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I have nothing new to present to my esteemed colleagues.
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In selecting my topic, I was thinking more of the students who are now also participating in large numbers.
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You are the ones who, so to speak, learned the quantum theory at your mother's knee.
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You know very well what the laws of the photoelectric effect are, and those of the Compton Effect,
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where the light quanta become almost palpably visible.
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And perhaps it is hard for you to understand how difficult it was,
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back when the laws of the photoelectric effect were not yet known, and the Compton Effect was unheard of,
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to find these quanta in blackbody radiation, where they are much more hidden,
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and what a mighty deed Planck achieved when he found these quanta in just this blackbody radiation.
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It was, as he said himself in his memoirs, a tortuous path.
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And I would like to show you how this path can be understood, at least psychologically.
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In Planck's scientific development, two different periods can be discerned, a very long period
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in which he was a thermodynamicist and followed in the footsteps of Rudolf Clausius, Rudolf Clausius
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who introduced the concept of entropy to physics in 1850, and then, very much at the end,
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a short period where he had turned to the Boltzmann way of thinking and attempted
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to transfer the concept of probability to blackbody radiation.
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Perhaps we would like to see this, in order to save time, by viewing slides.
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So, we start with thermodynamics and view a system's internal energy as a function of the entropy S and the volume W.
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We now know that all of these three variables are so-called extensive quantities,
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so that the property of a total system is the additive combination of the properties of the subsystems.
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The differential of the entropy or the increase of energy, we can view, as we know,
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as an increase of heat and an increase of work.
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And then, we express the second principle, the Carnot Principle by, like Clausius,
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setting dQ = TdS, where dS is the total differential of the entropy.
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And so we see that the absolute temperature T appears as the partial differential of the energy with respect to the entropy.
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T = dU/dS, partial differential taken at a constant volume.
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Now we will specialize by moving from the energy and entropy of the total system to a sub-system,
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for a unit volume because then we can introduce the entropy densities.
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The small s is the entropy density, which is the entropy for the unit volume,
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and the small u is the energy density, the energy for the unit volume.
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And then we can write this same formula, we can also invert it, dS/du (W) = T - 1, with small letters we write s/du = T - 1.
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But we can specialize even more by now also, just as Planck did, applying this formula to blackbody radiation
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and then also we can spectrally decompose the energy and the entropy of the blackbody radiation.
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And then we consider the unit of the frequency interval.
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And in this way then introduce sNu, the spectral density per frequency interval,
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and uNu, the spectral energy density per frequency interval, and then we can also write dSNu/duNu always = T - 1.
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But we can specialize even more, as the second picture will show.
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We know that in the year 1900, early in the year 1900, Lord Rayleigh showed how to count the standing waves in a cavity.
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You know that it is very easy for a cavity with a cube shape, you have done this as an exercise,
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and then Lord Rayleigh could show that for the unit volume, and also for the unit frequency interval,
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the number of standing waves in the cavity is given by the value g = 8Pi*Nu^2 - Nu is the frequency -
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divided by c^3 - c is the speed of light.
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And then we can also introduce an average energy for each standing wave in the cavity,
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which we call u1, and an average entropy for each of these waves.
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And then we can write uNu = gu1 and SNu = g*S1,
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and can now again write the fundamental formula for the average entropy
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and the average energy of such a standing wave dS1/du1 = T - 1.
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And so, Max Planck was not in the habit of operating with the standing waves in the cavity,
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instead he introduced the so-called Planck oscillator in that, based on the Kirchhoff Law,
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he knew that the thermal balance of the blackbody radiation does not depend on the nature of the walls,
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but instead only on the temperature.
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So he introduced very simple walls that were covered with harmonic oscillators of various frequencies Nu,
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and he could then show that these equations were not only the same for the standing waves in the cavity itself,
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but also for each harmonic oscillator in the wall, meaning with the same g,
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so that the average energy of such a harmonic oscillator is equal to the average energy of a Rayleigh standing wave.
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This shows us, incidentally, that we can treat these standing waves of the vacuum as linear oscillators.
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And then we can, and I want to do this, write this fundamental expression on the board, dS1/du1 or dS1/dE
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- where E is the average energy of the Planck oscillator - = T - 1.
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Or what we can also write is dS1 = T - 1 dE.
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And now, what was characteristic of Planck's considerations, was
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that he now attempted to calculate the entropy of the radiation, not as a function of the temperature,
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but instead as a function of the average energy of the oscillator,
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or what amounts to the same thing, the average energy of the standing wave in the cavity.
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And now the next picture shows how he worked from Wien's Displacement Law, which Wien established in the year 1894,
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where he was able to show that the average energy, that the energy density of the blackbody radiation must have this form.
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As we know, he discovered this by applying the Doppler principle to the cavity,
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or that u1 or E would have to = Nu, multiplied by a certain function of Nu/T.
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Now, we can invert this and say Nu/T must be a function of E/Nu.
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Nu/T a function of E/Nu.
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And if we now write this fundamental equation again, dS1 = T - 1 dE, here we can multiply by Nu by dividing Nu/T here by Nu,
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we have this expression, and because Nu/T must be a function of E/Nu,
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we see that the entropy S1 must be a function of only the variable E/Nu.
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Consequently S1 must have the form of some function of E/Nu, and I also want to write this on the board:
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S1 must be a function of E/Nu.
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I would also like to write the value of g on the board,
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which is the number of standing waves per frequency interval and per unit volume.
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Now we come to the next picture.
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In the year 1896 Wien specialised his law and proposed
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to represent the blackbody radiation by such a function, an exponential function.
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And we know that this was very well confirmed by the measurements in the visible range and in the ultraviolet range.
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So we can then write for E and u1 in this form: Alpha(prime) Nu exponential, always from this expression,
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where Alpha(prime) must be = Alpha * c^3 = 8 Pi.
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And if we now again apply the fundamental formula here, dS1/dE = T - 1, we can write 1/T,
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if we here explicitly calculate the 1/T by taking the logarithm,
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1/T = -1/BetaNu log E/Alpha(prime) Nu, and that must again = dS1/dE.
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Excuse me, that is of course a natural logarithm.
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In France we are in the habit of writing it so, not "ln".
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Now we can differentiate these relationships and obtain the second differential d2S1/dE2 = -1/Beta Nu1/E, or its reciprocal value,
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which Planck called "R", which is a linear function of the average energy of the oscillator
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if or when Wien's law is valid.
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Now it was just in the year 1900 that newer measurements by Rubens and Kurlbaum with residual rays in the far infrared showed
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that these measurements did not correspond to Wien's Law.
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And that it was more the case that the average density, the radiation density,
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could not be represented by an exponential function in a function of 1/T,
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but instead simply increased linearly with the absolute temperature T.
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This was shown by the work of Rubens and Kurlbaum.
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But at the same time, as the next picture shows, Rayleigh had shown, also early in 1900,
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by taking the equipartition theorem, by carrying Boltzmann's classic equipartition theorem over to radiation,
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that it should be that E = kT and in this case, if we now again set 1/T = k/E = dS1/dE
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and calculate the second differential, we see that there, where the Rayleigh-Jeans Law is valid for small frequencies,
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this expression must depend quadratically on the energy, and not linearly.
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We want to look at this together again in the next picture.
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So for large frequencies or small wavelengths, there where the energy of the radiation is small, this law is valid.
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And for the large wavelengths, there where the energy of the radiation is large, as this picture shows, this law is valid.
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Now Planck's approach was in attempting to combine the two laws.
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This means to find a law that results in Wien's Law for small wavelengths,
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as a limiting law, and in the Rayleigh-Jeans Law for large wavelengths.
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And in order to do this, he added, his approach was to start with this reciprocal value,
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in that he added the two terms, the linear term and the quadratic term.
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But at that time, that was a mathematical trick for him.
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Not until much later, ten years later, was Einstein able to clarify and show the physical meaning of this expression,
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that this expression has a physical meaning of the fluctuation of the radiation field,
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namely the first term is caused by the quantum nature and the second term by the wave nature of light.
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Now, this can also be written as 1/BetaNu multiplied by the difference of these two fractions,
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and then one can attempt to integrate it, which we will then do in the next picture. So, we have this function.
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And for Planck the integration resulted in just the logarithms,
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the natural logarithms, of E = kBetaNu, and an integration constant log Gamma, that had to be determined.
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Now, this can be written like this.
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And this, dS1/dE, now again has to be, according to the fundamental formula, Planck's formula here, = 1/T.
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And then one can now calculate this from here from E as a function of Nu and of T.
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And Planck stopped with this formula, which already looks very slightly like Planck's formula.
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Now we have only three constants, the Gamma, the integration constant the k, and the Beta.
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Now, in order to determine these constants, Planck used the two limiting laws. The next picture shows this.
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For a small Nu/T, this law must result in the Rayleigh Law, the Rayleigh Limiting Law,
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and that can only be possible, the exponent Epsilon then becomes = 1 + Epsilon,
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and that can only be possible if the Gamma is made = 1 or the log Gamma = 0.
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And at the other end of the spectral range for large Nu/T we must obtain Wien's law,
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excuse me, there is a Nu missing here, Alpha(prime) Nu exponential.
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And in order to obtain this, it appears that kBeta must be = Alpha(prime).
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One can consequently then replace the Beta with Alpha(prime)/k
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and finally obtain for the average energy of the oscillator or of the standing wave in the cavity this Plank's law,
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or then for the average energy, by multiplying by the g factor, this here,
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and this is Planck's law, which was then confirmed by all measurements over time.
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But Planck's work was not done yet, because his objective after all was always to find an expression for the entropy S1.
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And now the next picture shows this.
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Starting with this relationship, which was just up on the screen,
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but where we now set log Gamma = 0 and replace the Beta with k/Alpha (prime),
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this will now be integrated a second time and now reflects the average energy per oscillator by this function,
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where now again an integration constant comes into play, but that can be determined now
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because after all we know, it is up here on the board again, that the S1 must be a function only of E/mu.
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And that determines here then the integration constant and Planck entered this value for S1,
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where we see that S1 really must be a function of E/Nu or, better, of E/Alpha(prime) Nu.
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And this is the formula that Planck obtained from thermodynamics.
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But he was not at all satisfied with this derivation, because he himself said, "That interpolation was a lucky guess."
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And now he tried to capture the physical meaning of this formula.
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And then he turned to Boltzmann's way of thinking, which he had before always viewed very sceptically, or dismissively,
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because he wanted to understand entropy as a rigorous quantity and not as a probability quantity.
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But now he applied Boltzmann's way of thinking.
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And the next picture shows you his problem.
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Boltzmann had shown how one can calculate the probability of a distribution,
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namely that then one must deal with a content and a container, one must deal with a content and with a container.
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For Boltzmann, the content was a gas and the container was Maxwell's velocity space.
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And for Boltzmann, this velocity space had a continuous structure, but for him the gas was composed of molecules.
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But in his time this was not so obvious,
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because we know that Boltzmann had to fight a difficult battle against Ostwald and against Ernst Mach,
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who did not want to believe in molecules.
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But for Boltzmann, the gas was already composed of molecules,
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which means that the content had a quantum structure, but the container did not. And what did Boltzmann do now?
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In order to bring about a distribution,
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he also had to divide the container into elements of finite dimensions of the velocity space,
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meaning small elements, but finite elements.
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And these elements he called the cells, the cells of the velocity space.
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And then he was able to calculate a number of complexions, meaning a probability.
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But we do not want to look closer at Boltzmann's problems, and instead we turn now to Planck's problem.
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For Planck, the content was the radiation energy and preliminarily it had a continuous structure.
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And the container for him was the cavity.
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And Rayleigh had shown that the cavity oscillations consist of the standing waves
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and had shown how one can count these standing waves, how one can determine the g.
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And so for Planck, this cavity already had a discrete structure.
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And in order now also to be able to produce a distribution of radiation energy over these modes of the standing waves
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or the oscillators, he first had to do the opposite of Boltzmann.
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He had to make the content discrete.
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And so he introduced what he called "the radiation energy element", which he called Epsilon.
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And the next picture now shows, if we see the unit volume and the unit of frequency interval like this,
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we have the energy uNu, and this must be = g times the average energy of the oscillators or of the standing waves.
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But if we give the energy a quantum structure and the Epsilon, the units that are the elements of this radiation energy,
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we can also write uNu = N * Epsilon - N quanta of energy particles -, so that we have the relationship N/g = g/Epsilon = n.
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And now Planck searched for a formula that reflected the probability of this distribution
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and he was aware that these energy elements of the radiation are indistinguishable.
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So he did not turn to the Boltzmann formula, but instead took from combination analysis this formula,
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that we know today as the formula of Bose-Einstein statistics.
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But this formula is found in the paper from Planck from the year 1900.
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Working from this formula, and because N and g are large numbers,
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he could use Stirling's Approximation and write log WNu of the probability = g1 + n log 1+n-1,
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so that is now very familiar to you due to Bose statistics, and then use the Boltzmann postulate,
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namely the bridge between thermodynamics and probability considerations,
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in that he, following Boltzmann, set the logarithm of the probability proportional to the thermodynamic entropy.
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And actually Planck was the first to write out this formula, as he himself mentioned,
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Boltzmann never wrote it, and this Boltzmann constant k was introduced by Planck in this way.
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If sNu is the entropy of the unit volume, S1 is the entropy of the oscillator or of the standing wave,
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here we must divide by g, by the number of standing waves in the unit volume,
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and then finally obtain in this way the formula for the entropy of the radiation: S1 = k.
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And if one now compares the two formulas - the next picture please -
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the formula that resulted from thermodynamics was this one here,
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the formula that resulted from the Boltzmann probability considerations is this one.
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If we now replace the n by Epsilon and if one now, as Planck now compared these two formulas,
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he saw that the element of the radiation energy Epsilon had to be proportional to the frequency.
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Consequently Epsilon = Alpha(prime) Nu, and from now on he named the Alpha(prime) by the letter h,
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and that was then the famous Planck's action quantum, that one always views perhaps as erg*seconds
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or perhaps to be more precise, erg per unit of the frequency interval, erg per Hertz.
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And with the available measurement results, Planck was then able to calculate these numerical values for the h,
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that really comes out finite, small, but finite, and also for the k.
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And what is remarkable is that one not only obtains information in this way through the study of blackbody radiation alone,
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obtains information about the quantum structure of radiation energy, but also about the quantum structure of matter.
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One obtains not only the h, but also the k and from the k value,
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which is equal to the gas constant R divided by Loschmidt's number,
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Planck was able also to calculate Loschmidt's number and then also the elementary electrical charge.
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And he was very satisfied when a few years later,
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Rutherford obtained the same value for the elementary charge by counting the alpha rays.
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And at that time, those were the most precise values that were known at that time.
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We know that since then that has changed somewhat.
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I would now like to conclude, we can say the first formula,
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the formula from thermodynamics that is above, that Planck reported it, I believe it was on 19 October 1900.
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And then, on 14 December 1900, he announced this formula and this is where he introduced the elementary quantum.
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Traditionally, it is said that Planck is the father of quantum theory, and the child was born on the 14th of December.
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I would like to change this just a bit, without diminishing the great merit of Planck,
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but I think that every child should have a father and a mother.
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I would like to say that one hundred years ago,
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when Boltzmann introduced the cells of the velocity space, there he accomplished a quantum act.
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This was already a quantification.
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However, at that time he could not anticipate
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that the actual physical meaning of the cells was not the cells of the velocity space,
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but instead of the phase space, the phase space was first introduced by Gibbs,
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and that these cells of the phase space had very particular dimensions that are determined by Planck's action quantum.
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I would consequently like to say that Boltzmann contributed to the quantum theory with a very small spermatozoid.
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And I would like to say that Planck was the mother who then gave birth to the child on 14 December 1900.
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Perhaps you will allow me to show three portraits in conclusion:
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Here the grandfather Clausius, the father Boltzmann, with a fine beard, and Max Planck.
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Thank you.