Werner Heisenberg - Session IV

Kuhn:

I don't know whether you've had a chance to look more at the Rumpf paper.

Heisenberg:

I did look at the paper, but I didn't find out many things beyond what we had already discussed. I think that you asked me about this sum rule business. I think the sum rule was suggested at that time by the fact that only in this way you can get rid of these (square roots). So I would say, in a mathematical sense, the sum rule anticipated this idea of having a secular determinant. Then the sum of the diagonal elements must be independent of the magnetic field and all these things. The physical interpretation was rather vague. This idea of having only statistical conservation and so on was probably something which was in the air at that time. People spoke about statistical conservations but it was probably just what one called a rather foolish excuse for something which one didn't understand.

Kuhn:

Later we will discuss how people felt about the Bohr-Kramers-Slater paper. I wonder if you remember any discussions at that time about violation of energy or momentum conservation?

Heisenberg:

Well, I would say that most people would have, at that time, said, "Well, it looks as if energy and momentum are only conserved statistically." But they would not be apt to take it very seriously. They would say, "Well, we don't understand. We have the wave there, we have the light quanta, both things must be right somehow, but we don't know what somehow means. It looks as if the whole thing was only statistical, but it also sometimes looks as if it was not statistical, so we just don't know." It was only much later that one really could, by experiments, decide whether or not energy was conserved in the single acts. This was in the Bothe-Geiger experiments. That was, of course, a very important step. May I just come to this later period for one moment? I remember that already before the Bothe-Geiger experiment was really carried out that Bohr felt that it should come out the way it later did come out, that this statistical conservation was not the real point. So Bohr had in some way already talked himself out of this idea of only statistical conservation, probably because he had seen that this brought very great difficulties in many places in physics. So one simply came back to the old situation and one felt, "Well, it looks as if it was only statistical, but really when you check it, it is not statistical, and so we just don't understand what happens." It was only later on that one felt — that was also, of course, due to Bohr — that all these words don't really get hold of the real situation. So that the word light quantum, or wave and so on, each such word always implied certain pictures which are not correct. Therefore, one could not get the whole thing to fit.

Kuhn:

We are a little ahead of ourselves, but what you just said relates so closely to another question of this sort that I wanted to ask you. I think it is striking, although no longer surprising, but it certainly is surprising against the background of a methodology that you give up ideas because they have been proven wrong, that although not very long after the Bohr-Kramers-Slater paper there are the Bothe-Geiger experiments and then the Compton-Simon experiments, nevertheless, a large part of the basic ideas and the whole use of the Correspondence Principle formulated in terms of virtual oscillators goes on quite unshaken.

Heisenberg:

Well, I would say that everybody felt that paper contained an essential part of truth. I would like to formulate this essential part of truth in the following way. What Bohr, Kramers, and Slater did was to establish the probability as a kind of reality, that is, that the probability is not something about just counting numbers ... but that this expectation itself is like something real. That was, of course, something absolutely unusual from the classical point of view. That "something" had developed in the minds of many physicists at that time because physicists do acquire feelings by studying the experimental situation, and there one saw the waves and one saw the light quanta and so on. You could always see these things in some way. So one felt that by making the probability become some kind of reality, you get hold of something which is there. It was, at that time of course, very difficult to say what it was that you had gotten hold of. I would say only through the paper of Born did it become quite clear that one should say, "All right, the Schrodinger wave means the probability that an electron should be there." But the main point was that the probability itself was something real. It was not only in the mind of the people, but it was something in nature. That was a new feature; that was a real progress which helped. It was an invention of a concept which was proven to be very helpful in the whole discussion. This concept then was not lost. Nobody wanted to lose this concept even if the result of the experiment of Bohr and Bothe-Geiger came out contrary to this idea. Everybody felt, "Well, even if Bohr was wrong, and the Bohr-Kramers-Slater paper was wrong, even then this concept must be something usable." ... Up to that time people had two possibilities. One possibility was that the reality is a wave. There is an electric field, and a magnetic field acting upon an atom, shaking the electron, and then the atom does something, it makes a transition. All right, that is a thing that one can imagine. There is an entirely different picture of reality in which there is a light quantum coming on with the velocity of light, hitting the atom, and then something happens. But now the idea is that there is a wave. But this wave is not the reality. This wave is a probability — this wave is a tendency. It means that when this wave is present then the atom gets a tendency to emit light quanta. So this idea of the wave field being a tendency was something just in the middle between reality and non-reality. So you invented the old Aristotelian term of possibility or potential. At least it was something in the middle between the actual fact and the non-fact. And it was the tendency that a fact should happen. That was the striking thing about it, you know, this new invention of a possibility which was a reality in some way but not a real reality — a half reality.

Kuhn:

Did the Aristotelian notion play any role in discussions at that time or is that something you would have thought of later?

Heisenberg:

I think the latter one. Later on I discussed with Bohr that that was more or less what Schrodinger had also played with, and, of course, it interested him. I think we followed with some discussions about what the fact that old philosophical ideas again and again come recurring into modern science means. You can say many things about this old problem. It's a very nice problem. On the other hand, it's quite clear that this idea came up I think entirely without reference to Aristotle. It was forced upon the physicists by the experimental situation which we found.

Kuhn:

In order to do that paper, one talks not only about a tendency to absorb, and about the probability of emitting waves, but also transforms one's idea of the atom into a collection of virtual oscillators that operate between states.

Heisenberg:

Yes, that was it. This idea, of course, also was there already that an atom was really a collection of virtual oscillators. Now this idea of an atom being a collection of oscillators was in some way contrary to the idea of an atom being an electron moving around a nucleus. The obvious connection, the only possible connection, was that the Fourier components of this motion in some way corresponded, as Bohr said, to the oscillators. But certainly this paper then prepared the way for this later idea that the assembly of oscillators is nothing but a matrix. For instance, we can simply say the matrix elements are the collection of the oscillators. In this way, you can say that matrix mechanics was already contained in this paper.

Kuhn:

That's the sort of problem I don't want anyone to have to decide. That paper, the Kramers dispersion formula, your own paper is a route with elements of continuity, though there are other elements in that picture also to which we must come back. I think we must pursue some topics ahead of time as they come up this way. Very likely they are things you will not remember in that paper, but I am curious about the differences in that paper and the next one which you did with Sommerfeld — about the identification of the inner quantum number with total angular momentum. In the Rumpf paper you are rather pleased to notice that this formulation makes the inner quantum number total angular momentum except for triplets.

Heisenberg:

Well, it was at that time completely unclear still. Also, I would say that the progress of this paper was that one thought about the total angular momentum and thought about the two vectors being added up to a total angular momentum. But it was far from being clear that the total angular momentum was such an important quantity, which obviously should be so when you believe in conservation laws. But since one was away from classical physics anyway, it was always difficult to know what one should keep and what one should lose. I would say the main point is that one had the idea that one should add regular momenta to a total angular momentum. Sommerfeld was not too pleased that his inner quantum number was something which rather is an outer property of the atom but he didn't object to it. He said, "Well, all right, we don't know." Everything was so unclear that he didn't mind too much.

Kuhn:

Do you know whether you tried, while doing that paper, to do it in a version in which it would be the total angular momenta that was space quantized?

Heisenberg:

No, not yet. Somehow it didn't occur to me. Well, you know everything was so unclear at that time. I don't think that I tried at that time to quantize the total angular momentum with respect to the outer field. That was only later on.

Kuhn:

It's a strange one because what you do finally quantize is the average of the electron angular momentum. However, having averaged for its precession around the total angular momentum.

Heisenberg:

Yes, I mean it is just very poor classical mechanics — the whole thing. But nobody knew what one should do with classical mechanics. Only later on one acquired more stability about his concepts.

Kuhn:

Then just one other question about the paper. From the beginning, implicit in that paper is the problem of the Auswahlprinzip (Ausschlussprinzip) but there's nothing said about it in the paper. I wondered whether you had seen it at that time or if not, whether you remembered when it had first come to your attention?

Heisenberg:

No, I'm pretty certain that that was known at the time in Munich because Bohr's paper on the Periodic System had come out and they certainly were studied in Copenhagen. No doubt Sommerfeld always talked with Kossel and Grimm about the Auswahlprinzip (Ausschlussprinzip) about the Periodic System. I have heard many lectures at the colloquium about the Periodic System. Certainly, that was in my mind but I felt it was not too important for this special problem. Well, the idea, for instance, of having two valence electrons for the triplets came also from that side.

Kuhn:

I meant that particularly then the problem that the Rumpf of the next higher element will not correspond in its angular momentum to the appropriate state, presumably the ground state of the whole atom.

Heisenberg:

Ja.

Kuhn:

You needed extra factor of one half.

Heisenberg:

Oh yes, this extra factor of one half. ... Well, of course, Pauli then was at these discussions on the Rumpf model and he must have emphasized this problem very early because Pauli was never happy whenever something was still unclear and didn't quite fit well. Well, of course, it was the same factor 2 which also was in the X-ray spectra, so it was the same problem again.

Kuhn:

It's doubtful that that could have been recognized then. I mean they both get clarified by spin. But they certainly, superficially, don't look at all like the same problem.

Heisenberg:

Well, but it was the same factor 2? Was it not?

Kuhn:

Well, I would say it's hard for me to think, although perfectly possible, that it would have been seen that way then. In Pauli's paper on the two valuedness of the electron, then it is the same problem. ...

Heisenberg:

I know that Pauli was very proud of the exclusion principle paper just on account of the fact that he could disprove some of the things which Bohr had claimed. Bohr had this idea of making the electron shells by resonance between electrons, or the electrons should move with the same phase around the nucleus, or something like that. Bohr wanted an explanation as to why two electrons make a closed shell, and then eight electrons make a closed shell, and so on. And so he had a rather strange idea of families of ellipses going around the nucleus. Pauli never liked this idea because he always said, "Well, this resonance is swindle and I don't believe a word." The invention of Bohr was quite wrong for Pauli. As soon as Pauli had realized that all these papers of Bohr on the Periodic System were more the intuition of Bohr than actual calculation, then he started to become critical and to say, "Well, now let's look through the real arguments." He agreed that Bohr was probably more or less all right about the results, thus about the picture of these atoms, but he said, "Well, what are his arguments? This resonance between electrons is pure swindle because why should two electrons have resonance and three electrons not have resonance, but eight again have resonance; that is nonsense!" So he felt very soon that this thing should have to do with the number of possible states. So the doubling of the states and then this counting of the occupation of a state — that was practically at the same time. That occurred practically simultaneously. So these papers of Pauli must have been the result of rather long and critical discussions about the Periodic System, partly with Sommerfeld. I don't know whether Pauli had been in Copenhagen during that period. ...

Kuhn:

Let me ask you about that meeting in the summer of '22.

Heisenberg:

Oh yes, the summer of '22, which was, of course, very important for me.

Kuhn:

Who went up there and what was that meeting like?

Heisenberg:

Well, from Munich, of course, Sommerfeld and I came. Sommerfeld told me that he wanted to see Bohr and to listen to lectures, and he invited me to go there with him. So then there were the Gottingen people at the meeting, and I, for the first time, met all the Gottingen people: Born, and Franck and Pohl and Runge and Hilbert. Hilbert played a great role already at that time so I think he presided at the meeting sometimes. It was extremely interesting for me to see these many new faces. Yes, the mathematician (Cargiato). And I saw a number of younger people from Gottingen who were interested in general relativity. That was quite new to me and so I tried to talk to them and to learn from them. Now let me see — whom did I meet else? Maybe Fritz Hund already was there. I do not think that people like Schrodinger were there. May be that Richard Becker was, I couldn't tell.

Kuhn:

Were the mathematicians other than Hilbert much interested?

Heisenberg:

Yes, I think there was some interest. There was Bernays. Of course, the mathematicians had their own interest but they also felt that this whole development was something important and they did take considerable interest.

Kuhn:

Was Courant there?

Heisenberg:

Yes, Courant was, yes. Well, I'm sure that the mathematicians would all attend the lectures of Bohr. I think all the mathematicians were there and also the assistants and so. They were interest to hear about the new developments.

Kuhn:

Is it your recollection that it was a large group? Would you guess in memory how many might have been there?

Heisenberg:

Oh yes, well, I should say that in this lecture room where Bohr talked there would always be 100 people. Something like that. Yes, it's quite a big room. But the number of those who then took part in the discussion was very limited, of course. There were only a few that did take part in the discussions. The names should come back. Let me see. Kramers was not there. Some of the young Dutch people —. I might mention one man. That was a bit of a funny fellow with the name of Senftleben and I think later on he became a school teacher; he was a bit of a queer fellow. I don't think he ever did very good physics. He just took the examination for being a teacher. He occurs to my mind now just because he lives now in the neighborhood of Munich. I think he is now pretty old and not in good health; I think he lives in a hospital or sanitorium, but he occasionally comes to this institute and loves to talk about old times. In some way he's a bit boring, still he's quite a nice gentleman and I don't always send him out, I just try to give him time. It may just be that he could tell you a few names. He might be in the institute today, for instance. He sometimes comes in. It might be quite nice to see the thing from his side. And he certainly would love to be interviewed about the whole thing.

Kuhn:

Well, if he would just come I think it would be very much worthwhile.

Heisenberg:

And then there was Heckmann, one of the pupils of Born and Hermann was one of the pupils. I think they both must have been there.

Kuhn:

Rudolph Minkowski I know was there. He was one of the two people assigned to do the Ausarbeitung.

Heisenberg:

Was he? Well, then he can tell you a lot about it, yes. Did you talk to him?

Kuhn:

I talked to him. Although I must say it was not until I talked to Born and Hund that I realized quite how important those lectures had been.

Heisenberg:

Yes, yes. Well, at least in Germany they played a very great role in the development of the whole thing. I would say two things — the book of Sommerfeld and the lectures of Bohr — were the real start of that kind of atomic physics in Germany. Also the Dutch people in some way must have taken part to some extent. Well, I wonder if Ehrenfest perhaps came. That would not be impossible. ... I certainly have met Ehrenfest at many of these meetings, but whether he was at Gottingen, I couldn't tell.

Kuhn:

You speak of how important those lectures were and how much you learned there. Can you say more about what was really striking to you about them? You, after all, had Atombau and Sommerfeld behind you. To a Sommerfeld student, to one who knew Atombau very well, were these lectures like?

Heisenberg:

Well, first I would say a new feature for me was that one could speak and think about these problems in a very different way from the way in which Sommerfeld thought and spoke about them. One of the things that impressed me most was this kind of intuition of Bohr — Bohr knew the whole Periodic System. At the same time, one could easily learn from the way he talked about it that he had not proved anything mathematically, that he just knew that this was more or less the connection. When he talked about the closed shells at that time, of course, he didn't know why they were closed. One really saw two things. Bohr tried to give explanations for things he knew. One sometimes felt that these explanations were not too good, and he also said that they were not too good. Bohr was always very clear about the small degree of clarity he had. One could see from the way he talked that he took things extremely seriously and at the same time he saw that he could not really prove the things. He only knew that must be the connection. The intensity of imagination which he had — that made an enormous impression. The intensity by which he knew such and such must be the whole connection and that's the picture of the Periodic System, and it must have to do with this feature of quantum theory and this feature of the integral pdq and so on. Still, the whole thing was vague and I would say it was just this vagueness, in connection with this enormous force of imagination of Bohr which, of course, was enormously attractive for a young man who wanted to do some work himself. A young man learned that one can do a lot of work oneself — it's not finished at all. At the same time, it is almost clear that the whole picture cannot be much different from what Bohr says, but still, all the details have to be filled out and one essential or two or three essential features are still missing and we must find out about them. In this way also, it occurred to me that it was very interesting to raise criticisms just to listen to what Bohr would say to this criticism. I liked to see whether or not his answers would be more or less a kind of excuse or whether they would really hit some essential point and so on. And so, of course, I was especially glad about this discussion which I mentioned about the dispersion relations. That was, for me, perhaps the most essential conversation I had for many years. Because there for the first time I felt, "Well, this is again a point where I can perhaps go ahead myself," you know, because Bohr had not tried to go ahead at this point. He had found that it was too difficult; on the other hand, one has to solve the question, and so I wanted to do something with dispersion from that moment on. Then I learned about the papers of Ladenburg. Yes, Ladenburg — he must have been there. Yes, I think Ladenburg was at the meeting. That I would believe.

Kuhn:

You had not known the Ladenburg paper before that then?

Heisenberg:

I would think no. I'm not sure. It may be that people in Munich had talked about it. Do you know the time the Ladenburg paper came out relative to the time of the meeting?

Kuhn:

The first Ladenburg paper — I've got those dates in the other room. As a matter of fact, why don't I go get those? [Interruption] The Ladenburg paper was in 1921. And the Ladenburg and Reiche paper is '23. It's actually after the meeting.

Heisenberg:

So it was perhaps only the first paper. Well, I know from later discussions, but whether they have been at that time, earlier, or later, I don't know. But I know that my first impression on Ladenburg's paper was, when we discussed it, that that was too far away from the whole Bohr theory. That was really a paper, if I recall correctly, which was very closely attached to the Einstein paper and insofar, of course, was good physics. That everybody saw. But one didn't see the connection between the Ladenburg paper and the actual calculations of intensities. It was only the first Kramers paper which established this connection. You know the connection was between Einstein's ideas and the idea of the atom as a collection of linear oscillators which somehow had to do with the electron and this "somehow", of course, was to be explained. It was quite clear in Kramers' paper how one should connect the idea of the linear oscillators in the atom with the Einstein idea of the probability. That had not been so clear in Ladenburg's paper. And the second paper of Ladenburg and Reiche I don't know at all. ...

Kuhn:

You say that you decided at that point that you really wanted to work on dispersion yourself?

Heisenberg:

Yes, that from this moment on —. Well, maybe it was not decided, but I felt now that dispersion is a very interesting thing now and I thought that maybe I could do something with it. I would say that probably after that discussion with Bohr, when I came back to Munich, I discussed the problem with Pauli and would have his opinion on it, and then try to do something myself. But the Ladenburg paper was a bit too far away to me. It was entirely on the Einstein side, but still I felt that one must always keep in mind this paper although I couldn't do anything with it. I could not do anything with it before I saw Kramers' paper. When I had Kramers' paper, then I felt, "Now this idea of using the harmonic oscillators somehow in connection with the atomic model appeals to me." But that was actually to a large extent Kramers' idea, so I do not know whether I would have come back to dispersion if it was not with Kramers. Well, you never can tell what would have happened if —. The things happen somehow.

Kuhn:

You had yourself, between the time you worked again together with Kramers, and this time in Gottingen in the summer of '22, tried on occasions to find some way?

Heisenberg:

Yes, I would say I had thought about the problem. I had turned around the problem in my mind quite frequently but without getting a grip on something. I didn't get hold of something which I found interesting enough to really do something with it. Besides that, I think during that time I had to work on my doctor's thesis, which was turbulence. This problem was in an entirely different field. But I do remember that when we spoke about the difficulties of quantum theory the problem came up again and again. But then I think during that period also there came out the paper of Compton which, of course, occupied the minds of many people. This paper showed the strong reality of the light quantum picture. So all this contributed to the interest in the dispersion problem.

Kuhn:

What other problems presented issues for important discussions at the Gottingen meeting?

Heisenberg:

What did impress me also strongly was the enormous complication which one had to expect if one wanted to explain complicated atoms. I remember that Bohr showed some nice pictures, which you find in textbooks nowadays, of some inert gases — the radium emanation — some of the most complicated inert gases with closed shells — with I don't know how many electrons - - 86 or something like that. These pictures, of course, are quite nice pictures to have on a blackboard but if you imagine that you should do real physics with such pictures then one feels it's hopeless. So we all were quite impressed by the courage of Bohr to deal with problems of this kind, but still we felt, "Well, after all there is a Periodic System; somehow it must be explained." And it kept again our interest alive, so that we felt, "Sometimes you must try —."

Kuhn:

What about the helium atom? Did Bohr talk about that?

Heisenberg:

Well, yes, that I should mention that already at that meeting the impression was a rigorous calculation of stationary states from the integral pdq business was not possible except in hydrogen. Now I don't know whether Bohr actually referred to the paper of Pauli, but somehow at least this was ray impression. It may be that Bohr had not realized that so clearly at that meeting, but the general impression was, "Well, as soon as you get away from hydrogen, you get into trouble." We put it in this cautious form. We just felt as soon as you get into more complicated atoms, then you get in trouble. But that may, of course, be due to the great complications of the many-body problem of astronomy. So the normal excuse was this: "Well, if you have one electron then you have clearly a periodic motion, integral pdq is quite simple, and then everything works well." As soon as you come to many-body problems, then it is true that you have formulas by which you could possibly apply integral pdq for the periodical solutions which exist. But at the same time, everybody knew that the periodical solution is a very exceptional solution, that in an infinitesimal neighborhood of the periodical solution you always find non-periodical solutions. All these complications of the many-body problem then worked as a kind of excuse as to why it did not work for the helium or the other problems. And the Pauli problem with the hydrogen molecule ion was a special case because actually this was in some way still a periodic problem. Therefore that was the most disagreeable result at that time. But was it clear at the time of the meeting? Do you know when the Pauli paper appeared? That would be an interesting date.

Kuhn:

That would be a good date and I ought to know it. I will try to find it out by next time. It is not very long after this meeting that you and Born do the paper on the helium atom in which you finally come out and say this problem is not going to be solved by these techniques. It has not been clear to me to what extent that conclusion was really reached while doing that paper or to what extent you knew the conclusion already and that this was proving the point?

Heisenberg:

Well, we certainly had been influenced by the Pauli paper which has been earlier, or at least the discussions with Pauli had been earlier. So I knew, from many discussions with Pauli, that one gets into trouble already with this H₂⁺. Then it was, of course, a natural assumption to say, "Well, then probably it doesn't work for the helium either." So in some way, we anticipated that for the helium it didn't work and we wanted to show that it didn't work. So actually the result was more or less anticipated at that time probably on account of the Pauli paper. I don't recall what kind of excuse one had for the Pauli paper. One really should look up the Pauli paper. Did he find any excuse, or did he mention any possibility?

Kuhn:

I don't know. See one of my great disadvantages is having too little time to read all of the papers. I concentrate on the papers of the people I'm going to be talking to. I've not read that paper. ...But on problems of this sort, what about the cross-field problem? That's a perplexing one in retrospect.

Heisenberg:

Yes, well, of course, that was first the kind of mathematical sport of Lenz, I should say. The cross-field problem — why was that so important? Well, of course, I should say one tried to use the old technique integral pdq as much as possible and to see how far one can get. In the hydrogen one had been very successful with the Stark effect, one had been successful with the normal Zeeman effect, and so in some way one felt, "Well, even when one had put two fields together it should work, and that would be quite an interesting thing to try." And nobody knew the technique of doing it and then Lenz invented this technique. I think it was Lenz who solved the problem with the two cross-fields by means of these funny variables which Pauli has used, you remember, for the quantum mechanics of the hydrogen atom. The first paper on the cross-fields which really gave, from a classical point of view, the correct solution must have been the paper of Lenz.

Kuhn:

I rather think that was Epstein.

Heisenberg:

Epstein!? [Heisenberg locates a reprint of Pauli's paper on H₂⁺ in Ann. d. Phys. and reads through the concluding paragraphs.] ... He doesn't really state that the theory does not work, that the theory is in contradiction to the experiment. But, of course, implicitly it is stated. By the way, I see here that he mentions Kratzer so Kratzer was here.

Kuhn:

I notice in about 1927 in Naturwiss. you give a survey of the development of quantum mechanics. And you mention that prior to matrix mechanics and wave mechanics there had been some problems that weren't working out. You mention just two problems and I think one of them is dispersion theory, the other was the cross-field problem; you don't mention the helium atom.

Heisenberg:

(You see, the cross-field problem is a problem where it does not work.)

Kuhn:

Yes. Now, I think the classical solution was finally given, but my impression is that it wasn't really given until '25, '26.

Heisenberg:

Oh, I see. No, I don't agree. Because at the end of 1925 Pauli published his paper on the hydrogen, on the quantum mechanics of the hydrogen atom and that he did with a technique which had been used for the cross-field problems. So at least the general technique of dealing with the cross-field must have been known by that time. Well, that is quite certain.

Kuhn:

A lot of work had been done on it and a lot of techniques had been developed, but my impression was that none of them were felt to be satisfactory.

Heisenberg:

For the cross-field problem, you mean? Well, the cross-field in the case of alkali atoms didn't work because that would have been a problem of perturbation theory. But cross-field in hydrogen — that must have worked. Now, I can tell you it at once because I have here the Pauli paper on the hydrogen and he must quote the paper of Lenz. Now, let me see. Yes, he says, "In the general case of crossed electromagnetic fields, there are papers of Klein and Lenz," and I know that Lenz, at least, has developed this technique which he uses. And the paper of Lenz is from 1924, and that of Klein is also '24. So that must have been earlier. So this says quite clearly that Lenz is '24. It may be that Lenz has taken the idea from Epstein. That's of course possible, as you say.

Kuhn:

I must say that I have the impression that although those are important attempts on the problem, they were still not thought to be satisfactory and it's later that Epstein furnishes a satisfactory solution. I haven't read either of these papers, but I know of both of them. They don't set the problem aside because there is still later attempts on the problem. This is a very complex one, we've not followed by any means all the way through and I've never been quite clear why it looks so —. You know, as I say, in '27, you point back to it. You give just two examples of a real breakdown of the classical quantum mechanical theory and point to this as one. It may be that I easily, attribute too much significance to that.

Heisenberg:

Yes, I'm quite surprised to bear it because I thought, in the case of hydrogen at least, the cross-fields made no real difficulty and everything worked out. Of course, it may have been that I meant the cross-fields for alkali spectra and there the situation is different because there you don't use this technique. There you can only use the perturbation theory and there it actually did not work out. But, I'm somewhat surprised. ...

Kuhn:

One other problem that might well have been discussed at the Gottingen meeting and very likely earlier and which I didn't ask you about in connection with Munich itself was the Stern-Gerlach experiments.

Heisenberg:

Yes, the Stern-Gerlach. When did the Stern-Gerlach paper come out?

Kuhn:

That came out early in '22.

Heisenberg:

Early in '22, so it must have been known by the time of the Gottingen meeting.

Kuhn:

Even earlier than that there would very likely have been discussions with Sommerfeld.

Heisenberg:

Well, the strange thing is that I do not recall long discussions on the Stern-Gerlach effect from the Gottingen meeting. It would have been very natural that things should have been discussed quite thoroughly by Bohr. Well, maybe this Stern-Gerlach paper was not fully acknowledged as being correct at that time or I don't know —.

Kuhn:

There were certainly problems about it and you and Born talk about them in this first paper you publish on phase relations.

Heisenberg:

You know, this quantization in space —. Yes, there I should mention one point which is quite interesting. The quantization in space was looked upon in Munich and in Copenhagen very differently in the following way. For Sommerfeld, it was that kind of mysticism that nature just made it that an atom can be this and this and this in space; and Bohr would, of course, at once ask, "But what is the set direction?" I mean after all, space has the continuous rotation group, so what is the set direction of this quantization. That was a question about which Sommerfeld would be almost angry because he would say, "Well, if you ask that way then you would admit that actually an atom can have continuous number of positions and that apparently is not true." And so, before the Stern-Gerlach paper came out, in some way Sommerfeld simply claimed that there were only three ways of putting an atom, depending on the angular momentum, of course, into space. And Bohr would say, "Well, that must be complete nonsense because after all, we have continuous rotation in space and what should be the set direction?" Then Sommerfeld felt, of course, that it was a kind of triumph that now one could see that there was this quantization in space. Bohr, of course, would say, "Well, after all, of course, that just means that if there is an outer magnetic field, then you can tell what is the direction. Then you have a set axis. But if you have no outer magnetic field, then where do you get the set axis from?" That was probably a problem which puzzled us for a considerable time. I know that again these differences between the Sommerfeld way of looking at things and the Bohr way came out in a rather heated discussion which Bohr, Kramers and I had on the resonance fluorescence in sodium. Had Bohr told you about this thing?

Kuhn:

No, he had not. I have read your paper. I have been very much struck by the way certain of these problems emerged there.

Heisenberg:

Yes, and that was just what I tried to do at that time when I look back to it. I tried to rationalize it. I wanted to find the way just in between Bohr and Sommerfeld there. That is, I could not believe in the strict way of quantization as Sommerfeld had the idea, that just for some mystical reason an atom was always such and such, because what is the set direction after all? And on the other hand, I felt there was a certain strength in Sommerfeld's argument, namely the strength of quantization in general, so quantization must mean something. And so I again was inclined to think, "Well, both-sides must contain some part of the truth and it's very difficult to find out which part of the truth." And then there was this problem of the polarization of the resonance fluorescence in sodium. I was interested in the problem already from the Gottingen time because I had seen the experiment. I think Hanle did the experiments in Franck's laboratory; and Hanle had found out that there was a complete polarization for some of the lines; I don't recall exactly. I had the idea to explain this complete polarization. But first, Bohr had written something. That was again, as very often with Bohr, just a correct description of what it was, but it was very vague and he did not dare to make predictions as to how the polarization should be. Then I found out that one could say, "Well, all right, let us put an outer magnetic field just around the axis of the electric vector of the light. Then we can quantize, and then we see that we must have complete polarization. Why not afterwards let the magnetic field go away again, and then there we are." I remember that I tried to explain this to Bohr and Bohr first was impressed and he said, "Well, that looks quite all right." I asked him would he mind if I wrote something about it. I found it was a good idea and I wanted to write a paper. But then after I think one day, I was called to Bohr's room, and Bohr and Kramers were there, and they now tried to explain very seriously that it was all wrong, that this idea didn't work. And I was completely shocked. I got quite furious because there I thought I had something real, and now they tried to explain it away. So we had quite a heated discussion, but at the end I think I came out with a slight victory because in some way I had succeeded in convincing at least Kramers. It was easier to convince Kramers than to convince Bohr. Well, the end of this discussion was that they said, "Well, we must think about it again, and next day or so we all agreed that this was a possibility. But Bohr was considerably worried about this use of a magnetic field which didn't exist. That was, of course, the very point — that one invented the magnetic field which didn't exist just to derive some rules which then were consistent and then one left it out again. So in some way, I think we all were quite happy at the end of the thing. And I had, for the first time, the feeling that now I had been able to convince Bohr of something about which we had disagreed. So I was quite glad about it, and I think everybody was pretty happy.

Kuhn:

I'm delighted. [Kuhn gives Heisenberg an outline showing that Kuhn had noticed exactly this point in reading Heisenberg's paper.] ...

Heisenberg:

I have really, in this whole period, been in real disagreement with Bohr; and the most serious disagreement was at the time of the Uncertainty Relations. But this was the first time that I was really disagreeing with him and where I really got very angry about Bohr. You know sometimes, in some way, I was quite offended. So it was very nice and Bohr, of course, when he had really seen the argument, would always agree on it; there was never any difficulty of this kind. But that was a nice discussion and it took not very long. I think after three days everybody was happy about it and Bohr agreed that I should write the paper. We tried then to apply the same methods to other cases but there was no such good cases. It was really the only quite good and clear case.

Kuhn:

You were deeply convinced and ultimately convinced, at least partially, the other two. I want to ask to what extent you can remember something which you might say was the source of reasoning for your conviction. Was it that the numbers came out so well in that case, or was it simply that the arguments seemed to fit?

Heisenberg:

Well, it was certainly both. I mean the fact that the numbers came out was, of course, a confirmation that I was on the right track but also this kind of thinking that one could use the quantum conditions, so to say, to the utmost. ... I would say that I felt very strongly that this was the spirit of the real quantum theory and it was always the spirit we were after. Every physicist tried to get clearer and clearer feelings about how nature worked. So I felt strongly that this is in the right spirit — that is how nature works. So I was very happy to see that it actually did give the right numbers. That belongs to it; if it doesn't, you see that you are not on the right track.

Kuhn:

At the beginning of that same paper, where you go into this example, you treat this as an example of broadening the Correspondence Principle, and you talk about a number of other ways in which it might also be broadened. ...

Heisenberg:

I would say that all this is a part of the game to make the total table of linear oscillators be the real picture of the atom. One felt that in the Correspondence Principle, one should compare one of these linear oscillators with one Fourier component of a motion. Now keeping this in mind, one must ask, "Can one put that a little bit more accurately? That is a bit vague. Could we not try places where you can do it better?" So we tried wherever we could to push it a little bit further and this broadening of the Correspondence Principle means always that we should find places where we can use the Correspondence Principle in a precise manner, so that not only does it give vague indications about intensities, but so that it gives real laws about intensities. And just as we have selection rules according to which no changes in angular momentum by more than one unit are possible — that is a strict rule — so we should have the same strictness in the application of the Correspondence Principle. So I was very happy to see that in this case of the polarization of the fluorsence one could give a strict rule: polarization must be 100 per cent in this case. And I think there were other cases in which it was only 60 per cent and the figures again fitted quite well. So the whole thing was a program which one had consciously or unconsciously in one's mind. That is, how can we actually replace everywhere the orbits of the electron by the Fourier components and thereby get into better touch with what happens? Well, that was the main idea of quantum mechanics later on. One could see, more and more clearly, that the reality were the Fourier components and not the orbits. The Fourier components were more real than the orbit and still somehow their connection was similar to their connection in classical mechanics. So one tried to look for those connections between the Fourier components which were true in classical mechanics and to see whether or not similar connections are also true in quantum mechanics if one takes, instead of the Fourier components, the real lines. So this was the whole trick.

Kuhn:

This is a terribly important description that you've been giving. How generally was that held? Did Kramers and Bohr both feel that way?

Heisenberg:

Well, Kramers certainly did feel that way more or less. I had some discussions with Kramers about the dispersion formula. We did not agree to begin with, and I would say that Kramers did not take the problem quite as seriously as I did. But certainly he also tried in the same direction as I tried. I would say that both Bohr and Kramers tried in the same direction — Bohr perhaps less than Kramers and I. Bohr did not go so much into the details of the classical motion, and the mathematics of the classical motion was not such a center of interest to him. But Kramers, of course, would know that very well. Well, I might mention these discussions in connection with the dispersion paper.

Kuhn:

Yes, I wish you would.

Heisenberg:

If I recall correctly, it was so that Kramers had more or less already the idea that one should extend his paper on the dispersion and also to these Raman lines — on the lines of the different frequencies. Of course, if you have this idea then you can at once write down some formula which looks reasonable. But this formula was not very plausible — it had some faults, some defects. Now, what were they? I can't quite remember. Well, at least the end then of the discussion was this: that I did not try to discuss it primarily from the side of the Einstein laws and from physics, but I simply tried to say, "Well, let's see what classical mechanics does. Let's just assume that we have a classical atom like hydrogen and we put an electric wave on it and then see what happens to the Fourier component." Then I could see that I have certain products of terms and there, of course, the amplitudes must belong to those lines which are in the exponent as frequency. You know, in the exponent you have always the sum of two frequencies. We drew these pictures. First you go from this level to that and then from that level to here. Then one could easily see that in going from here to here you can go by different ways. That's like in Feynman graphs nowadays. [Interruption]

Kuhn:

You began to tell me about the disagreement with Kramers in the early discussions of the dispersion theory.

Heisenberg:

Yes. Well, as I recall there was a disagreement about a certain term in this dispersion formula which, from my mind, came out with necessity just because in the classical perturbation calculation you would always find that term. When you tried to interpret this term physically and see what it actually means, it looked, to begin with, as if this term could not be there. The term would actually mean that you get a strong interference at the point where you really wouldn't expect any strong intensity. The final solution then was that this happened only for a line which already existed as a spontaneous transition, that is, something happens when this secondary emission and a kind of scattering which Kramers considered — well, the frequency coincided with the spontaneous frequency which was already there. So it was again, say, the problem of the B coefficient which goes downward. I think the laser and the maser have very much to do just with this term on which we disagreed at that time. Well, the whole point is that it was just a subject of discussion on which we did not agree for some time — I would say at least for several weeks. And I always had in my head, so to say, the classical perturbation theory and had the feeling that it must be like in classical theory. Kramers, on the other hand, said, "I must understand it from physics. I must see what happens and if there is not a strong line then there cannot be such a resonance term." And then finally we found, by many discussions, the solution that the term is actually there, but that there is an interference between this term and the spontaneous transition and this interference makes the whole thing come out all right.

Kuhn:

This fascinates me and I think this is one of the most important points we've come to and we should go on with it. This points into a whole area of great importance to me because, as you know, Kramers only published discussions of the dispersion formula, prior to the joint paper, as two notes in Nature. The original one announces the formula and says almost nothing about how it's been gotten. Then the second one gives a little more argument, but still nothing like a full argument. It remains then very unclear to what extent your joint paper with Kramers represents not only an extension of the argument to the Raman term but also represents a presentation of the original procedures.

Heisenberg:

Well —.

Kuhn:

What you then say about your insistence on the classical [perturbation theory] —. I wonder how much of that argument was already there when Kramers did it in the first place.

Heisenberg:

Yes. I wonder myself. When Kramers had written his two terms, he must have thought of the classical dispersion formula. That is necessary. But in some way he hasn't insisted so much upon it as I had done. Then, at least when we came to these more complicated Raman terms, he was willing possibly to drop one of these terms because it didn't fit with the physical picture and I did not want to drop it. I was more closely attached to the perturbation theory and the connection with classical theory and correspondingly Kramers was more attached to the Einstein paper with the B and A. But Kramers must have known the classical perturbation theory at least so far as the non-Raman lines are concerned, the diagonal scattering is concerned. That must have been quite clear to him. Well, I don't recall all the details of the discussion. I only know that this was a controversy for some time.

Kuhn:

Did he say anything to you about how he had gotten started on the formula himself?

Heisenberg:

Probably we have spoken about it, but it's not in my mind anymore. He probably would have said that he of course thought about the classical dispersion theory and he also mentioned the paper of Ladenburg. And I agreed at once that I could understand his paper much better than Ladenburg's paper. I would say his paper was already different from Ladenburg's paper just in the same direction — more connection with the Correspondence Principle and the oscillators.

Kuhn:

Except that there really isn't a paper. There are only two notes which really give the result and a few hints. In a sense there is a promised paper. But when the promised paper comes out it is your joint paper.

Heisenberg:

Well, that may be. Perhaps the idea was this. Kramers had written this note, with the idea of later on writing a complete paper and giving all the arguments. Then there came this idea of the Raman lines which had not occurred to him to begin with. That idea was introduced by Raman and Smekal and these people. All right. Then he saw that he should try to work that into his theory. At that time already I took part in the discussions and I was very interested just in this side so that actually the first Kramers paper was not written as a complete paper. It was left as a note. So only this joint paper was written. For some time it was not clear whether we should write it as a joint paper or whether Kramers should write it alone. And actually — I don't know whether by Kramers himself or by somebody else — it was suggested perhaps it was just good that Kramers should publish it alone because most of the work was due to him. I was a bit hurt by this idea because I felt that by insisting on this business with the one term I had made an actual contribution. But I simply said, "Well, I leave that entirely to Bohr, as I had left it always to Sommerfeld. Bohr shall decide. I don't care which way his decision goes." Bohr then probably thought, "Well, after all, the young man has contributed a bit, why not put his name also on it. Everybody knows that it's really Kramers' theory because Kramers had written the first note." So this must have been Bohr's idea more or less. I think that was quite right. So that was the history of the paper. I also believe that Kramers first had though he would work out this whole thing alone but then the Raman lines came in as a kind of surprise which had to be worked into it. Since I always discussed these things with Bohr and Kramers, it was natural that I would come into the game. Then I had very early insisted on a special point which then turned out to be correct and in this way I came into the paper. But it was like it was in Sommerfeld's Institute — there were continuous discussions between all people who took these things seriously. It was always difficult to find out who has done what.

Kuhn:

How much was Bohr himself involved with this?

Heisenberg:

Well, he was involved at least so much that he would discuss things again and again. He listened to what Kramers said and to what I said; especially when Kramers and I disagreed, then he would very carefully listen. I remember once that we would both stand at the blackboard and Kramers would defend his scheme — he had a different scheme of how the formula should look — and I would defend mine. Bohr listened to the arguments. We did not reach a definite conclusion then. Bohr was very careful about it and always wanted to know the arguments but, of course, he only did agree when the final solution was found. And the solution was that my formula was correct but that one had to take the interference business into account. But that was again Kramers who found that this interference could save the thing. So it typically combined the work of all these three.

Kuhn:

One of the terribly exciting things reading that paper now, and I take it this was exciting at the time also, is the sense that here one has found a problem with which one can use the Correspondence Principle and have it all come out so that you are not left with anything you don't know how to interpret.

Heisenberg:

Yes. And for the first time we had something which every term could be explained and had a meaning. Well, there one should say again that that is almost true, but only almost. These terms were amplitudes and their absolute values were a very defined thing, but the phase was not so clear. That was also a new and very interesting feature. It appeared that not only the amplitude, but even the phase had some meaning. That made also a strong impression on me at that time. Then I saw the analogy to the Fourier components in classical physics was really very close, because not only the absolute value, but even the phase must somehow be there already and be well-defined and so on. Of course, only later on one learned that the phase was well-defined except for an arbitrary phase factor in every stationary state. But when you had an intermediate state, then, of course, this arbitrary phase went out and that was the important thing that you could have actually interference. Therefore, all this interference argument of Kramers was important. One could see that there the phase comes in and is important. So these amplitudes have not only an absolute value but also a phase.

Kuhn:

There is a strange feature in that manipulation which is terribly leading ultimately, but I wonder how it felt at the time. At a crucial point in the paper, you replace the differential by a difference — in an individual term.

Heisenberg:

Yes, yes.

Kuhn:

And now you sum, over all states, the difference between two terms?

Heisenberg:

Ja, ja.

Kuhn:

And then there's a bit of wonderful manipulation that works and I think that it has worked on no other problem. You manage to notice that if you take this undefined intermediate position in the Correspondence Principle and set it on the state that the atom is in, then these terms represent oscillations up from the state the atom is in and down from it. And then you say, "All right, we sum over all the excitation oscillators and we sum separately over all of the absorption oscillators." Now, this is, of course, the heart of the paper, but it's also true that there aren't enough emission oscillators. That is, in the two series you finally sum over you can't pair up the terms anymore.

Heisenberg:

No, no, of course not. That's natural, yes.

Kuhn:

Did that lead to discussion? Was that an important —.

Heisenberg:

No, I would say not, because at that time we felt already that such deviations from classical mechanics were quite justified. We shouldn't be surprised that such things happened in quantum theory. No, that wouldn't worry us anymore at that time. We would say, "Well, that is just the kind of correspondence we should expect. After all, quantum theory means that one has a finite number of terms and classical theory has an infinite number of terms." So I don't think that anybody was worried about this point. Though of course, you are quite right that this is a very important and strange feature of this formula.

Kuhn:

It is most leading because here is a very well-defined breaking point that gives you beautiful results.

Heisenberg:

Yes, and therefore one could learn already at this point how one should indicate classical mechanics by means of the linear oscillators and where one has to go away from the classical picture. So almost one had matrix mechanics, and one had practically the matrix multiplication at this point already without knowing it. Well, you can say the classical Fourier components are infinite matrices but the real matrices have corners and ends and therefore you have sums only over a finite number of terms.

Kuhn:

The Kramers formula fascinates and excites lots of people. Yet it's odd, because of this very misleading idea that goes into philosophy about the way scientists behave, that here is a formula that has substantially no experimental foundation. There's nothing to compare it with. It breaks both with some versions of the old quantum theory, it breaks with classical theory. Why was everybody so sure it was right?

Heisenberg:

Well, the answer would be because it looked right. But what does it mean — it looks right?

Kuhn:

Well, was everybody that sure?

Heisenberg:

I would say that most people were pretty sure that it was right because it was just again this reasonable compromise between the different aspects of nature which one already knew. I mean one had the light quanta, one had the Compton effect, one had the Raman effect, all these things —.

Kuhn:

One did not have the Raman effect yet. One had Smekal's remark that there ought to be such terms, think.

Heisenberg:

One had only the theoretical argument that they should be and not the fact yet. I see.

Kuhn:

One did have that Ladenburg was getting very good results with his formula but his formula didn't even have the second term of the dispersion formula.

Heisenberg:

Jay ja. No, but everybody agreed that Kramers' formula was better than Ladenburg's formula. Well, the Raman formula actually was only much later checked by experiments and so the experimental confirmation didn't exist. Still, one felt it was in the right spirit. I would say the essential point again was that one felt one had now come a step further in getting into the spirit of the new mechanics. Everybody knew there must be some new kind of mechanics behind it and nobody had a clear idea of it, but still, one felt this was now a good step in the right direction. So the feeling of how the new mechanics should look was rather widespread at least among ten or fifteen people at different places, in Holland, Cambridge, and so on.

Kuhn:

Who at Cambridge was interested?

Heisenberg:

Fowler was very much in touch with Bohr. They were good friends. It was perhaps mostly Fowler who would bring these things to Cambridge. Dirac was not yet in the picture in Cambridge. Yes, Fowler had very close connections to Copenhagen. When I came to Cambridge first I also lived in Fowler's house. It was in some way obvious that I should be invited by Fowler. He was the Copenhagen man in Cambridge, I should say. That again came through Lord Rutherford, I suppose. Fowler was the son-in-law of Rutherford.

Kuhn:

Would you remember anyone being skeptical about the formula? Those who were interested in it were all quite convinced?

Heisenberg:

Yes. I think all those who really knew about quantum theory like Pauli or the Dutch people and so on would be convinced at once. They would say, "Yes, that's just the thing!"

Kuhn:

I probably ought not now take you back to Gottingen. We must go back. I think it's been very good to skip over because this is a very important point and we needed to follow it when it came. You go into some problems in Gottingen and in any case the whole nature of your attachment to Gottingen is very important with an eye to what comes back. So we must go back and talk about the three semesters.

Heisenberg:

Shall we begin Friday morning?

Kuhn:

That would be excellent.