Llewellyn Hilleth Thomas

Uhlenbeck:

I think the main purpose of this project is that it is certain that if one do something about it, very much gets lost. Whether it is very important if it gets lost, people have different opinions, but I was very much impressed by ail the letters of Pauli. For instance, the letters of Kramers were really at the point of being simply destroyed. And these things give the most vivid kind of impression of what has happened. There was in Kramers’ correspondence a letter with Pauli’s reaction on the Thomas factor two. This change of mind of Pauli about electron spin occurred apparently because of this factor. It is of great interest, I think, to find exactly when that happened and what his reaction to it was. So this is all. One tries to get a little bit, so to say, in between the papers what — has happened in between the papers.

Kuhn:

And for this of course the most useful material of all is letters, notebooks, this sort of thing.

Thomas:

Well, I have no letters, essentially. Even if I could find any letters from that time, they probably would not contain any mention of this sort. I just didn’t write to people about physics.

Uhlenbeck:

Ja, of course Pauli was a great letter writer and so was Bohr, but others are not. I have also very few letters…

Thomas:

Well as far as factor two is concerned, it started with Professor Bohr and someone else — I can’t remember who — and Kramers. I don’t know whether Kramers was there when Bohr was talking about what you and Goudsmit had done. The question was of what are now called spin doublets versus relativity doublets, and the fact that they don’t go for the higher alkali metals as you would think on the basis of the Sommerfeld relativity theory. And you had suggested that it should be connected with the magnetic moment and angular momentum of the electron, but it gave something wrong by a factor two,

Uhlenbeck:

For the fine structure?

Thomas:

Yes, for the splitting. Double splitting. When this was discussed, I said I didn’t think the calculation could be done properly unless it should be done relativistically. And I can remember Kramers saying that he was not going to do it relativistically because that would just make a small collection and wouldn’t help.

Uhlenbeck:

That’s what everybody would say.

Kuhn:

Had the possibility of doing it relativistically, the chance that this would make some real difference been discussed?

Thomas:

At that stage I doubt that it was discussed.

Uhlenbeck:

No, very little. Sam and I certainly were glad that we knew how to compute it unrelativistically, because that was already in that stage perhaps not quite familiar. But perhaps it is a good idea to go back a little bit before we come to this. You were at Cambridge?...

Thomas:

Well, I had first of all a scholarship to Trinity College, and I took mathematical tripos as an undergraduate and specialized in advanced subjects in what they called applied mathematics. Here it would be called theoretical physics. Prof. R. H. Fowler, who at that time was a Fellow of Trinity and director of studies he had no professorial appointment at that time was my director of studies. He suggested what courses I should take. Among other courses I attended Professor Eddington’s lectures, just about a year before his book on the mathematical theory of relativity was published. Then later on I got his book. I was quite familiar with relativity theory at that point… Now, who took Eddington’s course at that time? Oh, six or eight of us I suppose.

Kuhn:

Did the course follow the book quite closely at that point?

Thomas:

I have notes of it. If you are really interested in that point I have notes of the course which I could let you have… You can’t say it followed the book, but it might be regarded as a selection of material which later on might have been expanded into the book, I suppose… It was a one-term course covering special and general relativity and tensor analysis. And it went on as far as Eddington’s more general ideas of geometry… Of course he published papers of these things before he wrote his book. I also took various courses of Fowler’s on quantum mechanics… It was of course the Bohr theory and applications to molecular spectra and to various other things. I think I do have notes of all these lectures that I took at that time… I write very illegibly. I can’t even read my own writing after some years, so what I did in most of these cases was to take notes almost verbatim during lecture and then afterwards I’d go over them and rewrite them. And it doesn’t follow that what I have in my notes is what the professors said, because it has had this editing, and I have interpreted what I might not have been able to read. Very often I didn’t do this rewriting for some months. Sometimes I did it immediately. Some notes I have never rewritten and they are almost unintelligible. I did perhaps also go to one or two courses which were physics. This is a question of who gave them. For instance I went to a course of Larmor’s. And it at first was one of the most boring courses I had ever been in. Just potential theory, treated from a rather old-fashioned point of view. But later on he suddenly started to do research on the blackboard, and he worked out on the blackboard in front of the class work which he almost immediately published about the gyration of radio waves in the Heaviside layer.

Kuhn:

Did he talk quantum theory at all?

Thomas:

No, oh no.

Kuhn:

You see he published at the very beginning of the century. There are a few interesting but very cryptic notes on the subject of the Planck problem and black body radiation. I wondered whether he carried on with this interest at all.

Thomas:

I don’t think so… Oh there were other people who did lecture on quantum theory and things advanced physics as it was at that time. But I don’t think I attended lectures by anyone else on quantum mechanics. Birtwhistle started lecturing on it, on the basis of his books a year or two later.

Uhlenbeck:

And how about the experimental side, Rutherford was already there.

Thomas:

Rutherford had been there a long time, but I had no direct contact I think with the experimental people, except that I would go to the various groups that met — colloquia. But I had practically no contact with experimental people. Kapitza was there at the latter part of the time I was there. And there was a group of people called Kapitza’s group that met about every second week in the rooms of various people — occasionally Kapitza’s, then someone else. It was even in my room… This was a group for graduate students… I may have attended a meeting before I went to Copenhagen. You see after I got my bachelor’s degree I went on one year for graduate work at Cambridge. I went for one academic year to Copenhagen. Then I went back to Cambridge.

Kuhn:

You were back at Cambridge until you came to this country?

Thomas:

Yes.

Uhlenbeck:

How about Dirac?

Thomas:

Well, Dirac was around… I knew Dirac, Yes. He had some of the same courses I think that he has now. He is a man of few words. If you ask him a question, he’d say oh, that’s very difficult. Then a week later he’d come back with the complete answer completely worked out.

Kuhn:

How early do you suppose you knew him there?

Thomas:

I’m pretty sure that I met him before I went to Copenhagen. But my recollections may be wrong… He is older than I am. About one or two years I think. But he was a senior in the University. He was a graduate student before I was.

Kuhn:

Did he make a major impression at Cambridge from the time of his arrival, or was he just a quiet man until he published his paper?

Thomas:

I would say that, yes. He was a quiet man until he published his papers.

Uhlenbeck:

Do you recall, at the Kapitza group or on other occasions, he talked on his early ideas about this?

Thomas:

About this I don’t know, because he was doing most of the early work on this while I was in Copenhagen, I think this is certainly correct, And so I must have seen him before that, I remember a talk of his. I don’t remember now whether it was at Kapitza’s group or some other. There were other groups. There was a Cavendish society which was a more formal group than the Kapitza’s group. And then there were various other groups I belonged to. I belonged to the Observatory Club, and I also belonged to the Trinity Mathematical Society, which did as much applied mathematics as anything else. At one of these societies, I think it was maybe Kapitza’s group. Dirac gave a talk about Bose-Einstein statistics. He said at the end he thought it would bear looking into if you went to the other extreme away from classical statistics. And this was before he did what afterwards became Fermi-Dirac Statistics… Yes, I think this must have been before I went to Copenhagen at that.

Kuhn:

Experimental physics would have meant then Rutherford’s group at Cavendish, at this point?

Thomas:

Yes, entirely. Of course after Kapitza came there, Kapitza was doing engineering experimental physics.

Kuhn:

How much attending of each other’s courses would there have been then? Did the experimentalists try to learn what was going on in theoretical physics as a matter of course?

Thomas:

Some of them would. There was nothing to stop them going to attend lectures if they wanted to. I think there was as much communication there as there is here, perhaps. But the point is the main dividing line then there was between mathematics and physics. Physics was experimental physics and mathematics included theoretical physics; whereas here the dividing now, which is much sharper than that line was, is between pure mathematics and everything else. Now let’s see. Who would be good examples? There was (Gaunt) who published some good papers. A man whose parents I think had been missionaries in China, and he regarded it as a calling to go and teach in some school in China. He was one year junior to me… The younger (Hartree) was two years senior to me, I think, Blackett was there, and Dymond… Well, these are graduate students, and junior staff members.

Uhlenbeck:

And, so to say, the center was all R.H. Fowler?

Thomas:

Of theoretical work. But at the Cavendish Laboratory with Rutherford and a lot of people was the experimental work. C.G. Darwin was there, in the earlier part of the time I was there… Not as a student, but on the level of Fowler. I attended lectures of Darwin also…, were the regular undergraduate lectures on statics, dynamics and so on.

Uhlenbeck:

Maxwell theory and such kinds of things, was that taught on the under graduate level?

Thomas:

Oh yes. But you see people taking them at that time were people who were working towards an ordinary degree. This is the sort of thing that you have here; two major subjects and three subsidiary subjects. And then there were people working towards an honors degree. And they did nothing but their specialty. While I was at Cambridge I did nothing but what I call mathematics, — for the Tripos.

Kuhn:

But would that have included electromagnetic theory?

Thomas:

Oh yes. Cunningham was the lecturer I took electron theory from, and then I also got some lectures from (Dolmetsch), lectured on the Heaviside methods applied to all kinds of more complicated problems.

Kuhn:

Where did Eddington fit in?

Thomas:

Eddington was director of the observatory and a professor. Also his lectures were part of the curriculum for the mathematics tripos. You see people gave advanced courses of lectures, and a committee made out a list of lectures which were available for various things… Now Cambridge at that time was made up of the colleges. And at Cambridge the colleges always had more — it’s not exactly power — but they were more effective there. At Oxford the colleges were more in the university. Because at Cambridge the colleges had the money and the university got any money for any university projects by taxing the colleges. Colleges were required to provide a certain number of fellowships for professors. Now Eddington was a professorial fellow of Trinity College at the time that I was there, and, being the professor he was, ipso facto, director of the observatory. By the way, there was another club also. It was called the Delta Square Club. This was a rather old organization. It didn’t meet very often — about two or three times a term but it had quite interesting papers. I think I still have the invitation card for a meeting of the Delta Square Club at which Eddington gave his first discussion of 137… Oh there were other clubs too. And some of these were ephemeral; some were of long standing.

Kuhn:

Did one tend, if one was in mathematics, to belong to all of them?

Thomas:

I don’t think so. For instance the Delta Square Club, you had to be invited to be a member. You were elected a member by the member… I don’t remember exactly, but I think there were about 20 members, maybe not as many… The Kapitza Club was much less formal and it met more often. And I suspect that it didn’t continue any longer after Kapitza left. There would only be about six or twelve at a meeting of the Kapitza Club.

Kuhn:

And the Cavendish club?

Thomas:

This tended to have meetings in the lecture room in the Cavendish with a larger number of people going to it. They published notices that they were going to meet, so it was more like a colloquium, The Cavendish Society I think it was called. The Trinity Mathematical Society was an undergraduate as well as a graduate group… Its meetings were not open. They had members. But I think only members of Trinity College would have become members of that society. Other colleges also had societies of that kind of their own; at least some of the larger ones did…

Kuhn:

Was there a good deal of overlap between the subject matters considered in these clubs, or were they relatively distinct in their function as well as sometimes in their membership?

Thomas:

Take the Kapitza Club or Kaptiza’s group as we always called it at that time. If you took the Delta Square V Club, there would be considerable overlap; or considerable overlap between Kapitza’s group and the Cavendish Society… The Trinity Mathematical Society was more undergraduates and also graduates. And what it did was to get people of the standing of graduate students and members of the staff to give talks on any kind of thing that they thought suitable… Staff members of Trinity would come to the meetings quite often. One of the fellows of Trinity named Pollard, who taught various mathematical courses, came and gave a talk to the mathematical society urging that applied mathematics should be instantly and radically revised. So (Soddy) appointed him a committee of one to revise the branch of applied mathematics. And he proceeded to give another talk about six months later on ways in which the teaching of geometrical optics should be revised. Or at least he was particularly critical of the arbitrary nomenclature that was used. But I don’t think what he did would have any effect. There were people who really did something about that, but that was not he.

Kuhn:

The Kapitza group was restricted to graduate students?

Thomas:

No. I mean, I suppose Fowler would come to meetings occasionally. He wasn’t at all the meetings, nor was any member.

Kuhn:

No, surely not. Were these, all of them, generally mathematical groups, or did they include experimentalists?

Thomas:

Well, Kapitza’s group included experimentalists, and so did the Cavendish society. But the Delta Square V Club was essentially applied mathematicians; at least what were then called applied mathematicians. For instance Southwell gave a talk on the question of stability of various hydro-dynamical flows.

Kuhn:

What about pure mathematics and its relation to applied mathematics, group theory, abstract algebras?

Thomas:

Well I didn’t attend any groups that went in much detail into that. At the Trinity mathematical society occasionally there were talks about various abstract kinds of things.

Uhlenbeck:

Was Hardy there?

Thomas:

Hardy was not at Cambridge while I was a graduate student. He had been there earlier, but I don’t think even at all when I was an undergraduate. He may have been the first year. I wouldn’t have known. And he didn’t come back to Cambridge until after I had left… I had already done when I got to Cambridge what is essentially the first year’s work — work for the first part of the mathematical tripos. So I started in on the regular curriculum for the schedule A as it was at that time — the second part of the mathematical tripos. This is the part on which degrees are really given. And I covered the material for that in two years in formal courses which were equally divided between pure mathematics and applied mathematics, Some courses I didn’t attend but got up the stuff myself. Then my third year I spent on more advanced subjects, which would be things like Fowler’s lectures on spectroscopic subjects. Also I attended Baker’s lectures on dynamical astronomy and various other lectures like that. The year I was a graduate student I took no lectures. None at all. I mean I went to one or two lectures. I went to Ramsey’s lectures on foundations of mathematics. And I may have gone to a course of lectures of Fowler’s on some things, but I think two courses of lectures would be all I took as a graduate student.

Uhlenbeck:

Now how did you get to Copenhagen? You got a fellowship from Cambridge?

Thomas:

I got a studentship from Cambridge for paying traveling expenses. And I was at that time an Isaac Newton student at Cambridge, and I could hold that studentship anywhere. And so with the Isaac Newton I went to Copenhagen. And because I was going abroad I got a (Rouse Ball) traveling fellowship which would pay traveling expenses.

Uhlenbeck:

Why did you go to Copenhagen?

Thomas:

Because Bohr was there… Either a year or two years before that, Fowler had been quite a long time at Copenhagen. And various other people were there. I mean this was presumably on advice that I went there… And I was there for the 1925/26 academic year… Well, now while I was in Copenhagen there were three problems principally on which I did work. I was already started on the first and finished it while I was there looking at the capture and loss of electrons by alpha particles passing through matter. And a paper was published on that… That I started on Fowler’s suggestion before going, and I went on with that… The main point there was that the experimental data had been wrongly interpreted. It was believed that the latent capture and loss of electrons by hydrogen was much larger than it in fact was. When I found that I couldn’t see any way that you would get a larger effect, I discussed this with Professor Bohr. Some people thought that the course of quantum theory would explain all sorts of things of this sort. But in this case it turned out that the observations had been wrongly interpreted. In the case of hydrogen they did not have the large values that they had for other things… And while I was there I looked at the experimental papers, and came to the conclusion that the experiments really supported the theoretical result, that the effect would be much larger for heavier things.

Kuhn:

Did you work closely with Bohr on that, at that point when you got back to the data?

Thomas:

… Now I don’t know, Bohr was always inspiring, but the first couple of months I was there, I saw very little of him. I think he was very busy at that time. And then either just before this or just after it must have been before Christmas, I think this question of the spin was brought up.

Uhlenbeck:

Before Christmas he was in Leiden… Bohr was in Leiden, and then we talked with him. This was for the occasion of Lorentz’ doctor’s degree. And after this Leiden visit he went to Germany, and he was then more or less convinced. Well, he was then in Copenhagen, which must have been about Christmas, I think.

Thomas:

As I recall, after this was talked about, I worked out the factor two in two or three days and brought it back to him.

Uhlenbeck:

Well, then let’s talk about it. Of course as you said, Kramers thought that it should be a very small effect. Were he and Bohr very soon convinced?

Thomas:

Kramers wasn’t there. He was away for some reason when I talked to Bohr about it afterwards. Bohr was almost immediately convinced. And Bohr introduced into the discussion some considerations which I hadn’t put in at all, and which I don’t know how relevant they are to getting the result. They are certainly relevant to understanding it. This question of whether you would regard the electron as a little magnet or as a rotating charge makes a difference in what you might expect the field to do to it.

Uhlenbeck:

Now, I have also certain memories about that time. One of the things which is so striking is just this fact that this composition of Lorentz transformations resulted in the precession. Now this was a consequence simply of the Lorentz group, which at that time one would have thought would have been investigated in great detail already. Apparently it wasn’t because I know or at least from hearsay, but I think it is true. — That even Einstein was not quite familiar with this; that so to say two Lorentz’ transformations in different directions were not a third Lorentz transformation.

Thomas:

Well I got the result at that time to apply it by directly taking the three dimensional rotation group and the (Roblagay) formulas, which essentially the group formulas constitute, or if you like Gauss’ rules. If you go around a circuit on the surface of a sphere, you have a spherical excess. All you had to do was to take that formula and change one of the variables — change say z to √-1 and you have x, y and t instead of x, y, and z.

Uhlenbeck:

Was that the way you found it?

Thomas:

Yes. Immediately.

Kuhn:

Well now was group theory, or thinking about transformations as groups, was a part of your standard equipment that you brought to the problem?

Thomas:

No, at that time I knew practically no group theory at all. I had to pick up group theory the hard way later. There were no courses on it given while I was a student at Cambridge. I know some people talked about it, and some of the students at Cambridge were griping that there was no one giving courses on it. They said that courses ought to be given.

Kuhn:

So you went back really to this result of Gauss’?

Thomas:

Well, to just the composition of rotations in the ordinary three-dimensional rotation group, which I knew very well.

Uhlenbeck:

The (three component) theory?

Thomas:

Yes.

Uhlenbeck:

I didn’t know that. I was under the wrong impression. I thought that that was the way Kramers interpreted it later, but that you had found it in another way.

Thomas:

Now, you see I was familiar with the general relativistic effects of the motion of the Moon’s nodes. That is given in Eddington’s book which of course was readily available at that time, and which I looked at in connection with this. Now this was exactly a similar problem.

Uhlenbeck:

Ja, but still it was not a problem in special relativity.

Thomas:

Well, I don’t know that there is this sharp distinction… The kind of mathematics you would use to discuss it was exactly there in the discussion of the moon’s nodes.

Kuhn:

You spoke in your letter to me of the paper of (Schouten.)

Thomas:

Well that paper of (Schouten’s) I had not looked at at that time, but this was the origin of the discussion in Eddington’s book.

Kuhn:

Did you go back to the original paper in the course of this, or did you simply take it from Eddington’s book?

Thomas:

What I had of that I took from Eddington’s book. Eddington’s book and composition of rotations through (???) Hamilton’s theorem was the basis of doing this. It only took a few days to get the result that way, applying just the business that you can — to a rough approximation on the basis of the old quantum theory get the effect on the spectrum if you can get the precession… Can we go back and pick up a little bit more of the structure of the development of the problem… Your early news, Professor Thomas, of the idea of electron spin came verbally from Bohr, and from discussions with Bohr.

Thomas:

Yes.

Kuhn:

In the original note in there is no mention of the difficulty over the factor of two…

Uhlenbeck:

No, then we didn’t know it yet. The one who told us that there was the difficulty of the factor of two was Heisenberg by letter…

Thomas:

I have heard it asserted — and this is just second hand or third hand report — that Kronig had gone through these things before that, and had come to just the conclusions you came to and gave it up because of this difficulty… That is, I heard this report at the time.

Uhlenbeck:

But with regard to us, it was that we heard about the factor two difficulty from Heisenberg in a letter. Arid that came a week or two weeks after the Naturwissenschaften note. And at that time we reconstructed it, because we had not even properly derived the fine structure formula. Then we found also the factor two difficulty of course, Heisenberg mentioned that surely to Bohr, at that time, And Bohr must have —

Thomas:

Well my recollection is Bohr and Kramers and probably someone else and mye1f were there at a discussion, in which Bohr talked about this and I said “Well, why didn’t people work that out relativistically and Kramers said, “Well, it will only make a small correction,”… It must have been after Christmas. This is I was not sure whether it was before or after Christmas.

Uhlenbeck:

Because your note I think appeared in February in Naturwissenschaften.

Thomas:

Yes. And the only notes I have of it are the ones which I have given you copies of.

Kuhn:

And those are presumably — many of them — not the first —

Thomas:

No, no. I would very likely throw away earlier drafts of an argument in favor of a later one… The paper in the Phil. Mag. came out after I had returned to Cambridge. But the paper was written, I think, when I was in Copenhagen, and only the appendix was written afterwards. But it takes a longer time to publish a paper than to publish a letter anyway. Those letters at that time to Nature were published almost immediately. I believe they still are.

Kuhn:

Going back to the first discussion with Bohr, did Bohr himself seem fairly thoroughly convinced at that point?

Thomas:

Bohr brought up the argument of the doubling of lines in the alkali spectra and so on as indication that in the hydrogen fine structure something like a spin must come in, and that it couldn’t be explained on the basis of the Sommerfeld theory of the hydrogen atom.

Uhlenbeck:

Alone, at least.

Thomas:

Yes, not alone. And Bohr was already convinced that there must be something of this kind with a spin. Now I’m not clear whether he would have ruled out the possibility of a nuclear magnetic moment (?????) having the effect at that time. That was also one of the suggested explanations, but one which did not give the right results… I think I had no difficulty at all in convincing Bohr that the relativistic effect is —

Uhlenbeck:

Was the factor of two instead of one over the square root of —

Kuhn:

Was Bohr himself at this point quite thoroughly in command of relativity theory? Or were people in general?

Thomas:

I don’t know what you can say about — Oh, there were plenty of people who were thoroughly in command of relativity theory. I was always amazed at. Bohr’s insight, and as I grow older I feel that insight is a matter of experience and knowledge. But Bohr was not so — how does one say — articulate, that you could really be clear about the logic of things.

Uhlenbeck:

Well, and he had certainly no interest in such formal things. But Kramers surely knew relativity theory, no doubt. Now I noticed that in your paper, you thank Pauli and Heisenberg.

Thomas:

Oh, this was either a letter, or they told it to Bohr, on some visit to them. You see by the scheme of this perturbation theory giving the splitting, you couldn’t say what should be the change in the ground state position, and this would be necessary to get the hydrogen spectrum correctly. And Pauli and Heisenberg asserted that the new quantum mechanics would then give the ground state shift correctly.

Uhlenbeck:

But you didn’t talk about the precession with Heisenberg or Pauli personally?

Thomas:

Not early. Later on both Pauli and Heisenberg at different times were at Copenhagen that year, and I talked to them about various things. Pauli and I had great difficulty with talking too at that time. He wanted to talk to me in German and I should answer in English, and I found this very difficult. Though, undoubtedly that would have been the best way to do it. But if he said anything to me in German, I would try to answer in German. And so he was reduced to talking to me in broken English.

Uhlenbeck:

It is I think historically quite interesting to know the reactions of Pauli, because the hearsay is that this convinced him.

Thomas:

This may be so… Weil, I have no knowledge as to what Pauli’s reaction was. I only saw him later, after things were settled. Of course this spin factor two was just a snail sideline really. I was spending most of my time at that time working on statistical electron distributions.

Uhlenbeck:

The Fermi-Thomas distribution.

Thomas:

Yes.

Uhlenbeck:

That was the third topic that you worked on in Copenhagen already?

Thomas:

Yes. Now I had thought about this. You see, I had met (Hartree) before that, and I had used (Hartree) (?????) fields for calculating scattering and things of that sort, and so I was interested. And then it was a question of whether one couldn’t get some kind of an approximation to a field for scattering by a smoothed-over electron distribution. One got an intuitive physical equation, (an equation for physical reasons for this.) It turned out that it had to be solved numerically. And I think this was the beginning of my knowing anything about numerical analysis.

Uhlenbeck:

Was this then really completed independent of Fermi?

Thomas:

Yes. More than a year before Fermi… Fermi did essentially the same thing, but he put in a little bit more. He analyzed angular momentum. Now I had not done this, because the analysis of angular momentum had been done on the Hartree fields before, and I thought that this would be regarded as following directly on. I actually did the numerical work while I was at Bohr’s Institute.

Uhlenbeck:

But did you know Fermi’s paper on the Fermi statistics for gases.

Thomas:

I don’t think I had that paper. I don’t know what date that paper was. This was on the basis that the Pauli principle says that you should have two electrons per —

Uhlenbeck:

Ja, but the main thing is that you thought of the Fermi sphere already in a sense.

Thomas:

Well, one knew, you see, that the Bohr orbits formed a uniform distribution in phase space. I mean that all this old business of (Kronecker’s) theorem and Berger’s theorem and so on — well one knew that in the general separable case, the Bohr orbits filled up the phase space uniformly. It was only a short step to saying that if you have two electrons per orbit you just distribute them in the phase space and you have naturally distributed them up to a definite energy. And the point was that this gave a quite reasonably good agreement to the older (Hartree) fields.

Uhlenbeck:

It is also a very little step to say that even if the nucleus is not there, that happens. You say, well, for the free electrons, you must have the same thing. This is the step that Fermi took, and I was just wondering —

Thomas:

Now I have the feeling — I’m not sure — that Sommerfeld took this step before Fermi.

Uhlenbeck:

No, no, surely not… Sommerfeld did it after Pauli. And Pauli came after — No, Fermi did it. I don’t know precisely the date… He did it in an harmonic oscillator field for simplicity, and then let the harmonic oscillator get very feeble so that he got a (vessel). And in that way he got the equation of state of the Fermi gas…

Thomas:

Now I was familiar with Dirac’s paper on this about as soon as it was done. But I don’t remember exactly what it —

Uhlenbeck:

No, Dirac’s paper is after quantum mechanics, so to say, because that was the antisymmetric wave function… Fermi didn’t use the quantum theory except just the same ideas that is the uniformity in phase space. But he didn’t take a central atom. He said well, a (vessel?) is just as good. He represented by a harmonic potential. And it is so close to this same idea really.

Kuhn:

How had this problem of the statistical field gotten started?

Thomas:

As far as my ideas were concerned, this was with the older Hartree fields. Now Hartree took the x-ray spectra and ideas about shielding and so on. He calculated the Bohr orbits in a smoothed-over field produced by other electrons in Bohr orbits in the atom. And this gave good representation — qualitative, I think within 10 percent anyhow, perhaps better of the x-ray levels. But it was a great deal of work to carry this through for any one atom, and the question was whether one couldn’t get by a simple averaging assumption; something which would be an approximation which would apply to all atoms… Then there was someone else who had been at Copenhagen the year before that who had done some work on Hartree fields, so it may have been at the suggestion —. But I had worked with Hartree fields before that, so that I had Hartree fields available to compare with them… The idea of doing it that way, of getting a statistca1 field, I did not have before I went to Copenhagen. But that took considerable time to do since it involved numerical integration. I didn’t know how to do numerical integration. I had to learn.

Uhlenbeck:

The correction amounting to a factor of l/2 must have been in the beginning a shock, but apparently there was no great resistance.

Thomas:

Well I think it doesn’t require any deep feeling for group theory to see that if you take rotations of an x, y, z space and you make z√-1 t, you get an x, y, t space, and you should have the same formula…

Uhlenbeck:

The non-commutativity of rotations, so to say. I see that was really your starting point.

Thomas:

Now I can’t say absolutely definitely what my starting point was. I mean I don’t know whether one very often can. The first thing, what I tried to do as to follow the electron around the orbit and see what the field looked like to the electron. This would make the electron process. But the precession should be in its own frame of reference. And then one has to carry out the combination of the rotations to do this… But the idea at the back was this general relativistic rotation of the moon’s nodes, and how you got —

Uhlenbeck:

Well, it is I still think so remarkable, because certainly everybody, even knowing a little bit of relativity, would say that it’s a tiny effect, if there is an effect. Apparently you immediately thought that it might be a big effect that the relativistic theory really here does something essential which, so to say, the simple considerations don’t give.

Thomas:

No, I don’t think really that at the beginning I thought it was necessarily a big effect. I thought that it would not be difficult to do it relativistically…Sommerfeld gave these same things as definitely a relativistic effect… I thought the calculation should be done seeing what would happen.

Kuhn:

Were you really looking for the factor of two, or did you decide to do this just on the basis that it could be done, and ought to be done, relativistically?

Thomas:

That is very difficult to say. You see I was not so familiar with spectroscopy that the factor two discrepancy was very striking to me.

Uhlenbeck:

I see, because that was for us an enormous trouble…

Thomas:

No, I think I probably said, “Well, you’ve not taken the relativistic effects into account, why don’t you take them into account?” But I don’t suppose that I had any ideas beforehand that it would just put things right… Then the answer came very quickly. It was a quite straightforward calculation once you had the idea of doing it.

Kuhn:

Who else was in Copenhagen that year, or at least for any length of time? Kramers was there?

Thomas:

Nishina was there, and (Shigiuto) was there, and (Kuhn?) was there. Now let’s see who else? Dennison was there… Goudsmit was there for a while… Then Heisenberg was there part of the time. I can’t remember exactly. Probably the same kind of short period of time that Goudsmit was there. Pauli was there just for a day or two. A short visit.

Kuhn:

What sorts of problems were principally under discussion?

Thomas:

Well, they were doing experimental work on various kinds of x-rays, and other things too. But, the things which were most under discussion were the things involved in the beginnings of quantum mechanics.

Uhlenbeck:

How was that discussed? Was it a colloquium? Or was it done privately, say two men at a time?

Thomas:

Practically no colloquia, I think, while I was there. Practically everything was that people would be around there in the library, and Bohr would have some time, and he might call one or two people in to talk to them, or he might talk to a large number of people.

Kuhn:

Can you capture any of your own or other people’s feelings about that whole series of papers. The first Heisenberg paper presumably had come out just before you got to Copenhagen.

Thomas:

What do you call the first Heisenberg paper? The one that was completely wrong, or the ones that were beginning on the right direction? There were precursors you know, by Heisenberg. Earlier papers which were trials that never came to anything.

Kuhn:

I mean the paper that was the immediate predecessor of the Born-Heisenberg-Jordan work.

Thomas:

The paper you mean is the one in which he takes the harmonic expansion of motion and says that you have to modify these terms and make them of different frequencies rather than the same… While I was at Copenhagen this was not really advanced — I never understood any of it… Dennison was interested in the applications in molecular spectra and extension of Kronig’s ideas on what they called the sharpened correspondence principle. And all the ideas that I got of matrix mechanics I think were in connection with simple molecular theory.

Uhlenbeck:

I see, then that must have been later.

Thomas:

And I never understood anything about any of these things I think until — I didn’t get really any sense out of the Heisenberg papers or the Dirac papers, but I did when the Schrodinger papers came out. Then I began to understand things… These began while I was still in Copenhagen… About March…I only stayed the academic year. I think I left about the beginning of June or something like that.

Kuhn:

Do you remember discussions or reactions to either of the papers?

Thomas:

Well I don’t — I can say. I felt that I understood nothing until the Schrodinger papers. And the Schrodinger paper I don’t think I read while I was — maybe at the very end of the time I was there. But I remember there was a question — You see the matrix mechanics — No, I must have heard something about it, but one felt that one couldn’t calculate anything with it. And when the Schrodinger paper came out, one saw how you would calculate things. I think this is essentially the difference. One felt before that that it was just ad hoc assumptions brought into each problem, largely because the mechanics of solving the equations was different in each case, and horribly complicated if you looked at Dirac’s papers on the hydrogen atom. While Schrodinger’s paper made it fairly easy to calculate things, and related it to mathematics when you —

Kuhn:

Did one have any sense with the Heisenberg papers initially of fundamental —

Thomas:

One had a sense that something of this sort must be a direction in which things were going, but one just didn’t — One regarded some of these things as coming out of the sharpened correspondence principle business. But this may simply be the influence of Dennison’s method of taking it from that point of view at that time…

Uhlenbeck:

Did Dirac come to Copenhagen in your time?

Thomas:

No, he wasn’t there while I was there at all. I think he came there the next year. I’m not sure.

Uhlenbeck:

You didn’t get in touch with him when you got back to Cambridge?

Thomas:

I may have seen him, but not to talk about these things.

Kuhn:

Was there gigantic excitement over these papers, first the matrix mechanics — then the Schrodinger?

Thomas:

I think only a few people understood anything about it, and I don’t think there was any gigantic excitement.

Kuhn:

That would be true also of the Schrodinger papers?

Thomas:

I think that’s true too. Now the de Broglie paper of course was earlier, and that was something again that I could make no sense of at all.

Kuhn:

Had you and other people around you tried? I mean, was this a paper that one really tried to work out? People knew about it, but I couldn’t make any sense out of it. What was the sense of taking a Bohr orbit, and then saying that a wave had to have a whole number the wave didn’t fit into anything? But it wasn’t long after that that the experimental, diffraction of electrons?

Added Note By Kuhn After Conversation With L. H. Thomas:

As he left, Professor Thomas indicated that it had been a peculiarity of Fowler’s that he had told him not to come to his — that is to Fowler’s elementary lectures. One set was on thermodynamics, which he had then worked up for himself from Bryan’s book on the subject. The other set of lectures was on differential equations, which again Thomas worked up for himself. One other interesting remark was made in connection with the discussion of the course I forget who gave it — on integral equations. Apparently there was a good deal of complaint here and there from people that that course was not designed in a way which indicated the physically applicable material, so that people didn’t know how to use it. It was too classical they felt, in its approach.