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