Lothar Nordheim

Nordheim:

Ja, these where the old days of the quantum mechanics, when one perfected the methods, but — one saw that one was running into some kind of a wall.

Heilbron:

Can we go back a little further and discuss the origin of your interest in science?

Nordheim:

Ja. I went to a humanistic gymnasium, where one had lots of Latin and Greek and languages and so on, and this I didn't like very much. It was a relief when we had mathematics and science. I was very good at that in school. I did not have very much outside stimulus in that. I had no contact with any scientists except through the school teachers at that time. But I felt that this was the thing I was interested in. And then I finished school and I was for a few months in the army, and then the First World War was over and at the same time the University of Hamburg was founded. The same time I was ready to start university studies, and I stayed there because I lived there at that time. This was quite a revelation for me. At the beginning there were no regular professors. There were teachers from the gymnasium and from some institutes they had in Hamburg, but I felt at once it was very stimulating. And I was not divided at the beginning what I would do. And then after one year there came the mathematician (Blaschke), who works in differential geometry, and in absence of a theoretical physicist, he gave a lecture in analytical mechanics. And then he gave in addition a seminar on it, in which he took as a text Sommerfeld’s Atombau und Spektrallinien, which just had appeared. That was the second year at the University. And I volunteered for the quantum conditions. It was the most difficult part. This caught my fancy at this time.

Heilbron:

And then you moved on to Munich.

Nordheim:

This was partly for family reasons. These things are not always rational. Of course Sommerfeld was there. And I thought at the beginning that I would try something more practical, and try to do some experimental physics with Wien who was there, but I didn’t get any place. And then I attended Sommerfeld’s lectures and seminars, and I caught his attention there. I gave a talk at a seminar which was very well received and for which I worked very hard.

Heilbron:

What happened when one caught Sommerfeld's attention?

Nordheim:

I mean, he had a circle of people. Pauli was among them, Heisenberg, Wentzel and Kratzer and Herzfeld, and then some other students also at the time. There were seminars and he took an interest in the people personally. Then we were even talking about a thesis subject, and it was a subject which was not soluble at that time. It was on the Zeeman effects of band spectra. At that time you got an unsolved problem in your lap. But then he got a request from Hilbert in Gottingen who wanted an assistant in physics. He recommended me, although I had not finished my degree yet.

Heilbron:

You were one of the series of Hilbert’s physics tutors?

Nordheim:

Yes. He asked Sommerfeld whether he had one, and Sommerfeld recommended me, and so I went to Gottingen. And so that is the story how I came to Gottingen in place of finishing with Sommerfeld. It turned out that by myself I wouldn’t get any place with the thesis subject Sommerfeld had given me, and I asked his permission if I could not approach Born. So I switched over to Born for this purpose, and at the same time I was an assistant of Hilbert.

Heilbron:

You must have started your work on the Zeeman business before Lande’s g factor, or just about the same time?

Nordheim:

No, Lande's work was just known, and so Sommerfeld’s idea was, couldn't one do something similar for the band spectra. The normal band spectra had been analyzed by Kratzer and so it was a natural thing. Of course we know now it's complicated with spins and things like that.

Heilbron:

What were the attitudes of the students generally towards the choice? What would determine a man to go say to Born rather than to Sommerfeld, or vice versa?

Nordheim:

Interest in particular problems, accidents of geography. You would go to a professor after you had reached a certain stage. And you first go to his seminars and lectures. That is similar as it is here. And if you found that it was interesting what he said, then you might approach him…

Heilbron:

Would you say the interests were much different between those two schools, or they were concerned with the same kind of problems?

Nordheim:

They were concerned with the same kind of problems. Of course Sommerfeld also was interested in partial differential equations and problems arising from them — wave propagation and things like that. And Born was also much interested in the theory of crystals, and had a number of people working in that field.

Heilbron:

You arrived at Gottingen just shortly after Born himself got started there. Had things changed much in the year that Born had been there, do you know? Did he make a great difference when he arrived?

Nordheim:

I think he did, because I think there was no theorist immediately before him. Before that was Voigt, who was very old-fashioned, and I don't know exactly when he died. Then there was a new group. Also James Franck came about the same time. In mathematics Courant, and later, Weyl. I might not always keep the dates right, because I was there for about ten years, and many people went through there. I may mention too that at the same time Bohr was working out the first parts of his theory of the periodic system. And when I was in Munich with Sommerfeld he had a manuscript of his. In Gottingen shortly after I came or a year after that, he gave a series of lectures there; and this got everybody very excited.

Heilbron:

Gottingen shortly thereafter attracted many foreign students too. It seems that this occurred after Born's arrival.

Nordheim:

Yes, though this was mostly a little bit later. The combination of Born and Franck was particularly good for one.

Heilbron:

I think Fermi was there fairly early.

Nordheim:

I'm not sure when. He didn't like it particularly.

Heilbron:

I know. Do you know the remark that Laura Fermi makes in her biography, which he didn't like it there and nobody encouraged him. Is it true that he didn't get into a group with the other students?

Nordheim:

He didn't really. I saw quite a bit of him and we made walks. Hut at that time he wasn't settled down. He had some ideas which were new but not particularly fruitful. Very soon thereafter of course he settled down and then he really became Fermi —

Heilbron:

So he didn't impress people very much during his short stay there?

Nordheim:

I don’t think particularly. In the early days of course the problem was quantum mechanics and what you can do with it and how far does it go. When I came there I think it was not quite clear whether you would get anywhere, but one saw from Bohr's consideration of the system of elements that something must be there. But one could not get any good results for more than the one particle problem. There was Kramers' theory of the helium ground state, which didn't give agreement with experiment. And in this line Born proposed to me the next more complicated thing, the hydrogen molecule. The ionized hydrogen molecule, the ion having been by Pauli. And he intuitively drew some pictures with phase relations, out there was no direct handle for tackling this problem really. So I went back to the sources and studied Poincare... I got this one method out of him which I would translate into a quantum mechanical equivalent, which is essentially a reintroduction of Cartesian coordinates for action angle variables in place of action and angle variables. It is applicable in this problem you mentioned here of Grenzentartung or limiting degeneracy. So from the study of Poincaré I got his method, and with this at least the problem was traceable. One could prove that motions with phase relations would be the ones which could come out of the consent application of the Bohr-Sommerfeld quantum conditions. And that these would automatically select these simple types of motion.

Heilbron:

And the notion of the Grenzentartung occurred to you before when you first approached the problem, or did it come up out of reading Poincaré?

Nordheim:

Oh, I studied Hamiltonian mechanics, and celestial mechanics of Poincaré, and then it occurred to me that there was another type of degeneracy, which had not been previously noticed or mentioned at least, which obviously was there.

Heilbron:

In Born’s textbook Atommechanik, which was published in 1925, he says he owes a considerable part of the section on degeneracy to your own efforts. Did you write most of that?

Nordheim:

I wrote parts of it. And in a way I think this was an important development, because it made it possible to apply quantum mechanics to complicated systems, and it gave of course a negative answer. This method of course was not fully understood — why there are quantum conditions — but even if you accepted them, yet you did not explain things beyond the hydrogen atom, so to speak.

Heilbron:

And was that an unexpected result, or did Born think that that would be the result of your investigation?

Nordheim:

Well, at the beginning one hoped that one would get some place, but then there was no success in the helium ground state, and then later Heisenberg tried also the excited states of helium, and there was no resemblance to what was actually observed. So I think already in my paper on the hydrogen molecule, I say, "It seems to be unlikely that this will be the final answer."

Heilbron:

In the thesis you built up the hydrogen molecule adiabatically, and then at the end you say that one would hardly expect that to work anyway, because you can unbuild it just as readily. So was the result of your thesis the disproof of that possibility, which shouldn't have been surprising anyway?

Nordheim:

There my memory is stretched, and as I said, I haven't been able to look at this paper again. I don't even have a copy, I moved so much around. So I really don't know. Of course the situation is so: to build something stable you have to do something which doesn’t make it go back. But since one didn't know what the nature of the quantum conditions themselves was, I don't think that was a decisive point. The decisive point was that no model approached anywhere near the observed dissociation energy of the hydrogen molecule. And this was a similar result as Kramers’ theory of the ground state of the helium atom, which in a way was very similar.

Heilbron:

And so by 1923 say, people at least at Gottingen were pretty well-convinced that something fundamental was missing?

Nordheim:

Yes. And there were some hints. This spread out over a number of years. One had the rules on intensities in which one apparently could translate correspondence formulas in something which fitted more or less, and then Kramers' dispersion theory, which seemed to be a logical translation, and which vie know now comes out of quantum mechanics. And so it was an instance in which one could more or, less guess the right answer. In a way it was a frustrating time. One felt one is near to something, but, one doesn't quite have it.

Heilbron:

As long as you've begun to mention some other work, do you recall reactions people had to things like the Kramers theory? I listed a few things here that perhaps you might recall. For instance the Stern-Gerlach experiment.

Nordheim:

The Stern-Gerlach experiment was considered very highly, and at once accepted. And in a way it really proved the existence of quantum states. This was similar to Franck’s own experiments on collisions between electrons and atoms or molecules, for which he got the Nobel Prize. And these were thought of as fundamental experiments which proved to you that there exist quantum states, and so they do.

Heilbron:

And at that time there was still sufficient skepticism on that point?

Nordheim:

No, I don't think there was really — when I started to become aware of these things — doubt on that anymore. But the Stern-Gerlach experiment, the Franck-Hertz experiment, was considered as very fundamental experiments which proved the existence of quanta. I mean of course probably people like Lenard were not convinced yet.

Heilbron:

Wel1, the Stern-Gerlach experiment made some theoretical difficulties though, didn’t it? The objection that I think was first made by Einstein and Ehrenfest, that in the beam where one doesn’t have the collisions to change the orientation of the atoms, they should all classically precess at whatever angle they happen to be located at around the field direction. The mechanism of tossing into the quantized states was a problem to them.

Nordheim:

It was a problem to them. Of course this has been answered by the statistical interpretation of quantum and wave mechanics very nicely. The instrument selects the right case. This was one of those things where one didn’t know quite the answers, with which the theory abounded at that time. But one felt, as I say, to be very near but not quite there.

Heilbron:

And what about the Compton effect?

Nordheim:

The Compton Effect also was considered as a very fundamental experiment, and it did not surprise anybody, because I think there was Einstein's paper from 1917, in which he said that if there is a quantum of energy, then there has to be an analogue in momentum. And the quantum effect just verifies this piece and directly proved this part of Einstein’s theory.

Heilbron:

What about De Broglie's thesis, did anyone take notice of that at Gottingen?

Nordheim:

Not very much. I think it was more or less considered as a curiosity. It was not taken too seriously. I think Dr. Elsasser wrote a note — and at that time he was a young student — that if this were so there should be something observable…

Heilbron:

What about the Bohr-Kramers-Slater theory — the statistical energy conservation in radiation processes?

Nordheim:

Oh, I think this was generally disliked. Wasn’t that resurrected later when the apparent non-conservation of energy in beta decay became known?... There was another instance of this idea in the days before the Bohr-Kramers-Slater theory with non-conservation of energy before quantum mechanics. As I say, I don’t recall very much about that. So I believe that it did not evoke very much of a response — particularly in Gottingen there were Franck’s experiments which directly showed conservation energy and transmission in collisions between these quantum states.

Heilbron:

It interested me, in connection with the realization about 1923 that things were fundamentally wrong, that Fritz London, who came from an entirely different faculty knew that things were about to happen in physics, or at least so the story foes, and so changed.

Nordheim:

This is difficult to say. He switched from philosophy to physics at that time.

Heilbron:

Because things were going to happen.

Nordheim:

Ja. And it was an exciting time. There were these experimental confirmations of the quantum ideas which you mentioned, Stern-Gerlach, Franck-Hertz, and the Compton Effect, and there were the beginnings of Bohr’s theory. There was the beginning of the understanding of the multiplets. Of course they were really only understood after the spin of the electron was discovered. But one had regularities — so it was a very exciting time.

Heilbron:

What did you think of the notion of electron spin when you first heard about it?

Nordheim:

I myself probably didn’t think very much. I was not very active myself in the theory of spectra, from which this of course originated I remember one remark of Pauli. He said (“Bohr reisst herum um eine Irrelehre zu verbreiten.”) That was that at the beginning he didn’t think very much about the spin, and the story gives that he talked Kronig out of publishing. But then it became very soon accepted. And then of course Pauli himself gave the wave-mechanical formulation of it, he was the first to show how to treat this.

Heilbron:

Where did you learn your mathematics? Just in the few years that you were in the University?

Nordheim:

Oh, in the universities I had calculus and theory of functions still at Hamburg, and differential equations from Lindemann in Munich, in which I didn’t learn anything. He was a very old mathematician. Then I picked up quite a few things as Hilbert’s assistant. And then, as you go along in the study of physics you generally have to look up some mathematical methods. Eigenvalue problems I learned when I learned wave mechanics.

Heilbron:

So by the time already that you arrived at Hamburg you were able to take a course in differential equations?

Nordheim:

No, I started with calculus, and then the theory of functions for complex variables. And this I learned there, and then as I said in the third term we had this mechanics course from Blaschke which introduced lots of differential equations as of course they do in classical mechanics.

Heilbron:

How did it happen that during the time you were writing for the Handbuch you didn’t write the article on the perturbation theory instead of the one on the straight mechanics?

Nordheim:

I don’t know; and I think it was a good thing, because these formal developments, though I had contributed to it, didn’t really interest me so very much. But the principles of mechanics, and Hamiltonian theory, those I liked very much. And these I had learned from Hilbert quite a bit. The principles of mechanics, this I did entirely on my own.

Heilbron:

That fancy derivation of the Hamiltonian equation in the article in the Handbuch articles anyway? Were people just approached by the editors?

Nordheim:

That is essentially Hibert’s, yes. Today I would write this article differently. I was very young then, young and enthusiastic. Today I would write it more simply, but I don’t think I will.

Heilbron:

How did it happen that you got involved in writing the Handbuch articles anyway? Were people just approached by the editors?

Nordheim:

I suppose people were approached by the editors. At that time the number of physicists was not very large, not as large as it is today, and it was generally common knowledge who was interested in what.

Heilbron:

Was it considered an honor to be asked to write an article?

Nordheim:

To some extent. It was adding to your stature if you wrote a good article. Nearly everybody wrote something or else… because at that time it was more or less a purely German affair, not as this new edition, which is international. I think this was originally so that I got the article on the principles of mechanics and Fues on Hamilton-Jacobi theory and perturbation theory. And then I branched out so much into the Hamilton-Jacobi theory, because I found this interesting and I had sometimes worked in this field, that then it was separated in the three parts.

Heilbron:

When the concern with the older quantum theory began to crystallize at Gottingen, what directions did people think one should strike out into find the true mechanics?

Nordheim:

This was really the thing, I mean this one didn’t know. One tried more things to do, to see how far it goes. For instance there was this paper I wrote on the collisions in an attempt to drive the old quantum mechanics as far as it can be driven. And then the correspondence principle of course seemed to be the only guidepost there was, and this remained so until finally Heisenberg got the bright idea that the real translation from classical to quantum theory is by introducing the kind of modification which is really suggested by Ritz’ combination principle.

Heilbron:

Heisenberg went at it through dispersion, whereas you were looking at the collision problems. Did Born have any ideas about the direction to proceed?

Nordheim:

I don’t think he had any original ideas. He suggested this application of collision theory to me, and that’s why there was the first paper in the Naturwissenschaften to show that. But then I think he left on some trip, and I worked out the details of this theory.

Heilbron:

Jordan worked on that too, didn’t he? He signs it.

Nordheim:

He signed the paper. I think he did not really do very much in this respect. He worked at that time I believe with Franck on Franck's book on the Anregung von Atomen durch Stosse. I don't recall that he contributed to this particular paper very much or anything.

Heilbron:

In the Naturwissenschaften article, you give the energy as worked out in the classical perturbation theory already, and you also write down the transition probabilities in terms of the same quantities you use later on in the more expanded article. How much of that work was already done with Born?

Nordheim:

Born originated the original perturbation theory and then I took over. I suppose — I am not clear about that — this paper in the Naturwissenschaften was after the structure of the work began to emerge and Born was leaving on leave of absence.

Heilbron:

I was just looking at that paper again — your second, expanded version. At the end of that you say that in the case the correspondence principle is pretty good, that it gives…

Nordheim:

I mean, it gave a fairly reasonable description, things which have been later derived. The correspondence principle gives qualitative answers; it always did. It did not give quantitative answers. It is a qualitative principle, and so this work was quite reasonable. And if the quantum mechanics had not been invented later on and the theory of collisions elaborated on the basis of that, this paper probably would be a very well-known paper!

Heilbron:

Yes, you go on to say that you think in the case that you have considered there, no basic modifications seem to be required.

Nordheim:

I don’t recall it, but it is a correspondence translation of the theory of collisions…

Heilbron:

Now, Born at this point, at least in his lectures on atomic mechanics, says that one of the big problems with the current theory is that it involves all these non-observable quantities. Was that a prevalent attitude, that one of the basic difficulties with the Bohr-Sommerfeld theory was that it required one to consider all sorts of non-observable characteristics of the orbit?

Nordheim:

Yes, of course, the mechanical description went much farther than one ever could hope to observe, but I don’t recall this particular passage. One felt of such a detailed mechanical description, that there is something wrong with it. That it is very good in the limit of large quantum numbers where you get continuous transition into classical mechanics, but then you have to replace differential expressions by difference expressions, and why that is, of course one had no clue really.

Heilbron:

What interested me is that Born was teaching this approach that is so prevalent now, that Heisenberg himself mentions in his first paper, that there are all these unobservables around and that’s bad.

Nordheim:

This I can’t recall, how much this is due to Born and how much to Heisenberg. It was of course formulated first by Heisenberg in his first paper, who said one observes only frequencies and amplitudes and intensities, and so the theory should be built up in terms which only involve these. I think this was fairly clearly stated in his first paper.

Heilbron:

But it is also stated in Born’s printed lectures, the first column of the Atommechanik.

Nordheim:

I mean I was not a member of Born’s circle at that time, because was Hilbert’s assistant. And I realized even at that time I was not as bright as Pauli and Heisenberg were.

Heilbron:

Do you think that the continuous use of the action-angle variables influences this sort of thinking of Born and Heisenberg? That is, one goes to a set of coordinates and gets the frequencies and so forth, and you don’t really need the unobservable quantities, or you try not to work with them.

Nordheim:

Yes, I believe so, because there was of course an inconsistency in the old quantum theory which is very deep. The old quantum theory tried to describe all the systems as multiply periodic systems. On the other hand, one was very well-aware that actually many systems were not of that nature and that these series therefore, which one obtained by perturbation theory, were actually not converging for infinitely long times. The nature of motion really was different. Of course in some cases the quantum conditions selected the periodic motions, period in the sense of Poincaré. But in more complicated cases they really did some violence to the mechanical system, and I think one realized that. I think I realized that, and so did Born of course. And on the other hand, that what one saw were frequencies and amplitudes and intensities, and so from hindsight it’s not so difficult to argue what other things you could work with.

Heilbron:

How did you feel about Heisenberg’s paper when you first read it or heard about it?

Nordheim:

I found it somewhat strange, but it seemed to work well. And then of course at once Born and Jordan rushed in and erected the formal theory, gave the formal theory which was very incomplete in Heisenberg’s paper. And then Heisenberg came to Gottingen for some time, I believe became convinces only after Schrodinger’s equation was found, and the connection between the wave-mechanics and matric mechanics became established. That was at least for me — the clinching point. I mean this was really something which was true and was the real answer to the problems which plagues physics before.

Heilbron:

How did other people at Gottingen react to this matrix mechanics? For instance, how did the experimentalists react? Do you know what Franck thought of it?

Nordheim:

I don’t know. I can’t speak for Franck. He has a very deep feeling for physics. He feels instinctively the right thing. I think in several quarters the reaction to the matrix mechanics was that this was something very formalistic and a typical Born product. He likes to write down matrices and indices and so on, I think a number of people thought that. I found it hard to follow at the beginning, until it later became very familiar. For instance, Born writes in a difficult style, so at that time things were fairly clear, when he wrote the second volume called “Elementare Quantenmechanik.” That’s far from elementary!

Heilbron:

What about Hilbert's circle? How did Hilbert react to this business?

Nordheim:

Ja, about Hilbert I have to say several things. He was quite ill in those years, and not the same man he was before he had pernicious anemia. And he was just saved in time by the development of the liver treatment of that… He could only understand things if he did them more or less himself. He could only understand things if he did them more or less himself. He could not bend his thoughts in the way others did. Of course I was not of those to teach him quantum mechanics and wave mechanics, of course by the process learning these things myself. It was also the duty of his assistant to help him prepare his lectures which was sometimes rather strenuous. You had to make clear outlines of the number of sessions and so on. And my interest in statistical mechanics comes from the first semester I came there, 20 years old. He gave a course on statistical mechanics, on which he had mostly old notes and so on, but that interested me quite a bit. Then he gave while I was his assistant a course in the old quantum mechanics in which he went through the Hamiltonian mechanics, and mechanics in which he went through the Hamiltonian mechanics, and then taught at the end things on wave mechanics and matrix mechanics.

Heilbron:

Did all the physicists at Gottingen, at least the theoretical physics students, attend Hilbert’s lectures regularly?

Nordheim:

Not always. I am not sure. This is difficult to know, since no formal attendances are kept.

Heilbron:

Did Born and Jordan discuss what they were doing during the writing of those papers with others? Were there seminars or colloquium or anything?

Nordheim:

There were of course seminars and colloquia, but these things were brought only more or less after they were finished. And as I said, I didn’t belong to the close circle of Born, and I did not work with him directly. You asked in your notes how close the collaboration was with him, which depended on the problem of the person. At that time I did not work very close with him. Then I did my thesis, he gave the idea and as I said he drew some pictures and then he left me more or less alone with it. I talked with him about every two months or something like that. And then I found, as I said, the right method, and made several mistakes in the calculations. I have very great difficulties in making complicated calculations. I force myself to do them, of course but I make very many mistakes, and it is a very great strain for me.

Heilbron:

I meant to ask you, were the doodles that Born drew when he first proposed the problem close to the final diagrams?

Nordheim:

Yes, they were the models which came out of the quantum conditions at the end. I mean, it was fairly obvious each electron went around its nucleus and there were certain phase relations, they had to move in phase, at least in the ground state. They could move either in planes or at an angle and so on. These turned out to be the types of motions if you hold the nuclei fixed which would come out of the proper application of the quantum conditions.

Heilbron:

Do you know when the probability interpretation began to be worked out, when Born and Pauli hit on that interpretation of Schrodinger’s psi function?

Nordheim:

I think that was mostly Born’s work. He was the first, I believe, who very clearly stated that explicitly in one of his papers.

Heilbron:

And how was that interpretation received?

Nordheim:

Lots of people were disturbed. However it was, I think, at once accepted by the people who really contributed very much to quantum mechanics, Pauli, Heisenberg, Bohr, I think accepted this at once. I think Sommerfeld didn’t like it and Schrodinger didn’t like it. I know for the immediate people who were concerned with the development of quantum mechanics, that this was more or less immediately accepted.

Heilbron:

And the final elaborations later at the Como Conference in 1927, when Bohr and Heisenberg both talked about complementarity, was that also taken so easily?

Nordheim:

The Gottingen people I think fully appreciated that and concurred with it. Again, as you know, it had some resistance from other people. I think all of the Gottingen group, or of the people who went through Gottingen, accepted this as the correct interpretation.

Heilbron:

How did people feel about wave mechanics at Gottingen, this being a foreign element to bring to the home of matrix mechanics? I heard a story somewhere about Heisenberg first not liking it, or being irritated that Born had taken it up.

Nordheim:

That is possible, I don’t know that. Of course then pretty soon the equivalence of the two methods was proven independently by Eckart and by Schrodinger. And after that of course everything was… I think before that — which was not a very long time after all — the Schrodinger equation was talked about, and it was thought to be something very ingenious, since it gave the correct results for the hydrogen atoms. And also then the de Broglie paper was recognized as something meaningful so to speak. But of course the real success came and everyone was convinced after the equivalence of matrix mechanics and wave mechanics was proven.

Heilbron:

How did people react to Dirac’s positron?

Nordheim:

This is a good bit later. At the beginning apparently Dirac thought this might be the proton, but then while in Gottingen around 1930 he proved there must be symmetry and that it could not have a different mass. Then this theory remained a curiosity until the positron was found experimentally. I think in his first paper on the negative energy states, Dirac suggested that they were the protons, and then I believe it was Weyl who showed there ought to be symmetry, and I don’t know whether Dirac wrote anything after that on the subject anymore, I don’t recall. But I am not absolutely sure.

Heilbron:

But it was regarded merely as a curiosity…

Nordheim:

… Of course connected with the difficulties of quantum electrodynamics. It was about the same time that one started to worry I think about infinities in quantum electrodynamics. The first paper by Pauli and Heisenberg on quantum electrodynamics I believe were around 1930.

Heilbron:

To return to your own work now, I wanted to ask you again about this correspondence principle procedure. You evidently viewed it that one needed a translation scheme to go from…

Nordheim:

Yes. I mean, in a way the Bohr-Sommerfeld quantum conditions are such a translation scheme. If you use the action and angle variables, then the Bohr-Sommerfeld conditions give integral values to the action variables and those you say are the stationary states. The frequencies are the energy divided by h, of course. And this guarantees for you that for large quantum numbers and small differences it goes over into classical results.

Heilbron:

Was the opinion that "We have now one or two translation principles, and one should look for more of this same nature”? Was the "true" quantum mechanics thought to be, before Heisenberg, a series of translation principles which would enable you first do the problem classically, and then so to speak "translate” it?

Nordheim:

And this is what one did by default, by lack of anything better. With quantum conditions as written in the old way, you ask always why does (nature do) this? People felt and knew that there was something in the physics behind it which we didn't know. If the quantum conditions had worked in more complicated cases one would still have asked this question.

Heilbron:

So the solution was not thought to be in principle an extension of these translation schemes?

Nordheim:

Oh, at the beginning one had hopes, maybe you can also treat more complicated systems with the right mathematics, and therefore all this effort in perturbation theory and so on.

Heilbron:

My question about your paper with Hilbert and von Neumann you've already in essence answered; you were really in Hilbert's circle and not in Born's.

Nordheim:

Yes, and at that time the first papers on what we call now the transformation theory in quantum mechanics appeared by Dirac, and in a very tortuous form by Jordan. I tried to give this in a form that could be easier understood. In one of the courses of Hilbert's I tried to prepare this for him, and then at that time von Neumann came to Gottingen, and he helped a little bit in the elaboration of it. This paper I think is quite elegant, but it is absolutely inacceptable mathematically. Such integral operations over improper functions and all these things occur, but they stimulated von Neumann to develop his theory. He really brought the proper mathematical tools to bear on the mathematics of the transformation theory and the Hilbert operators and so on.

Heilbron:

Who had worked out the mathematics of the complete operator, was that Hilbert's?

Nordheim:

I am not sure that I recall this. He had of course Hilbert space, called after him, and operators and Hilbert space transformations were very familiar to him. The way in which it fitted into the quantum mechanical scheme in this particular paper was worked out by myself, by translating ideas of Dirac essentially into this language. And then von Neumann helped very much polishing it up and bringing it in its final form. Then I think it started von Neumann to look for more appropriate mathematical tools to do it, and he did it.

Heilbron:

You published that in the Mathematische Annalen. You didn't send it to the physicists. Did the physicists react well to this?

Nordheim:

Yes, I think they generally accepted it. Of course most physicists don't worry about convergence or anything.

Heilbron:

But they weren't offended at this axiomatic approach you adopted?

Nordheim:

I don’t think so. And as I have said, I don't think that the paper by Hilbert, von Neumann and myself made a very great splash, because it was soon superseded by the better papers of von Neumann, mathematically on a much more solid foundation.

Heilbron:

Had von Neumann been concerned with quantum mechanics before he arrived, in Gottingen?

Nordheim:

I think that he got his start in this field in Gottingen, and being so tremendously fast, he got hold of it very soon. Yes, he had one of the fastest minds I have ever met. His brains always worked on roller bearings.

Heilbron:

What sort of things could he do in his head that were so impressive?

Nordheim:

Well, he could take hold of anything very shortly and then sit down one night and write a thirty page long paper on it, and with all the trimmings. You know, scientists like to give puzzles to each other; this is a pastime at parties and so on. And it is said about von Neumann, there was a problem given: A fly flies between two pistons which are gradually pushed together, at a certain speed and so on. What is the total path length the fly flies? Of course there is a very simple answer. You only have to take the time and the velocity and go through it. But he summed the series in his head!

Heilbron:

It sounds apocryphal. You seem to have changed your line of work shortly after that, after the new mechanics. At least you got interested in electron theories of metals.

Nordheim:

And I had always some interest in statistical mechanics, as I have said, and the development of quantum mechanics was so fast and furious, that all I could do was to keep up with it and try to teach or to transmit some of it to Hilbert. Then I sought for a field in which there would be an application of quantum mechanics, and I chose this field. Of course as it turned out with quantum practically all fields of physics could be attacked. Having some interest in statistical mechanics and seeing the first Sommerfeld paper in the Naturwissenschaften I believe, I was very much intrigued by it and I thought, "How can one be so naive and get good results? There must be more to it, and one must do more to really understand this.” So I got thinking about it and that started me on that phase of my work.

Heilbron:

Do you know who first got the idea that an electron gas should show the usual degeneracies that you get for the Fermi-Dirac statistics at low temperatures?

Nordheim:

The first that was published was a paper by Pauli…

Heilbron:

So the whole idea of applying the Fermi-Dirac statistics to an electron gas in a metal was Pauli's?

Nordheim:

I suppose it was Pauli. And this was in hindsight a very natural idea, because one had just learned that the Fermi-Dirac statistics and the exclusion principle worked for electrons and atoms, and then it should have appeared to everybody that it should do also in metals. But I believe the first who published it, I think was Pauli.

Heilbron:

When you were in school, were you taught the electron-gas theory of metals? Was that a familiar idea to you when you saw it in Sommerfeld's work?

Nordheim:

It was not really taught then. I think I read something about it in the old book by Drude, but I may be mistaken. I think I started actually after the first communication by Sommerfeld. And then it occurred to me that you should worry much more about the motion of electrons in solid metal, than just using the Fermi-Dirac statistics. I worked out the theory of emission as an exercise without knowing anything that one had done it classically.

Heilbron:

When you calculate the probability of reflection of the electrons and do a square-well potential, you essentially discover for yourself the tunnel effect.

Nordheim:

I discovered this for myself. There may have been others; not many. I think the first actual discovery or the first application was done by Hund on the ammonia molecule, this two center problem. The ammonia molecule is such a thing with three legs, unsymmetrical, and the nitrogen possesses a very famous spectrum in the microwave region, this triangle going from one side to the other side, atypically quantum mechanically degenerate problem. I think in ‘27 Hund treated this case. You get interactions even if the potential barrier between the the states is classically not penetrable. But quantum mechanically of course the wave function leaks through, and I think this also was the first time this effect has ever been mentioned; I believe it was in '27 by Hund. I worked it out on my own without having conscious knowledge of this or what other people might have got on it.

Heilbron:

Now Gamow's famous alpha particle penetration of the nucleus was somewhat later.

Nordheim:

That was somewhat later, and he told me actually once that he was inspired to it by my paper. Of course the recognition in the alpha case that you have a transmission of the barrier, that was his. But how to calculate it and how to go through a barrier — I'm not sure. I think that was clearly stated arid the general formula was used in the paper by Fowler and myself. I think it is already started contained in my paper in '28, but I am not sure. At least it is certainly for a rectangular barrier. For a square barrier, I'm not sure whether I wrote it down. I might have written down only the general formula.

Heilbron:

That was an idea that was well received?

Nordheim:

Yes, then it has become common knowledge that things can go through potential barriers, I think almost universally.

Heilbron:

Was that paper written at Cambridge?

Nordheim:

It was already written at Cambridge. It was pretty soon after my arrival, and I was still struggling with the English language at that time, so I decided to send it to the Zeitschrift fur Physik.

Heilbron:

Why did you choose to go to Cambridge?

Nordheim:

You asked also how I came. Born and Franck had very good relations with the Rockefeller Foundation. One official in charge of these fellowships, Dr. (?)sdale, came, went around and talked with them. Just as in these days it was then considered desirable for young people of some promise to go on fellowships to a different place to learn more. Now most of the people tried to Copenhagen at that time. And Born said I should also try to go to some other place. And I was interested in statistical mechanics and just starting to think about these things, so it was very natural that I should go to Fowler, who was one of the foremost experts in statistical mechanics at that time. So I went to Cambridge and for an extra bonus after that for a few months to Copenhagen on this fellowship, after the end of the academic year in Cambridge.

Heilbron:

What were the mechanics of getting one of those? How did you go about it?

Nordheim:

As I have said, I think this official officer of the Rockefeller Foundation in charge of these fellowships came through Gottingen and talked with people.

Heilbron:

And just awarded them on that basis?

Nordheim:

And of course on the recommendation of Born and Franck I went there.

Heilbron:

You thank Fowler for his suggestions in that first paper you did on the electron theory; what did he contribute?

Nordheim:

I really don't recall. I suppose this was more or less a courtesy, since I was working with him. And also I did not talk very often with him. He was a busy person. Oh, once in a while we talked about things.

Heilbron:

But then you did two or three papers with him.

Nordheim:

One paper, on the emission in strong fields, and this actually he wrote. We talked about it and it was clear what was to be done, but then I went on vacations to visit my family in Germany, and when I came back he had already sent it off. He was also quite fast. He had worked out the mathematics, which was not very difficult, yet I was very much surprised when he had finished the paper when I came back after a few weeks. But we had talked about it before, and this was an effect which could be calculated, and it was very interesting.

Heilbron:

Yes, it seems a very reasonable problem to approach after the work you had completed. As was the next one, the image charge.

Nordheim:

Ja, then I felt of course that square potential steps are not realized by nature, and by that this effect might be much smaller which it turned out to be. And to some of these things come quite naturally, of course.

Heilbron:

By the time you wrote that paper, were you able to write it in English without much trouble?

Nordheim:

I was not very fluent when I started there, having had English in high school. But I picked it up pretty fast there… I first worried of course what happens in collisions. Sommerfeld simply introduced the mean free path, and this I thought was not quite consistent. And then the first paper which I wrote in English during my Rockefeller year with Fowler was in this subject. I wrote on the kinetic equation, in showing that even with collisions, if you introduce the Fermi-Dirac statistics, the collision equation changes, becomes of the second order; if there is no energy transfer then you fall back on the normal case. So this was one of the worries which I had, and removed. Then I thought to myself that the motion of the electrons must be similar as a light wave in a medium.

Heilbron:

You mentioned that Houston made some suggestion about using X-ray diffraction.

Nordheim:

There is a paper by Houston around this time, I think in the Zs. f. Phys., which was done I think when he was studying with Sommerfeld. Yes, in which an attempt was made to calculate the resistance by using formulas borrowed from X-ray diffraction, but I forgot that I quoted him, and if so I must have seen some preprints.

Heilbron:

Then you go on to apply it to the alloys.

Nordheim:

Yes, at that time it was fairly clear to me, and the elaboration actually was done in Bloch's famous paper. I didn't quite get there before him. That you had to consider this as a wave mechanical problem, a periodic potential; and just as in a light wave in a crystal, there would be no resistance in an ideal crystal, so there would be no resistance in an ideal crystal.

Heilbron:

Was this idea yours independently, that you should expect no resistance in a perfect crystal?

Nordheim:

I believe so.

Heilbron:

Was this behavior of alloys well-known, that there was a resistance independent of the temperature?

Nordheim:

Yes, that was quite well-known, Mathiessen's rule was quite well known. I think there is a lot about it in the Handbuch article by Gruneisen. But this was a fairly well-known fact, that you had a temperature dependent part and a residue part which depended on past history and pureness and so on.

Heilbron:

Was it much of a problem? Were people worried about this irregularity in the behavior of alloys?

Nordheim:

I don't know whether they were worried about it. I mean it was a phenomenon, of which before one could bring the quantum theory to bear there was no real understanding. In the old day of Drude and Lorentz, the classical theory, one thought the mean free path ought to be of the order of atomic dimensions, but then with Sommerfeld's theory one could easily find out, that the mean free path may be several hundred atomic distances. Then if one brought in the wave side of electrons, it was not very far-fetching to think of the propagation, like in a crystal, in that only the deviations from regularity would do it. I am not sure how much there is said about this in Houston's paper, but I think it is quite clearly expressed in my paper. And then I found that one can explain the alloys. In pure metals the residual resistivity was very low; in alloys it was very high. So you discover things. You look at things, and think about them. And afterwards I worried about how this simple Sommerfeld picture can be relatively so good. And the final work I did is in paper number 22 of my own bibliography, which I sent to Dr. King. But in it I left out this letter to Naturwissenschaften with Born and Jordan, which I had forgotten about, and v a popular article in the Metallwirtschaft, where I couldn't find the reference.

Heilbron:

Was there much difference to you between Cambridge and Copenhagen?

Nordheim:

Oh yes. I had generally a good time in Cambridge. I got together with all kinds, but it was not concerted effort. In Copenhagen everybody came together, and one had seminars all the time and people came in and brought the newest ideas. This was an enormously stimulating atmosphere. In Cambridge everyone worked for himself, and so did I, and had occasionally talks with Fowler. And, oh, I played chess with Kapitza. Kapitza had a kind of private seminar held sometimes in Thomas' rooms, which was the only one where people got into contact with each other. This was a private seminar, not organizational. It was at the University rooms… Ja, Dirac was there, and I met him occasionally. But he worked by himself, and Fowler worked by himself, and Kapitza worked by himself. It was not that common spirit as it was in Copenhagen; you were sure to get always the newest scoop there and so on. I went several times again to Copenhagen during vacations, as many people did, on my own.

Heilbron:

And there were always seminars going, to which all visitors were invited?

Nordheim:

Yes. You had to be, so to speak, admitted. But the world of physics was a relatively small world at that time. Once admitted you were in away a member of the family. Ehrenfest used to come there; Kramers also, and others.

Heilbron:

Shortly after the completion of this first set of work on the electron theory of metals, you began to work on semi-conductors?

Nordheim:

No, actually I did very little work on semi-conductors. In 1932 I wrote a paper on rectification, using the semi-conductor picture.

Heilbron:

That's the one I had in mind. Well, in any event, the diagrams you use in that paper and the notion of the energy gap in semiconductors is…

Nordheim:

This diagram was really Brillouin's, and the idea that in semiconductors you have a very small energy gap, that was Wilson and unfortunately not me. It would have been very natural for me to do it too.

Heilbron:

Was that the reason for your going to Russia?

Nordheim:

No. The reason to go to Russia was just curiosity. I wanted to see how it was like, and we had relatively good connections with Russia. Several mathematicians from Russia studied at that time in Gottingen, had fellowships there. And then this was arranged by Tamm that Walter Heitler went there for a few months or for a semester, and then I went there.

Heilbron:

Had you any intentions of staying there at the time?

Nordheim:

Not at all. No, it was also a very uncomfortable time. It was a great experience, but an extremely uncomfortable time. It was a great experience, but an extremely uncomfortable time. Oh, I could talk about it a long time. And, when I came back, that was in January, which was just the time when the Nazis took over in Germany, and I was struck so much by the similarity of the propaganda methods. Some of the Russian colleagues are awfully nice people, but I felt very uncomfortable there, both physically and mentally.

Heilbron:

You mention the Nazis. Things were much less difficult at Gottingen than at other universities, weren't they?

Nordheim:

The University was entirely unbiased, and of course Born, Franck, and Courant were Jewish or part Jewish, so there was at least in the natural sciences absolutely no discrimination of any type. And also after the Nazis came to power, there were no riots and no physical discomforts, but of course the laws, and application of the laws, was the same as everywhere else.

Heilbron:

Well you didn't stay very long in Germany then. You went right on to Paris, did you not?

Nordheim:

Ja, the Nazis came to power early in '33, and then the people who had some taint were asked immediately not to teach, and so we didn't do anything. We had a private seminar mostly with Franck, and sometimes Weyl. Everybody was looking of course around, and we all were affected by the time of course, which also involved our personal fate and the fate of the nation. So this was a relief among all this turmoil, this little private seminar which was organized by Franck and by those people, and some others who dared to mingle with this crowd. Then one wrote — there were organized efforts in many respects to help and place these people. And I knew Brillouin, who was impressed by my papers on the theory of metals in which he was himself very much interested. The Rockefeller Foundation, as I was a former Rockefeller fellow, tried to help and they actually provided the stipend for me to work in Paris with Brillouin. And Born already left that summer. Then there was Lindemann, the later Lord Cherwell. He made an effort. He had interested the English industrial firms to give stipends, and he traveled around and talked to people to offer some of them fellowships. And he talked with me too, but he was against the Fermi-Dirac statistics, and since I had worked on this field, I didn't go to England.

Heilbron:

Now why was he against those?

Nordheim:

I don't know. I mean, he had some electron theory before of his own, and that was against his grain. He didn't believe in that

Heilbron:

How did it happen that you went to Holland then?

Nordheim:

Fokker offered me a Lorentz fellowship there.

Heilbron:

Your appointment at Paris was just for a year?

Nordheim:

It probably could have extended for another year, but it was not a permanent position. At that time one of course didn't expect that.

Heilbron:

What was the general feeling among people leaving Germany at that time? Were they convinced that they were leaving for good?

Nordheim:

Yes, absolutely. I mean, so long as the Nazis would stay in power, it was quite clear to I think everyone that there would be no return, and very soon that there would be war.

Heilbron:

So people were interested immediately in getting permanent positions?

Nordheim:

Yes, certainly.

Heilbron:

And was the United States viewed as the most attractive place, or England, or France?

Nordheim:

I don't think you can say that. Ultimately the most ended up in the United States of course, and secondly in England. Very few stayed in France, and could not very well acclimatize themselves there. The French system is very difficult for a foreigner unless you marry the daughter of a professor, to get a permanent hold in France. And some went to Russia, and of course in '33 they were welcomed, but in '35-‘36 they were thrown out.

Heilbron:

How did it happen that you got involved in translating that book of Jeans?

Nordheim:

Oh you know so many things about me. The first Mrs. Weyl was very literary, and I think she was asked to do the translation. She was a fairly well-known person in literary circles. And she wanted the help of someone who had more technical knowledge than she had, and so I was asked.

Heilbron:

How did it happen that this particular book was selected?

Nordheim:

This I don’t know. It is a fairly nice popular book, and the first thing I knew was that I was asked whether I would like to help in this. And, there is never too much money!

Heilbron:

Then you got involved in another one, or you improved a translation of someone else of another Jeans book, didn't you?

Nordheim:

Oh yes. You are right. You know more about me than I remember.

Heilbron:

Would you make some remarks about the early days of cosmic ray work and high energy physics?

Nordheim:

Ja, I started with my friend Heitler on this, and he was of course with Bethe one of the authors of high energy radiation by particles: Bremsstrahlung and pair production. And he wanted conditions for production of pairs by particles and involved me, and so I got started in this. It started rather slowly because this formal problem never has been completed; but it is not important. But then we made a qualitative attempt also to calculate the double quantum effect, or the higher order effect and so on. And then we went different parts. And then again I was led astray, because I thought I had experimental evidence that the formulas for radiation would not hold at high energies, and of course we now know that quantum electro-dynamics is valid for very high energies. But I thought I had empirical evidence that it was not the case. I published a little note on that, and tried to work out what happens on the basis of that hypothesis. In the meanwhile Bhabha and Heitler, and Carlson and Oppenheimer worked out the theory of cascade showers. So as soon as this was out I immediately found one thing, that you could explain any spectrum shower on the basis of a suitable primary spectrum. That means that the spectrum you observe in the atmosphere, anything you would find, could be explained in principle by a suitable primary distribution. And so I tried to analyze the cosmic radiation and so on, and did some mathematical theorems on it which were quite useful. But really the one thing which proved later to be of importance was one that I recognized, though this was a little bit after the time I think you are interested in, that if you consider the hard component you know cosmic rays are — in the old days you distinguished between the soft components which were electrons and photons and which behaved according to the theory of cascade showers, and hard particles you find at sea level and underground and which we know now are mu mesons. I found what I call the production anomaly, that they didn't seem to interact with anything as many experiments proved, and they were quite copious. So there was a difficulty and this I was the first to recognize, that these things are in a way incompatible, that if they are produced by something in the atmosphere then they ought to show greater interactions with matter. This has been resolved. I was not clever enough• to see that that was the pi-mu chain. The pi's interact with matter and then decay to mu’s which don’t interact with matter.

Heilbron:

Tell me, how did people react to the discoveries of these new particles? How did theoreticians feel about them? I'm thinking of the positron and the first meson.

Nordheim:

This is very curious. The positron, once it was established, one said this is fine. This is a consequence of Dirac's theory, and it should be there. Then Yukawa predicted, the meson as the carrier of nuclear forces first in 1935, I believe, and later more elaborated. And people liked the idea very much from the beginning. And then what we know now is the mu meson was found, and so people thought everything is good in this best of all worlds. Every thin has its place. But this didn't last long, and the mu meson didn't have the properties it should have had, and the theory became rather difficult. Ja, and we still are at it; it's a question of these particles, strange and otherwise. We know quite a bit of them, but very little rhymes. After Yukawa's theory and the discovery of the mu meson, you could speculate that really thing have to be like this. But not anymore. You need the mesons to keep the nuclei together; you need the photons of course for electro-magnetic interactions; and you need even the neutrino and the Fern interactions, because otherwise the world would be dead. There would be no energy evolution.

Heilbron:

By 1930, was atomic physics considered done, at least in principle?

Nordheim:

Ja, in the early 1930's one recognized one had probably most of the phenomena in molecular and atomic physics well understood, at least quantitatively, or one had the conviction that quantum mechanics was adequate to account for all molecular and atomic phenomena. And this I think was pretty well established very early in the '30's. Not that there wasn't an awful lot to do, and that things might be very complicated, but one really was convinced that quantum mechanics was the key to all atomic phenomenon. And so naturally the interested people gravitated to newer things: cosmic radiation and nuclear physics at that time and now also the discovery of the neutron, and then of artificial radioactivity. And of course in the development of the first accelerator, so that you could make experiments with greater certainty and more easily than with natural radioactive substances.

Heilbron:

We haven’t talked much about the large number of people whom you no doubt knew and met, and what you might recall of their attitudes and interest in physics, or anything relevant to the history of the period.

Nordheim:

I think I talked about quite a few people. The most important people for that period and for the development of quantum theory were Bohr, Born, and Sommerfeld, because practically every theoretical physicist learned their physics from them, and outside that, Ehrenfest, who was the Socrates of the theoretical physicists, who asked the questions. Ehrenfest was regarded very highly. He was very welcome to everybody, and it was generally a source of pride if you were invited by him to give a colloquium at his seminar in Leiden and put your name on the place under the blackboard where all the famous physicists had their names. There was a removable blackboard, and there was a wall in there, and simply everybody who gave a talk there was asked to write his signature on this wall, and so this was filled with the handwriting of all the famous physicists who ever visited Leiden.

Heilbron:

Was Ehrenfest always a little peculiar in his behavior, was his eventual end surprising to people?

Nordheim:

Yes. Of course he was known as an eccentric, but everybody had the very highest regards for him, and he was a very lovable person. And he was deeply affected by the Nazi business, and probably took it harder than practically anyone else. Of course I didn't know about his son. People when they learned about the circumstances about his death were not surprised.

Heilbron:

May I ask if you have anything more you'd care to enlighten us with?

Nordheim:

I believe at the moment I am running dry about the olden times.