## fredag 7 mars 2014

### Planck's Nobel Lecture: The Story of the Fictional Quantum

Planck received the 1918 Nobel Prize in Physics in 1919 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". During the selection process in 1918, the Nobel Committee for Physics decided that none of the year's nominations met the criteria as outlined in the will of Alfred Nobel (because energy quanta did not meet the criteria?).

Planck tells in his 1920 Nobel Lecture the story how he gave birth to the quantum mechanics of modern physics in fundamental break away from classical electromagnetic wave theory by resorting to probability arguments earlier used by Boltzmann in thermodynamics,  based on a smallest quantum of energy or quantum of action (Elementaren Wirkungsquantum) named Planck's constant $h =6.626\times 10^{-34}\, Jouleseconds$:
• The duty imposed upon me today is to give you the story of the origin of the quantum theory in broad outlines and to couple with this, a picture in a small frame, of the development of this
theory up to now, and its present-day significance for physics.
• 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.
• For this reason, I busied myself, from then on, that is, from the day of its establishment, with the task of elucidating a true physical character for the formula, and this problem led me automatically to a consideration of the connection between entropy and probability, that is, Boltzmann's trend of ideas; until after some weeks of the most strenuous work of my life, light came into the darkness, and a new undreamed-of perspective opened up before me.
• Nevertheless, the result meant no more than a preparatory step towards the initial onslaught on the particular problem which now towered with all its fearsome height even steeper before me.
• The first attempt upon it went wrong, for my original secret hope that the radiation emitted from the resonator can be in some characteristic way or other distinguished from the absorbed radiation and thereby allow a differential equation to be set up, from the integration of which one could gain some special condition for the properties of stationary radiation, proved false.
• For the numerical treatment of the indicated consideration of probability, knowledge of two universal constants is required, both of which have an independent physical meaning, and whose subsequent evaluation from the law of radiation must provide proof as to whether the whole method is to be looked upon as a mere artifice for calculation, or whether it has an inherent real physical sense and interpretation.
• The first constant is of a more formal nature and is connected with the definition of temperature. This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it - a peculiar state of affairs, which can be explained by the fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant.
• The explanation of the second universal constant of the radiation law was not so easy. Because it represents the product of energy and time (according to the first calculation it was 6.55 x 10-27 erg sec), I described it as the elementary quantum of action.
• Whilst it was completely indispensable for obtaining the correct expression for entropy - since only with its help could the magnitude of the "elementary regions" or "free rooms for action" of the probability, decisive for the assigned probability consideration, be determined - it proved elusive and resistant to all efforts to fit it into the framework of classical theory.
• Either the quantum of action was a fictional quantity, then the whole deduction of the radiation law was in the main illusory and represented nothing more than an empty non-significant play on formulae, or the derivation of the radiation law was based on a sound physical conception. In this case the quantum of action must play a fundamental role in physics, and here was something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking, built as this was, since the establishment of the infinitesimal calculus by Leibniz and Newton, upon the acceptance of the continuity of all causative connections.
• 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 forwardthrusting 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.
We see, that Planck recalls his original view from 1900 in his derivation of Planck's law of radiation of the quantum of action as a fictional quantity in a empty non-significant play with formula,  and then indicates that he eventually with pushes from Einstein and experiments changes view, but still does not consider his "special derivation" using the quantum as significant.

In the previous post we saw that Planck's constant $h$ enters only in high-frequency cut-off of frequencies $\nu > \frac{T}{h}$ with $T$ temperature in Kelvin and then with a value of about  $4\times 10^{-11}$ which relates to frequencies in the visible range of about $10^{14}$ and temperatures around $4000$. As such, Planck's constant can be determined from macroscopic experiments and the high-frequency cut-off can be seen as an effect of finite precision wave mechanics explained as Computational Blackbody Radiation.

Today Planck's constant $h =6.626\times 10^{-34}\, Js$ remains as a smallest quantity of action used in Planck's derivation of Planck's law as an "artifice for calculation" and a physically non-significant fictional quantity.

In his lecture Planck struggles to convince himself seeking support in relativity theory and wild speculations:
• After all these results, towards whose complete establishment still many reputable names ought essentially to have been mentioned here, there is no other decision left for a critic who does not intend to resist the facts, than to award to the quantum of action, which by each different process in the colourful show of processes, has ever-again yielded the same result, namely, 6.52 x 10-27 erg sec, for its magnitude, full citizenship in the system of universal physical constants.
• It must certainly appear a unique coincidence that just in that time when the ideas of general relativity have broken through, and have led to fantastic results, Nature should have revealed an "absolute" in a place where it could be least expected, an invariable unit, in fact, by means of which the action quantity, contained in a space-time element, can be represented by a completely definite non-arbitrary number, and thereby divested itself of its (until now) relative character.
• But numbers decide, and the result is that the roles, compared with earlier times, have gradually changed. What initially was a problem of fitting a new and strange element, with more or less gentle pressure, into what was generally regarded as a fixed frame has become a question of coping with an intruder who, after appropriating an assured place, has gone over to the offensive; and today it has become obvious that the old framework must somehow or other be burst asunder. It is merely a question of where and to what degree.
•  If one may make a conjecture about the expected escape from this tight comer, then one could remark that all the signs suggest that the main principles of thermodynamics from the classical theory will not only rule unchallenged but will more probably become correspondingly extended. What the armchair experiments meant for the foundation of classical thermodynamics, the adiabatic hypothesis of P. Ehrenfest means, provisionally, to the quantum theory; and in the same way as R. Clausius, as a starting point for the measurement of entropy, introduced the principle that, when treated appropriately, any two states of a material system can, by a reversible process, undergo a transition from one to the other, now the new ideas of Bohr's open up a very similar path into the interior of a wonderland hitherto hidden from him.
• Be that as it may, in any case no doubt can arise that science will master the dilemma, serious as it is, and that which appears today so unsatisfactory will in fact eventually, seen from a higher vantage point, be distinguished by its special harmony and simplicity. Until this aim is achieved, the problem of the quantum of action will not cease to inspire research and fructify it, and the greater the difficulties which oppose its solution, the more significant it finally will show itself to be for the broadening and deepening of our whole knowledge in physics.