two constants that still have to be determined1. Alongside this expression, an analogous formula was given for the entropy
of radiation, which included the same constants; to justify these assumptions, the change in the total entropy during the exchange of energy to which the previously found equation for \(\dfrac{dU}{dt}\) referred was calculated. Reasoning analogous to that used by Boltzmann in deriving his \(H\)-theorem showed that, both in the case when \(\dfrac{dU}{dt}\) is positive and when it is negative, i.e. for energy transitions in any direction, the total entropy of the system increases. It reaches a maximum when the equilibrium state is established, determined by the vanishing of \(\dfrac{dU}{dt}\).
After this justification of the expressions adopted for the entropy, it was possible to take the next step, which consisted in applying the entropy principle to the exchange of energy between two resonators with different numbers of oscillations. In contrast to the interaction between a resonator and radiation considered earlier, in which each time one had to deal with only a single frequency, the latter exchange, which must be imagined in order to obtain the relation between energies corresponding to different frequencies, should be regarded as a virtual change of the system. One may imagine, however, that this exchange is accomplished through the mediation of some kind of matter. Everyone knows Planck’s “carbon grains,” which he often used in considering analogous questions.
The result of the new considerations was that, for two vibrators with arbitrary numbers of oscillations, at equilibrium the expression
\[ \frac{dS}{dU} \]
must have the same value. It is natural to identify it with the reciprocal of the absolute temperature, and the subsequent method consists in applying to the resonators the known thermodynamic relation
\[ \frac{dS}{dU}=\frac{1}{T}. \tag{4} \]
If, for \(S\), one substitutes the value (3), then one obtains the value of the energy of the resonator as a function of \(T\) and \(\lambda\), and then, from (1), the radiation function:
\[ u=\frac{8\pi bc}{\lambda^5}e^{-\frac{ac}{\lambda T}}. \tag{4} \]
Finally, comparison with observations gives the values of the constants \(a\) and \(b\). The result agrees exactly with the above-named law
radiation law of Wien, whose approximate validity had just then been shown by the subsequent works of Paschen, as well as of Lummer and Pringsheim.
Planck, however, remained dissatisfied with his conclusion. The need to eliminate arbitrariness in the formulae for entropy gave him no peace, and in an article published in April 1900 in Annalen der Physik: “Entropie und Temperatur strahlender Wärme,” he returns again to this problem. In doing so he once more begins with the calculation of the change of entropy that accompanies the exchange of energy between the vibrator and the radiation field, and here he uses his earlier formula. This time, however, no definite assumption was made concerning the relation between \(U\) and \(S\). The result obtained was as follows: if the energy of the resonator differs by an amount \(\Delta U\) from that value \(U\) which it should have in equilibrium, and if its energy changes by \(dU\) during the interaction, then the change in the total entropy of the system, which must be positive, will be:
\[ dU \cdot \Delta U \cdot \frac{3}{5}\frac{d^2 S}{dU^2}. \]
Obviously, the stability of equilibrium requires that \(dU\) and \(\Delta U\) have opposite signs; the energy of the resonator must, as a result of the interaction, decrease if initially it was greater than corresponds to the equilibrium state. Thus Planck assumes:
\[ \frac{3}{5}\frac{d^2 S}{dU^2} = -f(U), \tag{5} \]
where \(f\) is a positive function of \(U\), and accordingly writes for the increase of entropy
\[ - dU \cdot \Delta U f(U). \tag{6} \]
The result shows that, if one relies only on the law of increase of entropy, then for \(S\) one may adopt very different functions of \(U\). In order to obtain a definite radiation formula, it is therefore necessary to introduce a further restrictive condition, which Planck thought to obtain in the following way.
Let the system contain a large number \(N\) of resonators, all having identical properties and all constantly in one and the same state of motion. If for each individual resonator the value of the energy in equilibrium is \(U\), the deviation from this value is \(\Delta U\), and the change occurring in a short interval of time is \(dU\), then the corresponding quantities for the whole group will be:
\[ U_N = NU,\quad \Delta U_N = N\Delta U,\quad dU_N = NdU. \]
MAX PLANCK AND THE THEORY OF QUANTA
Since the processes that take place in the various vibrators may be regarded as independent of one another, the change in the entropy of the system will be \(N\) times greater than for a single vibrator. On the other hand, the reasoning that led to expression (6) can be applied to \(N\) resonators. Therefore the equation must hold
\[ dU_N\,\Delta U_N\, f(U_N)=N\,dU\,\Delta U\, f(U), \tag{7} \]
i.e.
\[ N f(U_N)=f(U) \]
or
\[ NU f(UN)=U f(U). \]
This proves that the function \(U f(U)\), when its argument is changed, remains constant, and therefore
\[ f(U)=Const \]
or, according to equation (5),
\[ \frac{d^2 S}{dU^2}=-\frac{\alpha}{U}, \]
where the positive constant \(\alpha\) may still depend only on the number of oscillations.
Hence it follows:
\[ S=-\alpha U\log(\beta U), \]
where the second constant \(\beta\) likewise depends on \(\nu\). Since from Wien’s displacement law it follows that \(S\) can depend only on \(\dfrac{U}{\nu}\), formula (3) is necessarily obtained, and with it Wien’s radiation law.
Thus the impression was produced that by theoretical means one could arrive only at this latter law. However, when in the following months the measurements of Rubens and Kurlbaum showed beyond any doubt the inapplicability of Wien’s formula, and various researchers proposed modifications of it, Planck also took his proof under doubt. He therefore made an attempt to improve the formula, and, as later turned out, a very successful attempt. In a communication delivered on October 19, 1900, to the Physical Society, “Über eine Verbesserung der Wienschen Spektralgleichung,” where Planck proceeds from the relation denoted above by (7), we read:
“In the functional equation, the expression on the right-hand side undoubtedly represents the named change of entropy, since \(N\) exactly identical processes proceed quite independently of one another, and the change of entropy in them is simply
are added. Nevertheless, I came to the conclusion—though one not easily grasped and in any case difficult to prove—that the expression on the left, generally speaking, does not have the significance that I had previously ascribed to it. In other words, the values \(U_N\), \(dU\), and \(\Delta N_N\) are wholly insufficient to determine the desired change of entropy, and that for this one must also know \(U\) itself. Following this line of thought, I ultimately arrived at the construction of entirely arbitrary expressions for the entropy which, although more complicated than Wien’s expression, nevertheless satisfy all the requirements of thermodynamic and electromagnetic theory to the same extent as the latter.
Among the expressions constructed in this way, one in particular surprised me, which in its simplicity comes next after Wien’s expression; and since the latter is insufficient to represent all observations, the expression just mentioned deserved to be tested in detail. From this expression one obtains
\[ \frac{d^2 S}{dU^2}=-\frac{\alpha}{U(\beta+U)}. \]
This expression is the simplest among all expressions that give \(S\) as a logarithmic function of \(U\) (to which the theory of probabilities leads); for small values of \(U\) it passes over into Wien’s expression.
Since, according to the displacement law, \(S\) is a function of \(\frac{U}{\nu}\), and consequently its second derivative has the form \(\frac{1}{\nu^2}F\!\left(\frac{U}{\nu}\right)\), \(\beta\) must necessarily be proportional to the number of oscillations \(\nu\), while the constant \(\alpha\) [which according to (5) is negative] must be proportional to the square of \(\nu\). Taking this into account, Planck arrives at the formula with two constants
\[ u=\frac{\varepsilon\lambda^{-5}}{e^{\frac{c}{\lambda T}}-1}. \]
At the same time he also obtained his famous formula for radiant energy. True, a complete derivation of it from general principles had not yet been given, but only a few weeks passed and Planck was already able to present this derivation. In the exposition that Planck gave of his theory, he first of all formulates anew, but now with greater decisiveness, the objection to the derivation of Wien’s law. “It is first necessary, in the series of conclusions that lead to Wien’s law of the distribution of energy, to find the link that permits changes; then this link must be removed from the chain and another found in its place.”
This weak point, upon renewed examination, he finds in the assumption lying at the basis of equation (7): “For an infinitely small
in a reversible change, in which the system of \(N\) identical resonators, placed in one and the same stationary radiation field, is almost in thermal equilibrium, the increase of their total entropy \(S_N=NS\) depends only on their total energy \(U_N=NU\) and its changes, and not on the energy \(U\) of the individual resonators.”
In fact, as Planck says, entropy presupposes “disorder,” and therefore precisely the inequality of the separate values of the energy must play an essential role. Proceeding further along this path, he turns to the calculation of the entropy of the resonators, and in doing so uses Boltzmann’s principle, according to which the entropy of a system in a given state is determined by the “probability” \(W\) of this state, with which it is connected by the formula
\[ S = k \log W . \]
Thus it is necessary to find the probability that \(N\) resonators together possess the vibrational energy \(U_N\). For this it is necessary (and herein lies the kernel of the theory of quanta) to imagine \(U_N\) not as a continuous, indefinitely divisible quantity, but as a discrete quantity, consisting of an integer number of finite, equal parts. If we call one such part, an element of energy, \(\varepsilon\), then at the same time we obtain
\[ U_N = P\varepsilon, \]
where \(P\) is an integer, generally speaking a large number, while for the time being we leave open the question of the magnitude of \(\varepsilon\).”
Planck next calculates the number of ways in which the distribution of \(P\) elements of energy among \(N\) resonators can be carried out, the number of “complexions,” as he called them, using Boltzmann’s term. From the theory of combinations this number is obtained as
\[ \mathfrak{N}=\frac{(N+P-1)!}{(N-1)!\,P!} \]
or, with an approximation sufficient for our purpose,
\[ \mathfrak{N}=\frac{(N+P)^{N+P}}{N^N P^P}, \]
The following hypothesis is laid at the foundation of the further calculation: “The probability \(W\) that \(N\) resonators in the aggregate possess the vibrational energy \(U_N\) is proportional to the number \(\mathfrak{N}\) of all complexions possible in the distribution of the energy \(U_N\) among \(N\) resonators.” Consequently:
\[ S_N = k \log \mathfrak{N} = \]
\[ = k\{(N+P)\log(N+P)-N\log N-P\log P\} \]
or, taking into account formula (8) and denoting the mean energy of the resonator \(\dfrac{U_N}{N}\) by \(U\) and its entropy \(\dfrac{S_N}{N}\) by \(S\), we obtain:
\[ S=k\left\{\left(1+\frac{U}{\varepsilon}\right)\log\left(1+\frac{U}{\varepsilon}\right)-\frac{U}{\varepsilon}\log\frac{U}{\varepsilon}\right\}. \tag{9} \]
It follows hence, according to (4) and (1),
\[ U=\frac{\varepsilon}{e^{\frac{\varepsilon}{kT}}-1}, \tag{10} \]
and
\[ U=\frac{8\pi\varepsilon}{\lambda^4}\cdot\frac{1}{e^{\frac{\varepsilon}{kT}}-1}. \tag{11} \]
Finally, the law of energy distribution is again used. From (9) it is directly evident that the energy element \(\varepsilon\) must be proportional to the number of oscillations \(\nu\) of the resonator; consequently,
\[ \varepsilon=h\nu, \]
where \(h\) is the second universal constant (alongside \(k\)). Then the formula for \(U\) is transformed as follows:
\[ \frac{8\pi hc}{\lambda^5}\cdot\frac{1}{e^{\frac{ch}{\lambda kT}}-1}. \]
This is the final form of Planck’s law, of which it may be said that its derivation will forever remain one of the most outstanding achievements in theoretical physics. In order to evaluate it properly, we must not lose sight of the fact that Planck could easily have said: since resonators can, as is self-evident, give up or receive energy in arbitrarily small quantities, reality corresponds to the limiting case to which we approach by making the energy element continuously decrease; in doing so he would have returned to the uniform distribution of energy. Instead, the fortunate conjecture came to him to regard the quantity \(\varepsilon\) as finite not only for purposes of calculation, but also to leave it finite. It should not be forgotten, however, that such fortunate conjectures are the lot of those who have earned them by hard work and profound reflection.
However, now that we already have the theory, it is easy to see that Planck’s hypothesis corresponded exactly to the purpose it was meant to serve: a simple argument shows that the finite magnitude of the energy element must indeed lead to a deviation
MAX PLANCK AND THE THEORY OF QUANTA
from a uniform distribution; namely, if \(\varepsilon\) increases together with \(\nu\), then these deviations will be in the sense that the higher frequencies will receive relatively less energy than the lower frequencies. One may, for example, in order to obtain the radiation formula, proceed in the same way as Boltzmann did in his derivation of the Maxwellian distribution of the velocities of gas molecules. The lottery in which both the molecules of the body and the resonators of various frequencies contained in it take part must decide the question of the distribution of a given amount of energy; at the same time the molecules are prepared to accept an arbitrarily small amount of energy, whereas the resonators, on the contrary, require finite portions, and portions that are the larger the greater their number of oscillations. It is easy to understand that, with a limited store of energy, those particles which are the most animated will in the end receive the least of all. Thus the finite magnitude of the element of energy, increasing with the number of oscillations, makes the degrees of freedom with higher frequencies less effective. If it is assumed that the intensity of the elastic or quasi-elastic forces corresponding to one degree of freedom continually increases, then the energy which the system receives in these degrees of freedom becomes ever smaller. Thus, in the end, we approach the limiting case of an unchangeable, motionless state excluding all motion.
To these recollections concerning the origin of Planck’s theory it will be permissible to add a brief glance at its fruits. Numerous physicists, old and young, in all countries are engaged in developing it, and Planck himself, continuously from year to year, even under circumstances when work was difficult for him, continues his investigations. He is above all interested in a profound grounding of the theory and in a clear elucidation of its meaning. The circumstance that the doctrine of energy elements was able to transform itself into a general theory of quanta must be attributed to the astonishing capacity for adaptation possessed by this doctrine. So long as one had to deal with simply harmonic oscillations, the original concept of an element of energy proved quite sufficient. Later, when it became necessary to “quantize” other processes, wholly or conditionally periodic, and sometimes even nonperiodic, one had to deal with the values of a “phase integral” or with the magnitude of a bounded region of “phase space.” The quantum conditions written for such cases always consist in the fact that the corresponding quantity can have only values that are integral multiples of some unit value, and into this unit value there always enters the constant \(h\). We have now already come to the point where this constant must determine not only the intensity of radiation and the wavelength at which it has a maximum, but where it plays a role—
among many other quantitative relations. In combination with other physical quantities it determines—to name a few examples—the specific heat of a solid, the photochemical action of light, the paths of electrons in the atom, the wavelength of spectral lines, the frequency of X-rays, the speed with which gas molecules can rotate, and even the distances between the particles of which a crystal is composed. It would be no exaggeration to say that, in our picture of the world, the quantum conditions are what hold matter together and protect it from losing all its energy through radiation. And that in all cases these are real relations is convincingly shown by the striking agreement of the values of \(h\) calculated from various phenomena—values which, moreover, differ little from the number obtained 25 years ago by Planck from the experimental data then at his disposal.
As for the connection with the old mechanics, what is especially gratifying is the fact that—as the theory of adiabatic invariants shows—it is precisely those quantities that are determined by the quantum conditions which preserve their values unchanged when the conditions of motion of the system are slowly varied. It is also no accident that the displacement law associated with this theory can be used for the most immediate determination of the element of energy.
Of course, the fusion of the new ideas with classical mechanics and electrodynamics is still only a dream of the future, and we are still far from such a quantum mechanics, at whose foundation discontinuity would be laid. Yet even in this direction highly promising attempts have already been made.
Planck had the good fortune, to the joy of all who value and honor him as a scientist and as a human being, to witness, with his creative powers fully intact, the influence of his ideas. May it be his lot for many years yet to rejoice in the successes of the theory of quanta.