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Philosophers

Mortimer Adler
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Presentations

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James Symposium
 
Max Planck

In 1900, Max Planck's hypothesized a quantum of action h and restricted the energy in oscillators radiating electromagnetic energy to integer multiples of , where ν is the radiant frequency. He then discovered a formula for the distribution of radiant energy in a black body at any temperature.

Bν (v, T) = (23 / c2) (1 / ( e hν / kT - 1) )

Planck solved the great problem of blackbody radiation by applying the statistical mechanics of the Maxwell-Boltzmann velocity distribution law for particles to the distribution of energy in a radiation field. Planck did not suggest that light actually came in quantized (discrete) bundles of energy. That was the work of Albert Einstein five years later in his photo-electric effect paper (for which he won the Nobel Prize), in which he proposed his "light-quantum hypothesis." For Einstein, the particle equivalent of light (later called a "photon") contains units of energy, where h is Planck's constant and ν is the frequency of the light wave.

Planck did not actually believe that light radiation itself existed as light quanta. His quantization assumption was for an ensemble of "oscillators" or "resonators" that were emitting and absorbing the radiation. Although the Lorentz theory of the electron was already complete, Planck did not accept electrons and instead described "the energy flowing across a spherical surface of a certain radius containing the resonator." He assumed the resonators could be described as having energy values limited to multiples of .

Note the resemblance to the Bohr theory of the atom thirteen years later, where Bohr postulated stationary states of the electron and transitions between those states with the emission or absorption of continuous waves of energy equal to !

Planck's assumption was simply a mathematical device to make the distribution of light as a function of frequency (and thus energy) resemble the Maxwell-Boltzmann distribution of molecular velocities in a gas as a function of velocity (and thus energy). In 1925, he called his work "a fortunate guess at an interpolation formula" and "the quantum of action a fictitious quantity... nothing more than mathematical juggling."

Note the resemblance between the distribution of blackbody radiation as a function of temperature and the Maxwell-Boltzmann distribution of velocities.

The similarity between the two is the rise to a maximum with a power law on one side and an exponential decline on the other. The difference is because the radiant energy (the number of photons) increases greatly as temperature goes up, but the number of molecules is held constant.

Planck in 1900 explained the spectral distribution of colors (wavelengths) in blackbody electromagnetic radiation by using Boltzmann’s principle that the entropy S of a gas is related to the probabilities W for the possible random distributions of molecules in different places in its container and with different velocities. S = k logW, where k is Boltzmann’s constant (so named by Planck. Boltzmann and Einstein used R/N) ). Boltzmann’s calculations of probabilities used the number of ways that particles can be distributed in various volumes of phase space. Planck used the same combinatorial analysis, but now for the number of ways that discrete elements of energy could be distributed among a number of radiation oscillators.

To simplify calculations, both Boltzmann and Planck assumed that energies could be considered multiples of a unit of energy, E = ε, 2ε, 3ε ... Plank regarded this quantum hypothesis as a mathematically convenient device, but not representing reality. He found the density of radiation with frequency ν to be

ρν = (8πhν3/c2) / (ehν/kT - 1).

Planck's "blackbody" radiation law was the first known connection between the mechanical laws of matter and the laws of electromagnetic energy. Planck realized that he had made a great step in physical understanding, "the greatest discovery in physics since Newton," he reportedly told his seven-year-old son in 1900.

In particular, Planck found that Boltzmann's statistical mechanics constant k = R/N, derived from the distribution of velocities of material gas particles, appears in his new law for the distribution of electromagnetic radiation energy. Boltzmann himself had never described this constant k as such. It was Planck who gave it a symbol and a name, although it is inscribed on Boltzmann's tomb in his famous formula relating entropy to probability, S = k logW

Planck established an independent and very accurate value for Boltzmann's constant. His blackbody radiation distribution law of course also includes the new Planck constant h. He called it the "quantum of action" because it had the units of position times momentum. Planck's formula led him to a value for Avogadro's number of molecules in a mole (the gram molecular weight) of a gas and an estimate of the fundamental unit of electrical charge. These gave Planck great confidence that his "fictitious" formula must be correct.

Five years later, Albert Einstein explained the photoelectric effect using "light quanta," discrete units of light energy, later called photons. Since the momentum of a particle is the energy divided by velocity of a particle, the momentum p of a photon is p = hν/c, where c is the velocity of light. To make the dual aspect of light as both waves and particles (photons) more plausible, Einstein interpreted the square of the light wave amplitude as the probable density of photons.

In fact, Planck fundamentally disliked the idea that physical quantities might be discrete and not continuous. He did not truly accept quanta of light until many years after Einstein had shown the quantization of light in his 1905 explanation of the photoelectric effect. Nevertheless, Planck's constant h lies at the heart of quantum mechanics, which introduced an irreducible and ontological randomness or indeterminacy into physics, first recognized by Einstein in his 1916 work on transition probabilities for the emission and absorption of light quanta.

Planck, along with Einstein, Erwin Schrödinger and others, opposed such indeterminism. Einstein called chance a "weakness in the theory." Planck remained convinced that determinism and strict causality were essential requirements for physical science and so must be true.

"Just as no physicist will in the last resort acknowledge the play of chance in human nature, so no physiologist will admit the play of chance in the absolute sense."

"the assumption of chance in inorganic nature is incompatible with the working principle of natural science."

"We must admit that the mind of each one of our greatest geniuses — Aristotle, Kant or Leonardo, Goethe or Beethoven, Dante or Shakespeare — even at the moment of its highest flights of thought or in the most profound inner workings of the soul, was subject to the causal fiat and was a instrument in the hands of an almighty law which governs the world."

Planck looked very closely at the problem of free will, and gave a rough version of the logical opposition between determinism and blind chance in the standard argument against free will.

"And here a question arises which seems to set a definite impassable limit to the principle of strict causality, at least in the spiritual sphere. This question is of such urgent human interest that I think it will be well if I treat it here before I come to a close. It is the question of the freedom of the human will. Our own consciousness tells us that our wills are free. And the information which that consciousness directly gives us is the last and highest exercise of our powers of understanding.

Let us ask for a moment whether the human will is free or whether it is determined in a strictly causal way. These two alternatives seem definitely to exclude one another. And as the former has obviously to be answered in the affirmative, so the assumption of a law of strict causality operating in the universe seems to be reduced to an absurdity in at least this one instance. In other words, if we assume the law of strict dynamic causality as existing throughout the universe, how can we logically exclude the human will from its operation?

Many are the attempts that have been made to solve this dilemma. The purpose which in most cases they have set themselves has been to establish an exact limit beyond which the law of causality does not apply. Recent developments in physical science have come into play here, and the freedom of the human will has been put forward as offering logical grounds for the acceptance of only a statistical causality operative in the physical universe. As I have already stated on other occasions, I do not at all agree with this attitude. If we should accept it, then the logical result would be to reduce the human will to an organ which would be subject to the sway of mere blind chance. In my opinion the question of the human will has nothing whatsoever to do with the opposition between causal and statistical physics. Its importance is of a much more profound character and is entirely independent of any physical or biological hypothesis.

The logical disjunction goes back to the critics of Epicurus' swerve.
I am inclined to believe, with many famous philosophers, that the solution of the problem lies in quite another sphere. On close examination, the above-stated alternative — Is the human will free or is it determined by a law of strict causality? — is based on an inadmissible logical disjunction. The two cases opposed here are not exclusive of one another. What then does it mean if we say that the human will is causally determined? It can only have one meaning, which is that every single act of the will, with all its motives, can be foreseen and predicted, naturally only by somebody who knows the human being in question, with all his spiritual and physical characteristics, and who sees directly and clearly through his conscious and sub conscious life. But this would mean that such a person would be endowed with absolutely clear-seeing spiritual powers of vision; in other words he would be endowed with divine vision.

Now, in the sight of God all men are equal. Even the most highly gifted geniuses, such as a Goethe or a Mozart, are but as primitive beings the thread of whose innermost thought and most finely spun feelings is like a chain of pearls unrolling in regular succession before His eye. This does not belittle the greatness of great men. But it would be a piece of stupid sacrilege on our part if we were to arrogate to ourselves the power of being able, on the basis of our own studies, to see as clearly as the eye of God sees and to understand as clearly as the Divine Spirit understands.

The profound depths of thought cannot be penetrated by the ordinary intellect. And when we say that spiritual happenings are determined, the statement eludes the possibility of proof. It is of a metaphysical character, just as the statement that there exists an outer world of reality. But the statement that spiritual happenings are determined is logically unassailable, and it plays a very important role in our pursuit of knowledge, because it forms the basis of every attempt to understand the connections between spiritual events. No biographer will attempt to solve the question of the motives that govern the acts of his hero by attributing these to mere chance. He will rather attribute his inability to the lack of source materials or he will admit that his own powers of spiritual penetration are not capable of reaching down into the depths of these motives. And in practical everyday life our attitude to our fellow beings is based on the assumption that their words and actions are determined by distinct causes, which lie in the individual nature itself or in the environment, even though we admit that the source of these causes cannot be discovered by ourselves.

What do we then mean when we say that the human will is free? That we are always given the chance of choosing between two alternatives when it comes to a question of taking a decision. And this statement is not in contradiction with what I have already said. It would be in contradiction only if a man could perfectly see through himself as the eye of God sees through him; for then, on the basis of the law of causality, he would foresee every action of his own will and thus his will would no longer be free. But that case is logically excluded; for the most penetrative eye cannot see itself, no more than a working instrument can work upon itself. The object and subject of an act of knowing can never be identical; for we can speak of the act of knowing only when the object to be known is not influenced by the action of the subject who initiates and performs the act of knowing. Therefore the question as to whether the law of causality applies in this case or in that is in itself senseless if you apply it to the action of your own will, just as if somebody were to ask whether he could lift himself above himself or race beyond his shadow.

In principle every man can apply the law of causality to the happenings of the world around him, in the spiritual as well as in the physical order, according to the measure of his own intellectual powers; but he can do this only when he is sure that the act of applying the law of causality does not influence the happening itself. And therefore he cannot apply the law of causality to his own future thoughts or to the acts of his own will. These are the only objects which for the individual himself do not come within the force of the law of causality in such a way that he can understand its play upon them. And these objects are his dearest and most intimate treasures. On the wise management of them depend the peace and happiness of his life. The law of causality cannot lay down any line of action for him and it cannot relieve him from the rule of moral responsibility for his own doings; for the sanction of moral responsibility comes to him from another law, which has nothing to do with the law of causality. His own conscience is the tribunal of that law of moral responsibility and there he will always hear its promptings and its sanctions when he is willing to listen.

It is a dangerous act of self-delusion if one attempts to get rid of an unpleasant moral obligation by claiming that human action is the inevitable result of an inexorable law of nature. The human being who looks upon his own future as already determined by fate, or the nation that believes in a prophecy which states that its decline is inexorably decreed by a law of nature, only acknowledges a lack of will power to struggle and win through.

In 1925, as the development of quantum mechanics began in earnest, Planck republished a series of articles as the book A Survey of Physical Theory. In an article on "The Nature of Light," Planck describes Einstein's insight in 1905 that led to Einstein's "light-quantum hypothesis." But Planck does not explicitly mention Einstein!

When ultra-violet rays fall on a piece of metal in a vacuum, a large number of electrons are shot off from the metal at a high velocity, and since the magnitude of this velocity does not essentially depend on the state of the metal, certainly not on its temperature, it is concluded that the energy of the electrons is not derived from the metal, but from the light rays which fall on the metal. This would not be strange in itself; it would even be assumed that the electro-magnetic energy of light waves, is transformed into the kinetic energy of electronic movements. An apparently insuperable difficulty from the view of Huygens's wave theory is the fact (which was discovered by Philipp Lenard and others), that the velocity of the electrons does not depend on the intensity of the beam, but only on the wavelength, i.e. on the colour of light used. The velocity increases as the wave-length diminishes. If the distance between the metal and the source of light is continuously increased, using, for example, an electric spark as the source of light, the electrons continue to be flung off with the same velocity, in spite of the weakening of the illumination; the only difference is that the number of electrons thrown off per second decreases with the intensity of the light.

The difficulty is to state whence the electron obtains its energy, when the distance of the source of light becomes ultimately so great that the intensity of the light almost vanishes, and yet the electrons show no sign of diminution in their velocity. This must evidently be a case of a kind of accumulation of light energy at the spot from which the electron is flung out — an accumulation which is quite contrary to the uniform spreading out in all directions of electro-magnetic energy according to Huygens's wave theory. However, if it is assumed that the light source does not emit its rays uniformly but in impulses, something like an intermittent light, it follows that the energy of such a flash, spreading outwards in all directions in uniform waves, would finally be distributed over the surface of a sphere so large that the metal considered would receive but little of it. It is easy to calculate that under certain circumstances radiation extending for minutes, even hours, would be necessary for the liberation of one electron with the velocity corresponding to the colour of the light, while, in fact, no limiting condition can be determined, for the duration of radiation necessary to produce the effects; the action certainly takes place with great rapidity. Like ultra-violet rays, Röntgen rays and Gamma rays give us the same effect, though, owing to the very much shorter wave-lengths of these rays, the velocities of the liberated electrons are much greater.

Planck here describes Einstein's "light-quantum hypothesis," his 1905 explanation for the photoelectric effect for which Einstein won the Nobel Prize. But Planck does not mention Einstein!
The only possible explanation for these peculiar facts appears to be that the energy radiated from the source of light remains, not only for all time, but also throughout all space, concentrated in certain bundles, or, in other words, that light energy does not spread out quite uniformly in all directions, becoming continuously less intense, but always remains concentrated in certain definite quanta, depending only on the colour, and that these quanta move in all directions with the velocity of light. Such a light-quantum, striking the metal, communicates its energy to an electron, and the energy always remains the same, however great the distance from the source of light. Here we have Newton's emanation theory resurrected in another and modified form. But interference, which was a bar to the further development of Newton's emanation theory, is also an enormous difficulty in the quantum theory of light, for it is difficult at present to see how two exactly similar light quanta, moving independently in space, and meeting on a common path, can neutralize each other, without violating the principle of energy

From this state of affairs arose the pressing need of the radiation theory for an investigation to find some way out of this dilemma, difficult from all sides..

What becomes of them later as light disperses — whether the energy of a quantum remains concentrated as in Newton's emanation theory or whether, as in Huygens's wave theory, it spreads out in all directions and gets less dense indefinitely - is another question of a very fundamental character, to which I have referred above.

As late as 1925, Planck (along with Heisenberg, Bohr, Schrödinger, and others) did not yet accept the reality of Einstein's light quanta.
So the present lecture on our knowledge of the physical nature of light ends, not in a proud proclamation, but in a modest question. In fact, this question, whether light rays themselves consist of quanta, or whether the quanta exist only in matter, is the chief and most difficult dilemma before which the whole quantum theory, halts, and the answer to this question will be the first step towards further development.
In the final article. "The Origin and Development of Quantum Theory," which was to be Planck's last writing on quantum theory, as he turned back to classical problems, Planck has a minimum reference to Einstein. as one of many "who made use" of his quantum of action, and still no mention of Einstein's photoelectric effect prediction which Planck described extensively in the previous article.
...the restless, ever-advancing labour of those workers who have made use of the quantum of action in their investigations.

The first advance in this work was made by A. Einstein, who proved, on the one hand, that the introduction of the energy quanta, required by the quantum of action, appeared suitable for deriving a simple explanation for a series of remarkable observations of light effects, such as Stokes's rule, emission of electrons, and ionization of gases.

Irreversibility
In 1909, Einstein first suggested that the elementary process of radiation emission may be irreversible.

In 1916, Einstein derived the Planck law of radiation and Bohr's two quantum postulates (stationary states and transitions between states with E = hν). By contrast, Planck's "discovery" of his law was accomplished by trial and error guesses at the mathematical form. Einstein derived it from more basic principles.

Planck did give Einstein more credit in his 1920 Nobel lecture, "The Genesis and Present State of Development of the Quantum Theory". He asked whether the quantum of action was a fictional quantity or would it play a fundamental role in physics, ... something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking, built as this was ... on the acceptance of the continuity of all causative connections. He gave the major credit for the second alternative to Einstein:

Experiment has decided for the second alternative. That the decision could be made so soon and so definitely was due not to the proving of the energy distribution law of heat radiation, still less to the special derivation of that law devised by me, but rather should it be attributed to the restless forward thrusting work of those research workers who used the quantum of action to help them in their own investigations and experiments. The first impact in this field was made by A. Einstein who, on the one hand, pointed out that the introduction of the energy quanta, determined by the quantum of action, appeared suitable for obtaining a simple explanation for a series of noteworthy observations during the action of light, such as Stokes' Law, electron emission, and gas ionization, and, on the other hand, derived a formula for the specific heat of a solid body through the identification of the expression for the energy of a system of resonators with that of the energy of a solid body, and this formula expresses, more or less correctly, the changes in specific heat, particularly its reduction with falling temperature. The result was the emergence, in all directions, of a number of problems whose more accurate and extensive elaboration in the course of time brought to light a mass of valuable material...

The production of photons by electron impact appears as the reverse process to that of electron emission through irradiation by light-, Röntgen-, or gamma-rays and again here, the energy quanta, determined by the quantum of action and by the vibration frequency, play a characteristic role, as could be recognized, already at an early time, from the striking fact that the velocity of the emitted electrons is not determined by the intensity of radiation, but only by the colour of the light incident upon the substance. Also from the quantitative aspect, Einstein's equations with respect to the light quantum have proved true in every way, as established by R.A. Millikan, in particular, by measurements of the escape velocity of emitted electrons...

Strangely, Planck does not give specific credit to Einstein for first seeing the problem of nonlocality that arises from wave-particle duality. Indeed, in 1920 most of the Nobel-prize winning physicists still had serious doubts about Einstein's light-quantum hypothesis and its wave nature as a "ghost field" or guiding wave for the quanta (later to be called photons)

There is in particular one problem whose exhaustive solution could provide considerable elucidation. What becomes of the energy of a photon after complete emission? Does it spread out in all directions with further propagation in the sense of Huygens' wave theory, so constantly taking up more space, in boundless progressive attenuation? Or does it fly out like a projectile in one direction in the sense of Newton's emanation theory? In the first case, the quantum would no longer be in the position to concentrate energy upon a single point in space in such a way as to release an electron from its atomic bond, and in the second case, the main triumph of the Maxwell theory - the continuity between the static and the dynamic fields and, with it, the complete understanding we have enjoyed, until now, of the fully investigated interference phenomena - would have to be sacrificed, both being very unhappy consequences for today's theoreticians.

References

Scientific Autobiography (1947)

Dynamical Laws and Statistical Laws

The Genesis and Present State of Development of the Quantum Theory (Nobel Lecture - 1920)

The Nature of Light

Improvement in Wien Spectrum, October 19, 1900a

Energy Distribution Law, December 14, 1900b

Distribution of Energy in the Normal Spectrum, 1901a

Original Papers in Quantum Physics, with Notes by H. Kangro

The Origin and Development of Quantum Theory (A rewrite of the Nobel Lecture)

Max Planck Biography, by K. A. G. Mendelssohn, from A Physics Anthology, ed. Norman Clarke, 1960

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