George Uhlenbeck – Session IV

Kuhn:

There’s clearly a whole shift of perspective on what it means to have statistics involved there. Didn’t it occur while you were at Leiden with Ehrenfest?

Uhlenbeck:

I mean my dissertation was on it, you see. That was all done in the fall of ‘26. You see, the spring of ‘26 was the spin, really, and the quantum mechanics, especially the Schrodinger equation. Ehrenfest and I worked extremely hard on the Schrodinger equation and the selection rule for the principal quantum number which I thought I had found but which was wrong… Sam was then for two months, as I told you, in Copenhagen. I think with the Lorentz fellowship. He tried to work on helium, without much success at that time.

Kuhn:

This was helium using the Schrodinger —?

Uhlenbeck:

Using the spin and always the model, you see. Because then it was clearly that the difference between the singlet and triplet level must be due to a difference of the spin orientation. And then the question is, “why is it so large? Because the magnetic interaction was of course very small. And then he and Bohr worked very hard to think of some kind of a model where you could magnify that, without much success. And quite about the same time, or maybe a bit later, Heisenberg — it must have been in April, May, something like that — gave the correct explanation, as an exchange attraction. So that year or that spring it was mainly Schrodinger.

Kuhn:

what about the Bose-Einstein paper?

Uhlenbeck:

That didn’t at that time involve us at all, at least not in that spring. That came then in the fall, you see, because then it was also more or less clear. Ehrenfest knew that very well of course, the Bose papers. He had been involved in the controversy, that this was not something which was obvious, but that this was really a new assumption. That was in his papers. And at that time also the Schrodinger paper appeared, which was really the paper just before the quantum theory series. I think it was in Phys. ZS. And that we studied also, and that made a great impression both on Ehrenfest and on me, because it was so clear. And then in the fall Ehrenfest and I said … that we must now put everything together; he thought that that would be a proper thesis for me, to make this nicely systematic and see how it can be all said from one point of view. And there was of course always these two aspects of these statistics. Either you look upon the radiation as quantized oscillators, and use the good old statistics. Or look upon it as photons, and then do something funny with the counting of states. At that time also — It must have been beginning ‘27 — the Dirac paper came out, which translated it in terms of the symmetric eigen function. We worked on it. It was mainly a systematic paper. You did everything the same way; you did it always with the method of steepest descents, with the (Fowler) method, although that was against Ehrenfest’s grain… Too many complex integrals. But I insisted on it, that we do it. Because I like these things, and it was also quite simple. And then to my great distress — I was just then starting — (Ehrenfest said, “Now start typing it up”,) And Fowler came out — R. H. Fowler with really about the same thing as we did. And that was a little blow, but, Ehrenfest says, “Well don’t bother. We will not write a paper about it, but you will write a thesis about it anyway.” And that was ‘27. I was all this time assistant of Ehrenfest, of course.

Kuhn:

You had Fermi statistics in this too.

Uhlenbeck:

Fermi statistics was there also. And there again, as with the radiation, you must either speak about what we called the spin oscillators. They could only have two states. Or you must do it with particles. Of course in a sense it was formal, because even if you do it with oscillators you must, for material systems, put in that the sum of the particles is n; which is of course for such an oscillator view completely unnatural. For the radiation you didn’t have that. If you then translate it to the Bose ideal gas, it was a bit formal, but still you could always say it this way.

Kuhn:

Was this aspect of the counting bothersome? Did it trouble you?

Uhlenbeck:

No. You see, you knew what had to come out, so it was clear how you had to do it. And the basic assumptions were on the microscale, that means on the energy levels in the box. You must say that the distribution of particles on these levels is, for Bose statistics, always one; for Fermi statistics it must be zero as soon as there are two particles, and otherwise it is also one; and Boltzmann is n factorial over — And already if you say it that way, you see clearly that the Boltzmann statistics are always in between the two results. Only at the end of that, and it was of course incorporated in my dissertation, it became clear to us that in a certain sense the Bose and the Fermi statistics were the only ones. Because they are the symmetric, and antisymmetric eigen functions, and there was so to say nothing in between that you could really do.

Kuhn:

That was partly due to the Dirac paper, I take it.

Uhlenbeck:

Partially to the Dirac paper. And then immediately we said, but how should you then say the Boltzmann statistics? And then it was very clear that you had to say, “Well, you have to take all the eigen functions. Not only the symmetric ones, but you had to take them all. Well it was a little exercise to show that that’s equivalent to Boltzmann statistics. And that Ehrenfest and I wrote a little note about. That was one of our collaborations. And that was an excitement; that was very interesting. That was all before my dissertation. Then Ehrenfest went away. I think he was in Paris, or something or other. And I got suddenly a typical Ehrenfest postcard. Maybe I still have it, or I gave it to Martin Klein. He says, “Fermi statistics means the impenetrability of matter.” Exclamation sign; exclamation sign. “See you in Leiden. Come 10 tomorrow morning.” What he had seen was the following — unfortunately on a one-dimensional model. If you take the combined wave function and require that the wave function has to be zero if the coordinates of two points are the same, then they cannot (penetrate); that’s the impenetrability. Then he thought that the Fermi statistics followed from that, which is true in one-dimension, but not in two or three. For a while we were very excited about it. It was typical for Ehrenfest that now he saw why. Of course! Matter with impenetrability is a fundamental physical action and that it had Fermi statistics as a consequence, that was a great thing. Then we wrote a paper immediately, also, on something which always bothered him. That was the Einstein mixing paradox. You take say two Bose gases or two Fermi gases, and you take a mixture of these, and then you think of it that particles become more and more alike. Now there was of course the well—known Gibbs mixing paradox, that the entropy changes by R, but this had no effect on any physical properties of the mixing. (Even things which were very close were of course physically undistinguishable in the old statistics from their equal, although the entropy made a jump.) That was an old question, and that we knew alright.

Kuhn:

Had this question been worried about very much?

Uhlenbeck:

Ja, in the old days people worried a bit about it…It was clear that it was due to the counting. The number of ways changed suddenly because you make suddenly two things indistinguishable, where previously they were distinguishable… We had heard of it, and Ehrenfest had explained why in his lecture, and that was all. But here then, something different happened. Now, as soon as the two particles were equal, in the Fermi gas, the Pauli principle begins to work, and begins to push them up to higher energies suddenly. And that means that suddenly the pressure will jump. That seemed paradoxical — the Einstein paradox. We wrote a paper on this, which was based really on this impenetrability idea. This was the reason for it; that they suddenly became impenetrable. Now all that was wrong. But one must say of Ehrenfest, he is one of the few people, when they write a wrong paper, write a note about it, telling that it was wrong. And he took it all back. And that is in his collected works, both papers, and also his retraction. It then became clear relatively fast afterwards that this in one dimension helped so much, but that in two dimensions, when you make it zero only on a little sphere, that doesn’t (change the frequencies) very much. And therefore it was not the correct (proof). But it was, for a moment an excitement. It was about April of that year, I think. Being Ehrenfest’s assistant, I was in charge really of taking care of several of the colloquia for the younger students. He had always this pedagogical principle that everybody had to teach the younger students.

It had always been done in colloquia. I had always to run some of those for the younger students. And I was anyway quite busy. It was clear that I would never write a dissertation if I stayed in Leiden. And then he sent me off with a Lorentz fellowship. He said now you go to Copenhagen and there you write your dissertation. And Sam was there then too, and then for two months we did essentially nothing else but writing. That was all. It was high pressure. Boy was it high pressure, writing this dissertation. And the last day of the academic year — it was already all arranged, Ehrenfest had made the dates fixed, so it had to be printed, and a correction had to be read, and the whole book had to be read — Well, it all came through, and we got the degree on the same day. At that time we had already the job in America, so we left in August, just about a month after our dissertation. Oh, and then there was also, with regard to all these statistics, there was also the question of the Einstein-Bose condensation, which of course Einstein had predicted. And that was a curious argument. I tried to do it a little bit better, and then thought that I could prove that it was wrong, that there was no discontinuity whatsoever, that there was no Einstein condensation. And Ehrenfest was highly impressed and he believed it too.

He had some correspondence with Einstein about it, which Martin Klein at the moment is studying. It was a very humorous exchange of letters. One of them had all such citations from Goethe in it. Well — I forget. Very funny — typical Ehrenfest…He was so to say kidding Einstein about it, that he made this mistake, you see. He says, “Ja, but when the great men so to say sleep, then there is something for younger people to do.” But Einstein did not give in. He said “No, it is not wrong. It is not wrong.” “You have not understood it,” he says. And he never completely gave in, although he says, “Ja, it was not quite right the way I said it.” And of course at present we know really that we were both right. Namely, that if you do the Bose statistics with a finite number of particles, there is no Einstein-Bose condensation. The condensation is always a limit property, for infinite number and infinite volume, but with finite density. And Einstein had always silently assumed that, without making it clear. And I don’t think he was clear that that was essential. At least he didn’t say it in so many words. I put it in my dissertation. There is a part in my dissertation in which I said there is no Einstein-Bose condensation. Then later when I worked on the condensation problem, then I understood really. When I finally understood that it was a limit property, then it was clear that it was still existent. So I was in a sense fundamentally wrong, but not so to say the mathematics. That was also part of that same period, that I found that out.

Kuhn:

George, I’ve got a question. I don’t know whether I can make it clear. Today one speaks of particles that obey Fermi statistics, particles that obey Bose statistics…In talking about the derivations with you, I just wanted to say, “Now what’s the situation? I’ll write down a different sort of probability formula.” You wanted to say, “It’s a question as to whether these obey Boltzmann statistics as oscillators or whether they obey the Bose statistics as photons.” Now, at the beginning how was that? Did one start very early thinking of these as different sorts of statistics?

Uhlenbeck:

Ja, well you see, Ehrenfest — I may even still have notes about it — each year had this course in statistical mechanics which ended up so to say in quantum theory, and then off into newer things. He touched upon them…

Kuhn:

Now this would be before ‘25? Before the Bose papers?

Uhlenbeck:

Well, maybe at the same time. That I don’t know. It was just a lecture. Because that was part of his controversy with Bose, and also with other people I think. And he hammered always that if it is photons and if you could believe for these photons that there were particles like the Boltzmann particles in the gas, then you would get the Wien limit law.

Kuhn:

That’s a very old derivation.

Uhlenbeck:

A derivation that we learned. Then he said that if you want to, with this picture, get the Planck radiation formula, you must do something really radical in the distribution function. And he considered, I think, the whole Bose derivation as a kind of an ad hoc case; in order to get the right answer, which I think it more or less was. I mean there were many confusions in it, but of course it helped very much that he knew what the answer was.

Kuhn:

What about the slightly earlier interpretation in terms of photo-molecules?

Uhlenbeck:

That I never heard of; that we never heard much about…

Kuhn:

It’s a fairly brief development, but it may well have something to do with Bose’s route to the statistics.

Uhlenbeck:

Ja, I don’t know anything about that, how Bose came about, because he was simply not known. An unknown man he was when his paper came out… But the only memory I have really very sharply is that the Schrodinger paper in Phys. ZS, was, even for Ehrenfest, clarifying. He may have known it, but it was for him a very clarifying paper; and certainly for me. That was the paper which was really the starting point for my dissertation. But about the photo-molecules, I never heard anything.

Kuhn:

In the early days of the Bose statistics Einstein talks about a material gas with the property of photons. Is that thought of as purely an exercise?

Uhlenbeck:

It sounded extremely speculative. And I think with Einstein it also was pure speculation in a sense. He says, “Why not particles too?” But I think even in my dissertation I thought there were no examples. I thought Bose statistics only photons; Fermi statistics everything else; which was not right, of course. Then there was the Elsasser theorem, which later Ehrenfest and Oppenheimer wrote about. Say if you take a hydrogen molecule, that would have Bose statistics because it has an even number of Fermi particles. That means in the translational motion it is Bose statistics. That I am sure came much later. That was I think ‘28 or something like that…I have correspondence with Martin Klein about that, about how that came precisely about. Because I remember talking with Ehrenfest about that during one of the summer schools in Ann Arbor, so that must have been ‘28, ‘29; and then for the first time there were really — at least a priori — cases where it should be so. Although of course there was no empirical argument.

Kuhn:

No, I asked about that because just thinking and talking about material particles does mean that you’re thinking about a sort of statistics which material particles may satisfy. On the other hand you might say, “Photons are not like material particles so we count them differently”. If you take the second of these approaches, then you’re likely not even to set up the Bose material gas…The Einsteins’ papers came after de Broglie,…and I think this was, so to say, the first realization with him of the wave —

Uhlenbeck:

— particle duality. If really particles have also wave properties, then gas particles should also behave in some sense like separate waves. And therefore the distribution should be similar to Planck. That was extremely bold, of course, that it was immediately connected with the wave-particle duality. And that was also clear in the Schrodinger paper. There was always the duality question.

Kuhn:

Let me just tack on one other question. There’s a reference in something of Planck’s in ‘23 perhaps in the Bohr Heft of Naturwissenschaften — about the interpretations of quantum mechanics which undermine causality.

Uhlenbeck:

Well I’ve no memory of it. I don’t think it was an issue for us. I mean these questions were — Of course Ehrenfest was always interested in fundamental questions always. But he was not a philosopher. I think as soon as it got so philosophical, then he was —. He wanted always to know is there a sharp point. Is there something by which it goes one way or another and one can decide? So no, I have no memory about that at all. I don’t even know Ehrenfest’s reactions to the uncertainty relations. He probably would have liked it. He was extremely proud, extremely proud, about one of his last papers, and that was the Ehrenfest theorem; the wave packet in general moves like a classical particle; You can deduce this from the Schrodinger equation by thinking of the Schrodinger equation as a heat conduction equation. (And so he immediately would know in general what the wave packet is.) And he then derived that average value of the velocity and of the acceleration and so on obeyed the ordinary classical mechanical rules. That’s nowadays called the Ehrenfest theorem. That was one of his last papers. It was after I had gone. It was I think maybe ‘29. And he was very proud of that. It was also an extremely acute paper. He was at that time so sensitive about, so to say, the big shots and the things they said. And they were all very nice about this paper.

Kuhn:

With all the personal factors that were involved in his suicide, did, do you suppose, any sense of the new physics having left him behind also play a role there?

Uhlenbeck:

Oh ja, I mean, he had such a feeling that he couldn’t do it anymore. And then he had the feeling that he was the professor, that there was only one, and that there were all these smart youngsters who should have his job. He was perhaps still good to teach, but he couldn’t take part in it. That was certainly an aspect of it. It was not the only one, but certainly an aspect, because he was very depressed, boy, was he depressed about these things. He couldn’t understand it so quick. He wanted to have such Gefuhl, such feeling. No, for him it was really very hard to get used to this sort of confusion without sharp models and without (Anschaulichkeit) without knowing what you precisely did. (That was terrible for him.)