Werner Heisenberg - Session VII

Kuhn:

O.K. I will now turn on the machine. It's been off for this preliminary. Do go ahead and say what you were going to say, then we will come back to it.

Heisenberg:

You just mentioned this attitude of (mine) toward the development of wave mechanics of Schrodinger and especially toward the paper of Born on collision processes. There the actual psychological situation for myself was that I felt, after we had written our Drei Manner Arbeit, at that time the mathematical scheme was definite and could not be changed again. And then this mathematical scheme did make quite definite statements about how to calculate energy levels and to calculate amplitudes and intensities and so on. So I felt that from now on it is just a problem of working things out, if one wanted to get the correct interpretation also for collision problems or for whatever else is to be found in atomic physics. Therefore, when Schrodinger things came out, I found it very interesting. But I felt from the very beginning that either the thing is identical — is just a mathematical transformation of our own quantum mechanics — in which case it is extremely interesting, but it does not really contain new things except for new ways of talking about things; or it is different from our own attempt, in which case either or both must be wrong. Therefore, I did not like so much that Born went over to the Schrodinger theory. Either that was just using a new mathematical tool instead of an old one for the same thing, in which case all right, why not do it from the old mathematical tool. Or it really meant that Born doubted that our scheme was right, and thought rather Schrodinger's scheme was right. Well, all right. That was, of course, an entirely new possibility and I realized that this was a possibility, but at that time I was not willing to accept this possibility. I was so convinced by our own scheme that I felt that the two things were probably identical, although I didn't know this for sure. I didn't think from Schrodinger's picture one can get any new information with respect to the physical content. This was a slight feeling of criticism against the paper. On the other hand, I saw that this paper probably contained a lot of interesting things; but it may be because of this, therefore, that for some time I didn't quote it. I would think that I probably have not quoted wave mechanics until the time that the identity of the two schemes was proven. Because at that time when it was proven, then it was quite obvious that it was an extremely useful tool. But I was so much afraid that by means of the Schrodinger mathematical scheme, a new interpretation of the thing would be brought in. Just because the interpretation was not perfectly clear at that time, I was very much afraid that now entirely wrong ideas could enter into the thing and actually have entered. Schrodinger, as you know, wanted to throw all the quantum jumps away and to say that there is no quantization, it's just all wave pictures and so on. I was so much afraid for this interpretation that I tried to avoid it from the very beginning. On the other hand, one must say that Born's paper was a paper definitely in the right direction because it did connect Schrodinger mathematics with a correct interpretation, namely with the probability interpretation. Born had a different interpretation from that of Schrodinger. And so far, one may say that Born's paper was just in the right position between the two schemes. It was just one step by which you could connect the two schemes. Well, this is just to explain things historically.

Kuhn:

This statement itself presents an issue which I'd like to pursue. I don't know whether to go ahead with them and then go back —.

Heisenberg:

Whatever you prefer.

Kuhn:

Let me pursue it then. You say you were worried that the Schrodinger equation, or Schrodinger's remarks might introduce a wrong interpretation. But one of the things which is terribly noticeable is the extent to which all matrix mechanics entered without any interpretative remark at all. In the Born-Jordan paper and in the Drei Manner Arbeit one referred back for the physical interpretation entirely to your first paper. But granting the sense in which this at least follows the Correspondence Principle, there is nothing in here about what sort of world matrices would represent things in.

Heisenberg:

Well, yes, that may be so. But may I say that first of all, of course I had stated in this first paper how you do calculate the energy levels. So it was quite clear that the eigen values are the energies which are measured, and the differences are the frequencies. Also it was quite clear that the amplitudes which you observe, for instance, by the intensity of the emitted light, should be those amplitudes which are given by the matrices. Now I always felt that this should already be sufficient to get the interpretation in all other cases, I knew that it had not been carried out, but I felt, "Well, that is now just a matter of doing the things properly, then you will find, it out." Well, in the same sense, I would for instance, say, Sir Isaac Newton, when he had formulated the theory that the acceleration is equal to the force and so on, then he must have been convinced that if one did it properly then even the three-body problem or the planetary motion of all planets and all perturbations will come out. And still, he, of course, could certainly not do it at that stage. And in the same sense I found, "Well, that's now all settled. And since it is settled, one must first try to find it out from the scheme. And one is not allowed to make any new hypothesis," — and that was a point. In the Born paper on the collisions, it looked as if he made now the new hypothesis, that we have to interpret the transformation matrices, or the Schrodinger waves, as a probability. I just felt that there was absolutely no room for any new hypothesis. Either the hypothesis follows from our own interpretation, then it's all right; or it does not follow, then either or both must be wrong. Perhaps our whole scheme is wrong. There's just no room to do anything new. That was such a strong conviction in me. Therefore, for myself — and this goes a bit into the later time — it was so important that I wrote a little paper on it. I don't think that anybody has paid much attention to it. That was a paper on fluctuations. ["Schwankungserscheinungen und Quantenmechanik", Zs. f. Phys. 40 (1926/27) (Copenhagen, 6 Nov. 1926)].

Kuhn:

Yes. I want to talk to you about that too.

Heisenberg:

This paper was about two systems being in a kind of resonance, and then one studies the fluctuations of energy in one of these two systems. From that paper I thought that one could see quite clearly that this probability interpretation is a necessary consequence of the whole thing. I only assumed that the diagonal elements are equal to the time averages of some quantity. That was, of course, the main point from the very beginning: to say the diagonal element of the energy is, of course, just the eigen value, and the diagonal element of, say, p² is, of course, the time average of p² in that state and so on. And so this part of the interpretation had already been given from the very beginning. Therefore, it was interesting that one could see that if one uses this interpretation then one has to interpret the squares of the matrix elements of transformation matrix as probabilities. That was unavoidable.

Kuhn:

I don't think you do that in that paper, Sir. Because, you see, one of the things that's very curious to me about that paper — I may be missing the point — but I think in that paper you suddenly state that —. Maybe I'm misinterpreting it —. I've got that paper and I'd like very much to talk with you about it.

Heisenberg:

For myself it was a very important paper. Perhaps it doesn't look that way.

Kuhn:

No, excuse me. All I needed to do was to look once more at this sentence and realize that that is the significance of it and that I have been misreading it. Because there is the sentence in question, and I've been reading it as a statement rather than as a consequence of the preceding No, I've been reading it wrong.

Heisenberg:

Yes, that was the essential point. I felt that this last sentence here, this one here, was just a consequence of the whole interpretation. That is, when I believed the old interpretation about the time average being given by the diagonal element and so on, then I am forced to make this assumption that the squares of the transformation matrix elements are probabilities. That was my very point, you know. And just on account of this, I felt, "Well, Born should have done that." And Born acted as if he had made a new interpretation. That was a point of disagreement. On the other hand, Born had made the correct interpretation, so I could not object to it in any way. I felt so strongly that there was no room left, you know. When you once have written down your mathematical scheme, then you are under an enormous restriction and you are not allowed to say,"Well, now let's make this or this assumption." This was the only point of disagreement perhaps between Born and myself, that, from the Dei Manner Albeit on everything is completely fixed. There's absolutely no free space. So I was dissatisfied about the idea of Born that he says, "Well, let's go over to the Schrodinger picture," which, of course, was quite all right, "and let's assume that the wave is the probability." And then I felt, "Well, nothing is to be assumed here. Either it's wrong or it's right — it just follows, you know." That was the main point.

Kuhn:

... That point right there would be an obvious point at which to cite Born's paper, which was surely out by then and well, after the equivalence of the two schemes is known.

Heisenberg:

Did I not quote Born's paper?

Kuhn:

There's, I think, no mention. This is the place where I thought the absence of a citation of Born's paper was marked.

Heisenberg:

Well, I had many discussions with Born at that time and we never had a real disagreement. Certainly there was no personal ill feeling about it, but it may perhaps have been the following thing: that I felt, "Well, if I quote Born's paper, I would have also to criticize this one point, namely the point that Born thinks there is something to assume and I thought that there was nothing to be assumed." So I didn't want to criticize it. And, after all, it was rather wave mechanics than quantum mechanics, and I wanted to do only quantum mechanics and not wave mechanics, so I felt, "I won't quote anything." I think if I had been ten or twenty years older, I probably would have quoted these other papers, just because I knew that after all, you should be as correct and as polite as possible; but being a young man, I didn't think too much about these things. It may be that in some way feelings have been involved, but, after all, I had so many discussions with Born. I hope that Born didn't object too much about it.

Kuhn:

No. Certainly Born didn't even notice it, so far as I know. If he did notice it, he hasn't said anything to me or anybody else that I know of. See, this is, so far as I know, the first time that anything resembling a remark about a probability interpretation enters anywhere in your work. This interpretation has already gotten into the literature. One can't, therefore, help wondering a little bit whether this paper isn't in some parts an attempt to show that you can do with matrix mechanics what already has been done.

Heisenberg:

Well, I think that is a perfectly justified description that you have given. The only difference is this: I felt that there was no freedom. I quite agree that actually the probability interpretation had not been done on the quantum mechanics side. That is, this definition of a transformation matrix as giving the probability had not been derived by quantum mechanics. But for me, it has always been an important point that when you once have fixed some part of the mathematics, then there is absolutely no way out. Then you are put down into one channel. May I come back now to quite modern times. For instance, this unified field equation which Pauli and I once wrote down and which I liked so much and which I still believe in quite firmly — that has just such a property. It may leave some freedom at some places, but at many other places it just doesn't leave any freedom whatsoever. That is, either it's wrong, or it's correct. You cannot just talk about it, in a vague manner. You can say, "Well, either you can do it that way or you cannot." There is still some freedom about the assumptions about the ground state and so on. So some freedom is left, and that has been defined. But in a mathematical scheme, you cannot expect to have just the freedom to do new things. You must be extremely careful of what follows, what is to be derived, what is to be assumed. Therefore, I quite realize that this probability interpretation had been given before by Born and so on. But also in some way it had, been given by Bohr-Kramers-Slater. I just felt, "Well, that either follows, or it does not follow. That has to be decided." And I felt that this paper showed at least that it did follow, so there was no room for any new interpretation. I quite agree that I should have quoted other papers.

Kuhn:

No, I'm terribly grateful to you for letting me be this rude. None of this is in an effort to prove you guilty of a mistake. Usually these things are about something that happened, and where I can use them to find out more about what happened then, until you throw me out, I shall try to use them.

Heisenberg:

That's very good, yes.

Kuhn:

I think you somewhat misunderstood a question that I asked in this same connection a little earlier. But before we go back to that, let me carry this on a little bit. You say, and I understand exactly what you mean and I'm delighted, that there was not room unless it was identical; in which case, "What was the point?"; there was not room for this sort of freedom which Born was utilizing when he elected to do a problem with the Schrodinger mathematics. Tell me what happened later? In this sense, in part by this time, it was already known that they were identical. Let us take it for granted that they are mathematically identical. But I know that when you first point this out, you also point out the limits of how far we may carry the mathematical identification when it comes down to the interpretation of basic matters. So I would ask you, in part, do you get the feeling still that they're identical? One of the things which is perhaps a great misfortune, but which happened historically, is that, on the whole, the wave approach has dominated the picture as a method for producing problem solutions.

Heisenberg:

Well, this side of the question, of course, is perfectly clear to me now. As a mathematical tool, this wave mechanics was enormously successful, much more successful than quantum mechanics in the beginning. Later on, perhaps, it was a bit different again because when you come to these abstract things of modern elementary particle physics then you need rather these more sophisticated methods of general matrices and group theory and so on. To begin with, there's no doubt that Schrodinger's equation has been an enormous success. It was extremely important for the whole development. At the same time, I would also say that for the first years it has also made it a bit difficult for the average physicist to understand quantum theory because it suggested the simplicity which was not there. It suggested a new picture, which then was taken as the real picture and prevented for some years the realization that one has to be so extremely careful with any word one uses. I would say also that it has been extremely helpful in the sense that we see how terribly cautious one has to be about the words. But generally, I would say that coming from the matrix mechanics, one was to begin with already more inclined to be careful with words than one was when one came from the wave picture. In that sense, I would now say that these two things are completely identical and I always feel them as being completely identical. At the same time, I feel that as a constant warning that one must be extremely careful about any words which one used. So whenever I say sentences about an electron moving in an atom or a wave moving there and there, I always feel unconsciously how dangerous every word is which I use. And this feeling of inadequacy in the language is the strongest impression which has been left over, but this impression is completely connected with the feeling of identity between the two schemes. I found it then, at the end, extremely admirable that a mathematical scheme can be so flexible, can be transformed so widely, that you can either make it look as a wave mechanics with waves in space and so on, or make it look like a classical mechanics, just by mathematical transformation from one scheme to the other. This comes about, actually, by doing nothing, because a mathematical transformation is just writing one letter instead of another one. But that's so interesting that you can do that. I have never come away from the feeling that these two things are completely identical. Now I don't know how other people feel about it. I would say that the young people nowadays, who learn in the textbook this whole thing, would feel it more or less identical.

Kuhn:

Oh yes, I think it is generally pretty much entirely taken for granted that the two are entirely equivalent. I express a sort of scepticism about this largely in order to get your own reaction. But I will press this a little further, still hoping for reactions. It seems to me that a good deal of what you are now saying when you say, "Yes, indeed they are identical," and, "After all, they've got to be identical because by a mere mathematical manipulation you could transform one to the other,"is certainly right. On the other hand, "Their identity is a lesson to one about how careful one must be about words." Now, that must mean in some part that they're identical as long as having learned that we can get one of them out of the other one by a mathematical transformation, we utilize that fact to keep ourselves from ever saying anything about one that would not also be true about the other. They've got that funny sort of identity that, if we're very careful we can keep them identical after we have learned that they're identical. I don't know whether I'm being totally obscure, but what I'm trying to say is that possibly if it had taken us longer to learn' about the existence of the other one and about the identity between them, the method of thinking that goes with one is different enough from the way of thinking that goes naturally with the other so that physics might have developed quite differently for a while.

Heisenberg:

Yes, for a while, yes, it might. Yes, that's quite true, yes. It was only at the end of this thing, and the end came by the papers of 1927, that one did realize that every word has only a certain range of applicability; that, of course, was the final solution of it, that one could use the words, but should always remember the limits of the words, of the applicability of the words. I would say that however this story would have gone, at the end this would have been the result and then people would have realized, "Yes, we must use the words with this restriction."

Kuhn:

We've been talking about this terribly generally. Can you tell me about conversations, about other peoples feelings at the time, about ways in which these attitudes evolved?

Heisenberg:

Well, I remember our struggles in the autumn of '26. Historically I think the development was like this: We had developed quantum mechanics by the Gottingen paper. Then there was the Dirac paper, which was very important and which also confirmed me in my belief that if once one has started on a mathematical scheme, there is no way out; two different people must come to the same conclusion. So I was quite happy that Dirac had come to the same conclusion as Born and Jordan had done. So I saw now this is now fixed. There is no room anymore, now it is fixed.

Kuhn:

Was that paper a big surprise to people? Had you met Dirac, for example, or heard of him when you were at Cambridge?

Heisenberg:

No, well, the situation was this: I actually thought I had met Dirac, but Dirac explained to me that this must have been a mistake. Actually, he was in Cambridge when I came to Cambridge. Perhaps we should go back into this history.

Kuhn:

By all means. I've also got a few left-over questions that are even before that, but I think we should save those for later and go on.

Heisenberg:

Let's speak about spring and summer of '25. The situation was this: I came back from Copenhagen. I had just finished the paper with Kramers on the dispersion and, as I told you, I went home with a book on Bessel functions in the hopes to guess the intensity of the hydrogen atom.

Kuhn:

Now this was home in Gottingen?

Heisenberg:

Yes, in Gottingen. I had gotten the book from Born's room. Born had so many textbooks. Born gave me one of his books and so I tried to study Bessel functions and to see whether I could guess the intensities of the hydrogen. Then I realized that I couldn't; it was too complicated stuff.

Kuhn:

How long were you at that? Do you know?

Heisenberg:

Well, I should say a few weeks, only a few weeks. Then I gradually saw that the first thing I must know is if I have all the amplitudes of a coordinate X, and of another coordinate Y, how can I get the amplitudes of the problem XY.

Kuhn:

It's not clear to me quite how that problem rose out of the problem of the hydrogen atom that you had been wrestling with.

Heisenberg:

Well, the point is that in the hydrogen atom you have rather complicated multiplications, rather complicated mathematics, and still at the end you have a Bessel function as an amplitude. But then you see that this Bessel function comes out of many very complicated processes in the calculation. And I felt, "Well, I should try to imitate these processes in the same way as I did imitate in the dispersion formula the processes of the perturbation theory, [that is], by multiplying amplitudes and so on." And then I saw I couldn't really imitate these processes because the mathematics was so complicated. Then I started thinking about the fundamental side of this process. And that was, of course, that the most trivial operations are addition and multiplication. There was another point in it. I had for some time thought about higher corrections to radiation. In order to calculate the radiation, say, you have not only the dipole radiation, you have the quadrupole radiation, and so on. There are then parts in the interaction between electrons and radiation which do not depend linearly on the coordinate, but on the square of the coordinate. And. I felt, "Well, even if I had solved my whole problem, how can I calculate the quadrupole radiation instead of the dipole radiation?" And in this way I came to this idea that one should first see, "Well, can one generally multiply X and Y? If X and Y are known as being such patterns of quantum theory amplitudes, can I then guess the pattern of XY?" Actually that was already done in the dispersion paper. In the dispersion paper we had practically the matrix multiplication already in it, without knowing it, of course. And so this was a point. And then I saw, "Well, that's all right. One can actually multiply XY." There's a very disagreeable feature about it that YX is not equal to XY. That I saw, and was very dissatisfied with this situation. But then I felt, "Well, after all, can I not try a problem on which I have to do with only very simple multiplications — not so complicated as the Kepler motion." And so I came to the anharmonic oscillator — the anharmonic oscillator being a thing where if you do the perturbation theory you just have a few powers of X, and that is, of course, very agreeable. And so I just felt, "Now I must try to see whether I can get a complete quantum mechanics of the anharmonic oscillator just by using that kind of multiplication which was taken from the dispersion paper." And that I did. I think in Gottingen I had to give some lectures at the-same time. That was in May, early in May, just at the beginning of the summertime, that I was engaged with it. And then I was a bit ill. I had this hay fever, a very bad attack of hay fever. I couldn't see from my eyes, I just was in a terrible state. I asked Born if I could not go away to this island Heligoland, where one can get rid of hay fever. At that time I was very deeply involved in this calculation, and I remember —.

Kuhn:

May I interrupt. There's perhaps a missing element here. One could certainly investigate and get an important suggestion from the dispersion formula — how do I multiply the X's or an X with a Y? But there's still the step that's involved in the stage in which you come out of treating this whole array of X's as a dynamical variable, applying Newton's law to X considered now as an array. That the dispersion formula certainly does not do.

Heisenberg:

I could not say that this has occurred to me in a single instance, which I would remember. I only would say that came gradually as something which is almost obvious. In that sense, one would say, "What does X(t) mean?" Now X(t) in an atom does not really mean an orbit because such an orbit has the wrong frequency and so on. So what does it mean? Well, it means certainly some kind of radiation, so all these frequencies which are possibly emitted do somehow represent this X(t). "Somehow represents" was, of course, a very vague term but it wasn't more than that. Still I felt, "Well, why shouldn't this representation of X(t) also obey the laws of motion, that mx is momentum, and so on." I saw that there was really no reason why it shouldn't. At least one should try. One should try. And then I thought, "Well, can't one find a simple example in which I could try, in which I could see how things work out?" The anharmonic oscillator is a trivial example which one should try in that situation. At the same time, I doubted very much whether it would work. So I felt, "Well, if I do that kind of thing, if I actually replace the X(t) by this kind of funny pattern of elements, had I any chance, for instance) to prove the conservation of energy?" So I first doubted whether the conservation of energy would hold. And I felt, "Now, that is an interesting point. That can be seen. If I now do a simple example, can I not simply see whether the conservation of energy is right or not?" That was, I think, exactly the situation at the time when I left Gottingen for Heligoland. And I was in the middle of these deliberations. Now I thought, "Well, now I have a problem on which I can work, where I really can do calculations, and maybe it comes out that the energy is conserved." Then actually I was rather ill. I took a night train to Cuxhaven, and I had to get a boat there in the morning. I was extremely tired and I was swollen in my whole face. So I tried to get breakfast in some small inn there and the landlady in the inn said, "Well, you must have had a pretty bad night. You must have been beaten by somebody." She thought I had had a fight with somebody. I just told her that I was a bit ill and she was a bit worried about me. I just told her that I would take the boat. So I took the boat, and I arrived at Heligoland, and again in the first days I was pretty ill. But then at the same time, I started working very hard and I got in a state of great excitement because I saw that it worked out so nicely. I remember that I had the scheme there from which I could derive the energy conservation, and so I worked all night and I made many slips in the calculation. Around two or three o'clock in the morning I saw that the conservation of energy was correct. I was extremely excited, and it was just early in the morning already. I decided that I would go out for a walk and so I did. I rather half-climbed on one of the cliffs of Heligoland just for excitement. And I felt, "Well, now something has happened." So then after a while I went back and I went sound asleep. Then I started writing on a paper.

Kuhn:

Before you got to Heligoland, you had solved the problem of the intensities of the anharmonic oscillator, but you hadn't gotten the conservation of energy?

Heisenberg:

No, I had on paper just the general ideas that I could use an X which is always an X of n and n minus 1. Then I had realized that an anharmonic oscillator had not only matrix elements from n to n minus 1, but also from n to n minus 2 or to n minus 3, depending on what kind of oscillator I took. And I had general formulas there — how to calculate X² if I had X and XY if I had X and Y. Then, of course, it worried me terribly that XY was not equal to YX. But then I said, "Fortunately, I don't need it; fortunately, it is not very important, because I have only the X and XX." That, of course, was a point. It never occurred to me that I should write down XP_x plus or minus P_xX. I had, of course, to think about the quantum condition. And that was an important point. But there I knew so much from Copenhagen how important this Thomas-Kuhn sum rule was. That took some time. That I think I had done in Gottingen, had seen how one could translate the Thomas-Kuhn sum rule into what I call a quantum mechanical statement, into a statement in which only differences occurred. I did not see that it was a commutation rule, but I just saw that this sum rule can be translated into a statement so that only differences between amplitudes or products of a amplitudes occur. That means that now I can bring this sum rule into my whole scheme and then this sum rule actually fixes everything. I could see that this fixes the quantization.

Kuhn:

Did you try any other ways to do that? Off hand, your scheme simply lacks a quantum condition and there would be all sorts of places one might look or all sorts of ways in which one might try to apply it. Using the Kuhn-Thomas rule is a stroke of genius but one supposes that there were a lot of other intermediate attempts.

Heisenberg:

No, I would say it was rather trivial for the following reasons: First of all, there was the integral p.q. Then one had seen that integral p.q. sometimes is ½ and sometimes is not ½. That played a role. Because then I felt that perhaps only the difference of integral p.q. between one quantum state and the next quantum state is an important thing. So I actually felt, "Well, perhaps I should not write down integral pdq, but I should write down integral pdq in one state minus integral pdq in the neighboring state." Then I saw that if I write down this and try to translate it according to the scheme of the dispersion theory, then I get the Thomas-Kuhn sum rule. And that is the point. Then I thought, "Well, that is apparently the way how it is done." Then I elected one more point. That was the lower state. Since the Thomas-Kuhn sum rule just is the difference between integral pdq here and there, it was clear that one additive constant was missing, a constant of integration. And then one could see that this constant could be gotten from the lower state by saying that there must be one lower state defined by the fact that the amplitude to the next lower state must be zero, because there is no lower state. Then that fixed the constant, and then everything was clear.

Kuhn:

When you saw that would come out, that must have been still a little later?

Heisenberg:

Yes, but the deciding step was that the conservation of energy — that was the main point. That was the point where I felt, "Well, this I —." In some way I could not see that it must come out. It came out as a kind of "Geschenk des Himmels." And I felt that if that comes out, it must mean that there is something in this scheme, it has its inner consistency. So this was really the point in which I was convinced that I had found something real. This Thomas-Kuhn sum rule business was a rather obvious thing, and then I was quite interested to see that now half quantum numbers came out from the an-harmonic oscillator. Then I don't know what I did first. I believe that the first thing was again to write a letter to Pauli. I'm not absolutely sure about it. But after all, I should have sent all my letters to Pauli, so if I wrote from Heligoland, this letter should be there among the others. Well, it's a good idea. I must look for that letter. Do you know whether such a letter exists?

Kuhn:

I don't really know because I don't really know that collection.

Heisenberg:

This is the old-fashioned way to remember something, you know.

Kuhn:

Do you still have your letters to Pauli?

Heisenberg:

Yes. Mrs. Pauli sent me microfilms of my letters to Pauli and I made a copy. I think these copies are at home. Well, I do think that I wrote to Pauli and perhaps I wrote also to one of the Dutch people, perhaps to Kronig about it. There may be a letter to Kronig. At least I did exchange letters about the thing.

Kuhn:

This was in May?

Heisenberg:

This was in the end of May. And I think that I came back from Heligoland on perhaps the third of June, or so, I don't know.

Kuhn:

You'd been there about three weeks?

Heisenberg:

Two weeks, I should say. Two weeks. Between two and three weeks. Well, van der Waerden told me that he knew the exact date when I came back from Heligoland. There was some interesting connection with a talk which Born had given in Hanover. But that van der Waerden had found out. That I simply didn't know. But anyway, when I came back to Gottingen, I had more or less already finished my paper and handed it to Born, and asked his opinion about it — whether it should be published or not. And Born at once approved of the paper. So he was, I think, quite glad about it. So I did give the final form to the paper and sent it to the Zeitschrift fur Physik.

Kuhn:

You were still in Gottingen?

Heisenberg:

Yes, at that time I was in Gottingen.

Kuhn:

I think he now remembers that you left the paper with him and went off to Copenhagen immediately, or something of that sort.

Heisenberg:

To England very soon, yes.

Kuhn:

But he rather now remembers that he thought it should go in and sent it in himself after you had already left.

Heisenberg:

That is possible, yes. That is possible. It may be that I just handed it to Born and said, "Well, all right, you do with it what you think is correct to do."

Kuhn:

Had you talked with him at all about this work before you had left for Heligoland?

Heisenberg:

Not before, no. No, I don't think I had because at that time everything was so vague. We may, of course, have talked in a general manner about what one could possibly do, but the only time when I really had talked about it was when he gave me this book on the Bessel functions. I don't think that in the meantime we had many discussions. When I came back I told him, of course, that I felt that something real was in the paper, but that I was very uncertain about it. So I left it to him. Actually I had to leave for Cambridge — for Holland and Cambridge. I wanted to go to see Ehrenfest, and then afterwards to go to Cambridge. And (van der Waerden) had explained all this to me. He knew all the dates. He knew when I had given the paper to Born and when Born had sent it to the Zeitschrift, when he had given a talk, and so on. So I think he can fix every date with accuracy. Then, of course, I had to go in Holland and Cambridge. When I saw Ehrenfest I told him about my attempts.

Kuhn:

You went first to Ehrenfest?

Heisenberg:

I went first to Ehrenfest, yes.

Kuhn:

Had you known him before?

Heisenberg:

I knew him from Copenhagen, yes. He had been in Copenhagen in the time when I was in Copenhagen; he was not there permanently, but he was a good friend of Bohr, and so he came to Copenhagen quite frequently. So I think that I had met him at least once or twice before. Perhaps I had met him in Gottingen already but certainly in Copenhagen. So I knew him well, and he had invited me to come to Holland and to see his young people. I stayed in his house, was in his family, and then we had many discussions in the institute. That was a very nice time, but not very long. I would say perhaps two or three days.

Kuhn:

Sometime, I'd love to get you to talk more about Ehrenfest.

Heisenberg:

Yes, I think he played a very interesting role in physics at that time because he had such a critical mind. By this criticism he always could stir up people and could get interesting discussions going. He had a kind of catalytic effect on physics to a very high degree. He was a ("Zaudrer") of a man, who always has his scruples and his doubts and his difficulties, but he had a very interesting and certainly a very clear brain. He was a very good physicist. Well, now to Cambridge.

Kuhn:

We will come back. Just tell me, you did talk a little bit about these ideas when you were in Leiden?

Heisenberg:

Well, I do think that I did at least tell it to Ehrenfest, but I don't remember whether I also told it to the other people. Probably Goudsmit has been there, I don't know — Goudsmit and Uhlenbeck and these people. They must have been there. I don't know.

Kuhn:

Yes, they would have been there then.

Heisenberg:

Well, I would assume that I had talked to people about it but probably in a very vague manner.

Kuhn:

You don't remember any reactions to it?

Heisenberg:

Not from Holland. I do remember reactions in Cambridge. I remember that I spoke with Ehrenfest about all the fundamental difficulties of quantum theory and Ehrenfest would always say how dreadful it was and that one couldn't understand anything. And he probably was interested when I told him that I was trying now in quantum mechanics — such and such a type, but we didn't go into the details. So he probably was just interested, but he didn't really come into the details of the thing. But in Cambridge it was different. I should perhaps tell one story and this is just a kind of foolish story, but I think it characterizes one type of situation. By that time, somehow, I was completely exhausted. First I had written the paper while I was in Heligoland, then I had (written a paper when I came to Holland), and then also there was a change of climate, England was very warm. So I was in a state of complete exhaustion when I was in England. Now I stayed with Fowler and I liked Fowler very much. It just so happened that after the first night when I had entered this house, he had to go away for a whole day for meetings in London or so. I don't know why, but apparently his wife was not there, so he asked his maidservant to provide me with breakfast in the morning, and then with the lunch, and then with tea, and then with dinner. I rose in the morning to have my breakfast and while at the breakfast table I just fell asleep. The maid came in and saw that I slept and she thought, "Well, that's fine. This young man is always asleep." So she took the breakfast away, and I slept. At twelve she came in and said that lunch was ready in another room. I didn't hear a thing, I just kept on sleeping. So she took the lunch away, and was a bit frightened. Then she made tea in the afternoon and also told me that the tea was ready. I said, "Yes," but I went on sleeping. And the same thing happened with dinner. About nine o'clock in the evening, Fowler came home. The maid was terribly upset. She said, "Well, this young man must be half dead. He has been asleep since this morning and he must be terribly ill." Then Fowler came in and I realized that he was there, and I said, "Oh, hello Fowler," and Fowler said, "Well, what happened to you? Are you ill?" I said, I'm just perfectly alright, I'm perfectly happy and in the best of health." In some way I had slept the whole day. That was a strange experience. Well, we had a very good time together. I liked Fowler. He took me out when he played golf, and we went for walks. Then he took me to the Kapitza Club one time. Now the Kapitza Club was a peculiar thing. There was Kapitza and he had a room in the college. Some younger people used to come together in his room in the evening at the fireplace and somebody would tell what he had done in scientific work and the others would discuss it, but in an absolutely informal way. Actually, we were sitting on the floor, there were no chairs. I was asked to speak about this new paper of mine and I explained all the details of the paper and I think the others saw that I was deeply engaged in this kind of thing. Fowler got very interested and told me, "Well, as soon as you have something ready, in print or so, could you send me the proofs of it?" Actually, Dirac was not in this group. I later thought he had been. There were some young people I didn't know, but Dirac was not there. Fowler had told me that there was a young man actually studying engineering. He was an electrical engineer at that time. He was an extremely good mathematician and he might perhaps be interested. But it seemed that I had not met Dirac this first time. At least I don't remember it. When I had sent the proofs to Fowler later on, then Fowler handed these proofs to Dirac because Dirac was interested. It was then actually on the basis of these proofs that Dirac then started to do his own work. But I didn't know anything about this young Dirac doing some work on it before I got Dirac's letter. This letter, I should say, came into Gottingen late in August or early in September of '25. It was a great surprise to everybody in Gottingen because it contained the complete scheme of this p and q numbers — q numbers he called them. It contained the proof of energy conservation and all that. So practically everything that had been in the paper of Born and. Jordan was also in the paper of Dirac.

Kuhn:

When you say letter, was this an advance copy of the article or was this a letter describing his results?

Heisenberg:

Well, I do remember that it was handwritten, and, in so far, I would say it was a letter, but actually it was a paper. It was a paper which contained everything of the real paper so it was perhaps an advance copy. So far as I recall, it was not typewritten. It was handwritten and written in Dirac's way of very nice writing. But I haven't got it anymore. It must have belonged to this famous batch of papers which have been lost. But I kept it very carefully. I thought it was a very interesting piece of work that a young man who had never done anything in this field, and just knew about a paper of somebody, that he would work it out so well and describe new mathematical ideas. So I was full of admiration for this piece of work. Of course, Born and Jordan also agreed that this was an extremely interesting paper. It may be that Born was a bit disappointed that somebody else now also had done these things. But still, after all, it was so obvious that the two things are entirely independent, and also it was identically proved that the whole thing is on the right track.

Kuhn:

At the Kapitza Club, Fowler was very much interested. Was there anybody else at Cambridge either among the students or others? Well, in order to be interested in that paper, one had to have a great deal of background in the troubles which existed in quantum mechanics, particularly with the great transformations in the Correspondence Principle.

Heisenberg:

Yes. Well, so far as I can recall, it was only Fowler, but Fowler knew it very well, because Fowler was frequently together with Bohr. You know that Fowler was the son-in-law of Rutherford, and since Bohr was very much in the family of Rutherford, so Fowler belonged to the family, so to say. Fowler was frequently in Copenhagen. I had met him there. Fowler was certainly the one who was deeply involved in these things. He knew about all the troubles. But I do not know whether there was anybody else who really was in these things. I have tried to remember names of those who have been at the Kapitza Club at that time, but I just don't know. One really ought to ask Kapitza about it. He is still alive.

Kuhn:

Yes, and I hope that we will. I have not been able to get an answer to a letter but Sir James Cockcroft has praised and has probably written. In particular, Kapitza has the minute book — the first book — and I hope he will send us a microfilm.

Heisenberg:

Oh yes, yes. That would, of course, be very nice.

Kuhn:

The second volume, which is less relevant for us, is being microfilmed at the moment. The first volume is in Russia.

Heisenberg:

Is in Russia, I see, yes. Well, let's hope that he has still kept it. One never knows what happens to these things. That would be very nice. There must be some notes about this lecture.

Kuhn:

I have found here a complete list of all the Munich Colloquia from 1908 until '38. Just the subject, but it's a full list and has some quite interesting things in it.

Heisenberg:

Well, there I would be most interested to see notes about Schrodinger's lectures. Is that in the book? You remember that I told you about these lectures of Schrodinger in the summer of '26.

Kuhn:

I don't know whether that's in the book or not.

Heisenberg:

Because it did take part in Sommerfeld's Institute, so one could imagine that it's there.

Kuhn:

It very likely to be.

Heisenberg:

Yes. That would be interesting to see. Has Sommerfeld himself written the whole thing?

Kuhn:

No. In any part of the book that I've looked at, there are no notes at a there is just the date, the name of the speaker, and the subject. It's a purely formal record of who spoke and on what subject. In this sense, it is not a minute book. But even at that, this can itself be quite useful in finding out when things were first talked about.

Heisenberg:

Well, I would suppose that this lecture of Schrodinger was around, say, July of '26. I did tell you about this lecture of Schrodinger. Then later on Schrodinger came to Copenhagen. That was in September, '26. But anyway, now we are at the time of the Drei Manner Arbeit. Well, we had gotten the Dirac paper, and then we agreed that we should do it quite properly and work all the mathematics out.

Kuhn:

Well, could you tell me what has been going on in between. Well, did you go back to Gottingen from Cambridge?

Heisenberg:

Well, I would believe that probably I didn't go back to Gottingen at once. Probably I went on a holiday. I used to go with friends somewhere to the mountains or to the sea or both. Now, that was '25. I don't know when I got the Born-Jordan paper. So I would believe that I came back to Gottingen not before the end of August of '25. I will try to find out about '25. Where have I been in '25? No, I just don't know. Just don't know.

Kuhn:

Well, has this progress of theirs —. Had Born written to you about their matrices?

Heisenberg:

Yes, yes. I certainly remember that it had made a great impression on me to see that instead of the quantum condition as the Thomas-Kuhn sum rule, one could simply write pq minus qp, and then one could prove the conservation of energy quite strictly. That I felt was an enormous progress, and so I was very glad about it. But I could not say at the moment now when I heard of this progress first, whether I had gotten a letter already in Cambridge —. I would rather think I only heard it at the end of August when I came back to Gottingen. That I would believe. And it was not long in between the end of the Born-Jordan paper and this letter of Dirac. So there was only a short time in between.

Kuhn:

But you think that probably the whole Born-Jordan paper was done without your knowing more than that something was going on?

Heisenberg:

Yes. I think that was completely without interference from my side. I had been. away. I was in Cambridge, I was in Holland, and then on a holiday. And then the paper was finished. That was certainly done quite by Born and Jordan alone.

Kuhn:

Did you feel bad about that?

Heisenberg:

No, not at all. No, I would say just to the contrary. I was extremely happy. You see, I was very excited about my first paper, but in some way I felt quite uncertain. I had the feeling, "Well, I have entered an entirely new thing and now I need some help to do the mathematics. Now we are at a very interesting point." I was very happy when I heard that Born and Jordan could have done so much about it. So at this point there was absolutely no ill feeling — just to the contrary. On the other hand, I was also a bit happy that Dirac could do it independently because again this did show to me that from this moment on it is settled. There is no freedom left.

Kuhn:

Do you know anything about this strange story that Professor Born tells about Pauli?

Heisenberg:

Oh yes, that Pauli was dissatisfied with the Gottingen formalism?

Kuhn:

Yes, that Born, when he recognized that it was matrix mechanics, and wanted to work on it, he realized that he was too tired, or something, to do this by himself. He needed somebody to work with. He met Pauli on a train, told him about this, and said, "Will you work with me on it?" I am perhaps mis-remembering what it in any case only a recollection. But Pauli said something like, "I won't have anything to do with it. You'll ruin it with your Gottingen formalism."

Heisenberg:

Yes, I heard that story from Born. I can imagine that it is more or less a following thing. You know, when Born and the Gottingen group had entered into quantum theory around '22, they actually tried to do it by studying very carefully all the fine details of Himmelsmechanik, Charlier, and so on. Thus Born and I had published this paper about the Bohlin perturbation theory, and so on. So in some way in Gottingen Born liked complicated, elaborate mathematical formalism. This was an idea which Pauli disliked definitely because he felt that the real problems are not those of mathematics — the real problems are those of physics. I might say that, for instance, in the present situation of physics, we have exactly the same kind of disagreement between two groups. I would say, to translate in modern terms, Born and the Gottingen group represented the Wightman group in our times, Wightman, Lehmann, Szymanzic. Those people try to do proper mathematical analysis with the most modern tools of mathematics and hope thereby to solve the problem of physics. Pauli always emphasized that these were problems of physics, and not mathematics: "you have first to solve the problems of physics. And only then you have to find the mathematical tool, but that's a quite trivial matter. It's quite uninteresting that the mathematicians can prove that solutions exist and the axioms must be so. That's quite unimportant." The criticism of Pauli must have meant this: "We are just in the beginning of an interesting development in physics. It may be that the problems of physics can now be solved. But if we come in too early with mathematical proofs and the idea of mathematical proofs, then we have a good chance to ruin it. Because then we have a chance that we unconsciously assume some mathematical axioms which are not fulfilled and thereby get into contradictions and all the difficulties again." I find it so interesting that in our present time we have exactly this same situation. Because again, there's one group sating that we must only do the mathematics properly, and then we will know what it is all about. And other people like Chew, or also myself, would say, "Well, on the contrary, we must first solve the fundamental mathematical difficulty, and not worry at all about mathematical existence and the convergence and so on. We are just not interested in axioms. We just want to solve the physical problems. Later on the mathematicians will prove what we need." And that must be the basis of Pauli's criticism, which, on the other hand, was completely unjustified at that very moment. Because at that moment, the only thing that was needed was a good mathematical formalism. In this way, Born did exactly what had to be done at that point. But psychologically, I think one can understand how the criticisms came about.

Kuhn:

Surely. Well, now the Drei Manner Arbeit itself. I have the impression that you were not in Gottingen very much while this was being done. A lot of this was done by correspondence.

Heisenberg:

Well, some of it was done by correspondence because, as I recall, I had again been in Copenhagen for some weeks in the autumn, perhaps in September or October. So there was some correspondence. But I do remember that in this period van der Waerden has done a lot of work, and he found out quite clearly which parts were written by whom. I think the chapters 1 through 3 are written by Born and then there's one chapter which was written by myself, and then the last chapter about radiation was mostly written by Jordan. Well, after these parts were written by the three different people, we, of course, tried to work it together; each one corrected the papers of the other one.

Kuhn:

To what extent did one decide in advance what parts were going to be written. Clearly you must have known you needed the multi-dimensional version of the theory, and you needed the perturbation theory. Surely, being able to do a perturbation theory was one of the conditions for doing the paper in full. Now there are some other parts of the paper that I would suppose came in because one had learned how to do them. For example, the last part on fluctuations, possibly even the application to angular momentum.

Heisenberg:

Yes, that part I had done. I was interested in this angular momentum side of the problem. We could look at the paper. Do you have the paper here? I can tell you exactly which part was by whom. We had first many discussions and then we agreed that Born would write this, and I would write another thing. Well, the introduction I did write. It was actually a new introduction which used some older parts which Born had written. Born had written an introduction, but I wasn't too satisfied, so I did write a new introduction. This introduction was my formulation. Then Chapter 1 is Born's. ... And Chapter 2. ... And Chapter 3. ... Chapter 4 is mine. That is the angular momentum that I did. I was interested in this, especially, that one indirectly got the half quantum number business. That I did. Yes, and now there comes the part of Jordan, this third paragraph of Chapter 4. And, of course, there had been discussions between the three of us, but that was essentially how it was written. I think it had been agreed about beforehand. Later on we did homogenize it. I made some suggestions and changes in Born's part,. Born suggested changes in my part, and so on. But we had agreed more or less that the whole thing should be written in that way.

Kuhn:

To what extent did you all work on the working out of what was in the parts? When you say that Born wrote these, did he also invent most of them?

Heisenberg:

Yes, he did. Yes. Undoubtedly. Especially this idea of the principal axes transformation. That was his idea. Well, he had such a good knowledge of mathematical schemes, so he knew that these things could be done by the mathematicians and so he applied it here. I would say that it was all just bad luck that he did not find at that time already the Schrodinger picture. He could have found the Schrodinger version especially where he treated this continuous spectrum. There it's only a short way to the Schrodinger picture, but still, you have to find it.

Kuhn:

There are a number of things about that paper I'd like to ask you. In the paper you start out developing things in the standard, or what the Born-Jordan paper would have you believe is now the standard, matrix mechanics. Then, later on, one introduces Hermitian forms and does the whole thing over again, and shows that you can, in fact, go further more easily. It's a powerful technique. Was there any discussion of the possibility of simply doing it all in Hermitian forms from the beginning? What was the reason for doing that much twice?

Heisenberg:

Well, that was more or less a historical reason. In some way, we did not feel that we should do it too systematically, but also we felt that it might be some help for other physicists to show in the one part of the paper how similar it is to those older things which one had tried. Then we could show that there's a new technique which seems to be more powerful, which is actually identical with the other thing, but which does it in such and such a way. So we felt, "Why not give everything in the paper?" We did realize that it was a kind of duplication of things, but still we felt that it may be useful for people — we don't know what will be the most convenient. We had the impression that there may be other still more convenient mathematical schemes behind it. One already felt that there is a great transformability in the whole thing, but we were not so far really to write down the general scheme. Well, the canonical transformations were actually in the paper in some way — in the Born part of it — but not in this very general manner in which one later on could use it to go over to Schrodinger's picture.

Kuhn:

No, certainly not that general. Again, I may be, in some part, missing a point here, and it may be that I just never lived closely enough with classical Hamilton-Jacobi theory. It is always said the S in SWS^-1 is the equivalent of the old action function. Now, I must say, by the time one is through proving the properties — and I am struck by the fact that there are some interesting similarities — it's by no means an "anschaulich" parallelism, even after the development is all done.

Heisenberg:

No, no.

Kuhn:

Yet, in this paper, in the development of the time dependent perturbation theory, that analogy plays an immense role. You say this is like the old S from Hamilton-Jacobi theory, and you then proceed to simply write down the form for the, time dependent perturbations without a remark about where the devil they come from. I'm not sure how easily I can put my hands on the relevant formulas. Yes, here are the formulas for the time independent case, which are worked out in some detail, and the thing is entirely straightforward. Later on, I think it's here; [p. 571. "Eine einfache Uberlegung zeigt..."]. I mean, this is of no great importance but it's darn curious to me as to what was going on.

Heisenberg:

Well, what is the question there?

Kuhn:

This is all very nice and clear if one has at one's fingertips the full parallelism between this not very obviously action function of Hamilton-Jacobi theory, and knows the Hamilton-Jacobi theory time dependent perturbation. Otherwise, there's an awful lot to be proved and it isn't very clear how one is going to go about proving what lies on the way to that formula. Now, this is not a request to work it out, I'm just wondering if perhaps it was all that much more obvious against the background of the familiarity that is now so rare with Hamilton-Jacobi technique?

Heisenberg:

Well, I would say that the analogy with the classical Hamilton-Jacob technique was the beginning of all this and this formula with the S and the S^-1 — that was a much later stage. So actually, what we first did was just to try to imitate the old Hamilton-Jacobi technique as closely as we could. Van der Waerden told me that he had found a long letter which I had written to Pauli, in which I had worked out the whole perturbation theory. Actually, I did already include this idea of S and S^-1 because I just got the other one by dividing 1 over S in the perturbation series. In this way, I had succeeded in getting a scheme which was, exactly the old Hamilton-Jacobi technique of the classical theory, and of which one could, at the same time, prove that it was the S^-1. So we went from the old technique. We discussed that among ourselves and had actually discussed it many times in Gottingen before. That was at the time of the Bohlin trouble and that kind of thing. And then, at that time, we first still stuck to the old techniques, but in the course of time we realized that this was just writing S and S^-1. So we realized that there was an enormous simplification behind it which was not existent. in the classical theory. That, of course, was an enormous encouragement — that in this new mechanics things became simpler than in the old.

Kuhn:

Now you know why I ask stupid questions like that — because I get interesting answers. I mean this is a very interesting answer — that you only backed in a fully classical approach to the S and S^-1. I think this does very much account for how early you insist on what is in fact now obscure, that is, an intimacy of the connection between this theory.

Heisenberg:

Well, you must remember that the whole thing was that we wanted to make more and more precise the ideas of Bohr and the Correspondence Principle, and that we finally came out with something which was simpler than classical mechanics. That, of course, was a great surprise and a great encouragement. But to begin with, we always felt that we must try to be as close to classical mechanics as we just can possibly be. And so I think the perturbation theory was just a very nice point where this happened. Van der Waerden gave me then a copy of this letter which I wrote to Pauli in which I worked all this out. At the end I just stated that actually I could get this by writing S^-1.

Kuhn:

There are a couple of problems which run through all of these early papers.

Heisenberg:

I'm sorry, that's quite a cold you have.

Kuhn:

It's all right. I think it would be better if I'd stop smoking. One of them is the need to symmetrize — the fact that you can't just take the old Hamiltonian and put in any old variable. That must have been a nuisance, a source of concern?

Heisenberg:

Yes, that made us some worries. It wasn't too bad, because those problems in which we were interested first, like hydrogen and so on, didn't contain the troubles. So we were lucky not to be forced to think too much about this issue. But it did already come up, for instance, when you moved over from rectangular coordinates to polar coordinates. In polar coordinates you had to decide something about the order of terms. Now in this case, of course, one said, "Well, let's first do it by rectangular coordinates, then we have no trouble. Then we see what it means in polar coordinates and then that makes it unique." Still, we felt, "Well, after all, in quantum theory apparently it is not quite trivial what the Hamiltonian should be." The only condition is that the Hamiltonian in the limit of h equals zero should go over to the classical one. But this leaves room for several Hamiltonians. That is one of the cases where one could see that the new mathematical scheme did leave some room. That's again the problem which we discussed before — there are always some places where a new mathematical scheme actually does allow for some space. But in the interpretation, I felt at that time that it did not allow for any space, so that the probability interpretation should follow from the scheme.

Kuhn:

But actually in both the Born-Jordan and in the Drei Manner Arbeit there are quite elaborate abstract techniques for producing from a classical Hamiltonian a symmetric Hamiltonian for quantum mechanics. The issue was not simply, "Do we have to do it in order to solve present problems." This is faced up to as an issue in the abstract foundation. And I should think it was a bother.

Heisenberg:

Yes, it was a bit of a bother. We had still so much of the classical physics in our minds that we felt that in classical physics we knew what a Hamiltonian was and we knew the problem. So then in quantum physics, we felt, the Hamiltonian is something which should come only from the classical side, but, of course, it was not unique. So it didn't occur to us that one should simply forget about the certain. These three and Niels Bohr and I — the five of us — all went together to Tisvilde to have a few days of recreation. I remember that Hardy, the mathematician from Cambridge, was very interested in the things Bohr had told him about my attempts, and he was interested to see that it had to do with very general mathematical schemes. Well, I remember that this was a very lively discussion, and Hardy started to think about what general aspects of mathematics could be brought into this new scheme and so on. But he did not really try to get into it, he just was interested and was amused that the physicists all of a sudden would be interested in the quadratic forms and linear algebra, and all that. I remember this visit to Tisvilde also because of two foolish stories. One thing was, we used to play Boccia in the garden and all the others were rather good sportsmen and liked to do it well. Besicovitch, the Russian mathematician was extremely poor at the game. Well, it so happened that we were divided into two teams and the two teams were equal at the end. We had played for two hours and we came to the final game. It was just a question of this last game which had to be settled. And, unfortunately, it was only poor Besicovitch who was left throwing the last ball. Everything pointed now to complete defeat of our team. On our side it was Niels Bohr, Besicovitch and I, with only Besicovitch left, and he would have no chance to hit the ball at such a distance. So in his despair, he did the following thing. Thinking now that the game was lost, he turned around and threw the ball over his back. Somehow he just hit right in the middle, and so we won our game! Well, that was one type of story. Then I remember when we went home by train this evening, we spoke about mathematical games and somebody spoke about the game with different heaps of matches. I think it's originally a Chinese game where you can develop a mathematical theory and Harald Bohr asked me if I could find out the theory of it. So I worked very hard in this train. In some way I was a bit ambitious. A real theory would use integral numbers — what we would now do with the computers. At that time, that was quite unknown to me. I could work out a small part of the theory, but not a complete theory. Then I was a bit shocked by the following thing. All of a sudden Hardy said, "Well, Heisenberg, you stop that! Harald Bohr, it's quite wrong that you give Heisenberg such a thing. You shouldn't put a man on such a problem so that he really tries to work hard. That is not good. That you shouldn't do." It made a big impression on me that he all of a sudden stopped the whole thing and said, "Well, that's not right." Well, that, I think, was autumn of '25. So we had many discussions about quantum mechanics at that time. Then I came back to Gottingen. I was then called to Copenhagen as the successor of Kramers in the summer of '26.

Kuhn:

Well, that's whole year.

Heisenberg:

Yes. Well, I went back to Gottingen. In Gottingen I gave my lectures in the winter term and during the time of the winter term of '25 to '26, the Schrodinger papers appeared. The first word I heard about Schrodinger's papers was through a letter by Pauli. Pauli told me there is a Schrodinger — well, I knew his name, of course — who has written some new ideas about wave mechanics and I should read it, I should look at it. Then the papers came out. I think it must have been January or February '26. That was at once a problem of concern to everybody in Gottingen: Is that right? Has it to do with our own papers and what has it to do with them?

Kuhn:

Now, you were back here in Gottingen. I understood you to say back in Copenhagen.

Heisenberg:

No, no. I'm sorry. I meant that I was in Copenhagen in the autumn, but then I went back from Copenhagen to Gottingen. So I was back in Gottingen; I gave my lectures in Gottingen — I don't know on what, but I've certainly given lectures there. During this time we had just finished the Drei Manner Arbeit and, of course, now we thought of new work. It was then we heard about the Schrodinger paper. I think in addition to the Drei Manner Arbeit, I wrote a short paper during this winter term together with Jordan on the anomalous Zeeman effect. I just wanted to settle the whole problem.

Kuhn:

You also wrote a sort of summary paper for the Mathematische Annalen?

Heisenberg:

Yes, yes. That was on account of Hilbert's interest. Hilbert had asked me to give talks to the mathematical section of the faculty on these papers. So I gave a number of talks and then I wrote the paper for the mathematicians. Hilbert found it was so important that it should be presented in a way which appealed to the mathematicians. I thought that I had perhaps succeeded — that more or less (Hilbert) was quite satisfied with it.

Kuhn:

Were the mathematicians in fact quite interested? Did they follow it at all?

Heisenberg:

Well, at this stage they did, yes. And there was some excitement about it. I remember that after the first lecture Hilbert was very satisfied and said very nice things about the lecture. After the second lecture he said, "Well, the second lecture wasn't good. That kind of lecture I could do myself!" Apparently I had not succeeded so well in explaining things in the second lecture. I was quite ashamed, and tried to improve my lecture. I did give a number of lectures on it and then I wrote this one paper for the mathematicians, which was, I think, quite well received.

Kuhn:

Did any of them follow up on this?

Heisenberg:

Well, I think Courant was interested. He tried to connect the different ideas of general linear equations. Of course, Courant was completely interested later on when he saw the Schrodinger picture come in. Then, of course, there was the problem of the Hilbert space which was exciting for the mathematicians. And that just began to come into the picture. I don't think Hilbert himself tried to do some work of his own. He was perhaps too old for that, but still there was definitely a strong interest in it. The mathematicians had the feeling that something had happened with the physicists now, and they should be interested. Then I wrote this paper with the Zeeman effect and I was excited about the fact that the old formulas of Voigt from 1913 actually did come out of quantum mechanics. That was also a point of great enthusiasm. Then I tried to do something about the helium problem, so I went into the helium.

Kuhn:

Now I'd like to stay away for a minute from the question of resonance, for I think we rush too fast if we get to that also already. I still wonder if there's. more you can say about the way people reacted to all of this! In any case, I certainly hope that we talk more about how they reacted before the Schrodinger equation and then more about the problems which emerged with the Schrodinger equation. For example, it is striking how exclusively the follow-up on matrix mechanics comes from Gottingen. Surely Copenhagen knew about it almost as soon, and you say they were interested, but they don't do anything with it.

Heisenberg:

No. It's true that the only one who really did something with it was Pauli. Pauli was not in Gottingen, but he did work on the hydrogen problem with it. He did this very quickly and quite correctly. Well, I feel that probably people in Copenhagen still thought, "Well, this mathematical scheme is very complicated and it's very difficult to handle it." So they just didn't dare to work on it at once. Perhaps they found it too difficult or too foreign, too unusual and so on. This was probably the attitude in Holland or in Cambridge. Well, in Cambridge, Dirac, of course, did work on it. But in some way most physicists felt, "Well, that's a very interesting development. That's very odd and very unusual. What should we think about it? We'll just wait and see how things develop."

Kuhn:

Do you remember people actually saying that sort of thing?

Heisenberg:

Yes. No, I cannot say I do remember a definite statement by somebody, but I had the feeling that this was generally the reaction. I couldn't remember any definite conversation with somebody. But that certainly was the reaction. Well, if Schrodinger's papers would not have appeared, I think people would have tried also now to work on this scheme. But it became easy to use the many different mathematical tools only after Schrodinger's paper had appeared. It was very lucky for the whole development that Schrodinger's papers did appear. Well, the time was simply too short between the appearance of the Drei Manner Arbeit and the Schrodinger paper to develop a wider interest in these things. After the Schrodinger paper appeared, then, of course, everybody was just too happy that the two things were identical or seemed to be identical. Then it was natural that they would try to work on it.

Kuhn:

Perhaps it was because they were identical and they had already learned how to solve the wave equations?

Heisenberg:

Yes, that is just it. Schrodinger had to use mathematical tools which were familiar to everybody who was a good theoretical physicist. Everybody knew how to solve wave problems in electrodynamics and so on. But the matrix techniques were unusual and a bit odd for a physicist. He had to learn so many new things, and that's not so easy.

Kuhn:

One or two people have said to me that they still remember their sense of disappointment when the Schrodinger papers came out — this is not what most people say — because now it meant it was just going to be the old stuff again. It was so exciting while it looked as if it were going to be different.

Heisenberg:

Oh I see, yes. But that can only have happened to people who believed that possibly Schrodinger actually had put back the thing to the old scheme of wave problems.

Kuhn:

Well, I think they were thinking more just in mathematical terms. This is before the identity of the two.

Heisenberg:

Oh yes, I see, I see. Well, may be because a new mathematical tool is a point of interest to many people and so it was perhaps nice that here one would need quite new tools, while Schrodinger's were the old tools. I was worried about the Schrodinger theory in the sense that I felt now an entirely wrong interpretation may come into a discussion and will probably induce many people to believe they could go back to the old things. So that was a kind of a struggle-from that time on. I was not disappointed that Schrodinger had found such an interesting mathematical tool. On the contrary, I was very happy to see that now one could really calculate the matrix elements, for instance in the hydrogen or even in more complicated schemes. I used the Schrodinger technique with great pleasure, for instance in the helium case. But I was very much afraid of the interpretation. So I remember one letter which I wrote to Pauli where I said, "Well, Schrodinger's approach is, of course, extremely useful to do the mathematics, but of his physical interpretation, I don't believe a single word." I was very upset about his way of interpreting things.

Kuhn:

How did Pauli feel about it himself?

Heisenberg:

Oh, he agreed with our interpretation. Pauli was always completely on our side, but at the same time he thought that Schrodinger's development was extremely interesting and maybe also extremely useful. But I think Pauli never doubted the complete identity of the two schemes. In so far as I know, he also proved it himself — he didn't publish it. It was published by Schrodinger, but Pauli knew that the two things were identical already before Schrodinger knew it. Well, there was also a paper by Born and Wiener which went into this direction, which was pretty close to the Schrodinger theory.

Kuhn:

Yes, it certainly was pointing toward it.

Heisenberg:

Yes. By the way, was not Born away during this winter term perhaps in America for some time? So I was alone in Gottingen.

Kuhn:

Yes, you were alone there I think for the entire winter term.

Heisenberg:

Yes, that's quite possible, yes. I might mention one thing. I had started to tell it last time. That is, how I sometimes got ideas just from some discussion; the discussion drops the idea into one's mind and there it rests for many years, and then all of a sudden it comes out. I mentioned one case — that of Born's lecture on atomic physics where he spoke about ferromagnetism, and said that one could not understand the Weiss interactions. The other case was with the Uncertainty Relations. I think it was in that winter term that a good student friend of mine, Drude, the son of the physicist Paul Drude, came frequently into my room. One day I was in bed on account of the flu — just the kind everybody has now in Munich. I had a rather high temperature. He started arguing with me about quantum mechanics. I always said that there are no electronic orbits — the whole thing is more abstract. He came more from the experimental side and also he was younger and was still entirely in the classical ideas. He told me, "Well, I don't believe a word of your story about non-existing electronic orbits. First of all, the electron comes out, and therefore it must have been in the atom. Then you say there is no orbit. But if you would take a very good microscope, for instance a gamma ray microscope, why shouldn't it be possible? Then you could see the orbit." Now this remark made me feel terrible. I thought, "Well, after all, it's true. The gamma ray microscope has a resolving power which should do it." And I remember that when my friend had gone away, I started worrying about it. But then I forgot it again. Well, I had to do the mathematics and so on. Two years later, or one and a half years later, in the discussions with Bohr on the Uncertainty Relations, this came up again and I knew how to deal with this problem of the gamma ray microscope. I think that was, for me, a kind of tool for the development of this whole philosophy of the Uncertainty Relations. But it's quite funny that just such a remark, which was not meant so very seriously, started the development going. I did tell the young Drude later on that that was actually his question. He was quite amused about it.

Kuhn:

Well, now he can't have been the only one who said, "I don't believe a word of it."

Heisenberg:

Certainly such remarks didn't worry us at that time, because we knew how absurd the whole thing was. We knew that it was very different from anything in the old physics. It was a most natural attitude that somebody would first say, "Well, we don't believe a word." But then also most people said, after they really heard about the story, "Well, this is a very interesting attempt. You go ahead with it and let's see what comes out." Then, of course, the picture changed through the Schrodinger papers. (That part is now well known.) During the winter term in Gottingen when I wrote this paper on Zeeman effect and —. Did I write any other paper but that on Zeeman effect?

Kuhn:

I think in the winter term, until quite late in the spring, it's quite noticeable that the papers suddenly slow down quite a lot. I believe that I'm right in saying that until quite late in the spring those are the only two papers. Is this the first time that you have really been teaching?

Heisenberg:

No, I started teaching —. Now let me see. I started teaching in the summer term of '25, that is, in May of '25. Then I had come back from Copenhagen and that was the first term in which I actually taught.

Kuhn:

The paper for the Mathematische Annalen which you submit in December, 1925, and the one on the Zeeman effect [with Jordan, Zs. f. Phys. 37], which goes in the sixteenth of March, are the only papers after the Drei Manner Arbeit, which goes in on the sixteenth of November.

Heisenberg:

Yes, probably I had to do quite a lot with my teaching.

Kuhn:

Then in June, it starts all over again. The first of the resonance papers, [Zs. f. Phys. 38 p. 411], is the eleventh of June, the helium paper [Zs. f. Phys. 39 p. 499] on the twenty-fourth of July and it picks up. Particularly for the immediate aftermath of as big a breakthrough as this, given the rate at which you have been publishing, it is a surprising slow-down.

Heisenberg:

Well, it is probably partly due to the absence of Born, partly due to the teaching which I had to do and partly also a kind of exhaustion. One felt, "Well, now let's wait for a time." Then, of course, a new impetus came from the Schrodinger development. The paper which was most important for me was this paper on the fluctuations in the summer, [Zs. f. Phys. 40 p. 501, submitted from Copenhagen, 6 Nov. '26], because I first wanted to get straight that everything is uniquely fixed. Well, when did Born's paper on the collision problem appear?

Kuhn:

There's a little preliminary announcement in June and then the big paper in July.

Heisenberg:

Well, I did publish this paper on the helium. That I do know. But what else did I publish before the helium paper? You told me one thing which I did publish.

Kuhn:

[Hands Heisenberg a copy of a chronological bibliography of Heisenberg's papers.] ...

Heisenberg:

"Mehrkorperproblem und Resonanz in der Quantenmechanik" [Zs. f. Phys. 38, p.411], yes, I see. Oh yes, I see now. Yes, that was always a problem of having interaction between two bodies and so on. Yes, it was a preparation for the helium paper and the helium paper was worked out in detail and did give the spectrum of the helium more or less all right. Not the normal state. The normal state I couldn't do with that technique, but I could do the excited states on account of the perturbation calculations. I did use already in this perturbation calculation the methods of Schrodinger. I used Schrodinger to calculate the matrix elements. Yes, I remember. This paper in Naturwissenschaften [14 p. 989-95], September '26, I can't recall. That was probably just a kind of general survey.

Kuhn:

That's a survey. I want to ask you a few questions. It raises some of the questions of interpretation that come up more in terms of the big paper which we will not get to today — the Uncertainty Principle paper. Let that one go for the moment. There are a few little questions I wanted to ask you about that. Mainly it fits in with the question of the interpretation. Now I think would be a point to come back to the question I meant to have asked you earlier and somehow got misunderstood. With Born's paper in the summer of 1926, and with your own critical remarks about Schrodinger's own interpretation and certain of your attempts to replace it, now questions of interpretation —. [Secretary interrupts. Recorder turned off for a few minutes.] Well, I was just going to say that by the summer of '26, particularly in connection with the Schrodinger equation, I take it, under pressure from the things that Schrodinger is saying that you feel are wrong, people are now beginning to worry in writing about the interpretation of matrix mechanics. But it's gone a long way without anybody writing down anything except the formalism. Now when I said something about this before, you said that you knew you had the mathematical scheme. Now I mean what is the physics of the mathematical scheme? This sort of issue has, at least in the writing, been postponed. Is that fair?

Heisenberg:

I think that's quite fair. I would say that what one knew at that time was how to use the mathematical scheme in certain cases. For instance, you knew how to calculate stationary states. But how you would use the mathematical scheme in all cases puzzled us. For instance, if you wanted to calculate the orbit of an electron in a cloud chamber. There you can see the electron in the cloud chamber. Why does it go that way and why has it any orbit at all? That was quite unclear. The point was that this was even unclear after Schrodinger's formalism. To begin with, that was a problem which had been discussed also at this meeting in Munich here about which I told you. Schrodinger said, "Well, after all, we have no electron; there is no particle, we have just a wave." And when we have an electron going through a cloud chamber, we have just a wave packet going through the cloud chamber. Then, of course, we emphasized at once that this wave packet would spread out. It doesn't stick together, while the electron apparently does stay together. That was a problem of deep concern for Schrodinger and he tried to avoid it. Schrodinger was very proud that he could show that in the harmonic oscillator, the wave packet did not spread. He said, "Let's do the calculation for the harmonic oscillator. There we have the wave packet and then the wave packet stays together." Now, that was a point which we had discussed in some way already earlier. That is, I knew that the harmonic oscillator was always so much simpler than any other system and there, since we have just one frequency, it was very natural that it would stay together. So I put a very great emphasis on the point that the wave packet would spread while people like Wien would say, "Well, everything is perfectly all right and clear in the case of the harmonic oscillator. And in other cases it's not so clear yet, but Schrodinger will probably settle this point and find out how it is." While I, of course, tried to emphasize that this is a point which just can't be settled, it's a very deep point and shows that the interpretation is not so direct as Schrodinger wants it to be. So the general attitude of people was, "The interpretation is still quite uncertain. There is a new interpretation by Schrodinger. He might be right, but there are these other people like Heisenberg and Bohr and so on who object to this interpretation of Schrodinger. It may be that Schrodinger will find out how it's to be done. It may also be that there are deeper difficulties that one doesn't know."

Kuhn:

You and I may be misunderstanding each other or maybe you might have made a point which I then want you to make explicitly. I want to use the term 'interpretation' here in a way in which I would want to say, "Schrodinger introduces a new formula, which looks different from the formulas of the Drei Manner Arbeit and also introduces an interpretation which the authors of the Drei Manner Arbeit don't like. But they haven't produced any interpretation to go with theirs."

Heisenberg:

Well, no, no, with that I would not agree. For instance, we did say how to calculate spectra lines and we would for instance say, "If an atom is in an excited state — say a hydrogen atom is in the first excited state — experimentally that means that from a Franck-Hertz experiment we have found there is an atom in just that state." All right. Then we would say, "In this state the atom does emit a radiation according to Einstein's law — a spontaneous radiation — and the amplitude, or the intensity, of that radiation is given by the matrix element which is there. Now this is different from Schrodinger's interpretation. Schrodinger would say, "If the atom is just in this first excited state, then it doesn't emit any radiation whatsoever because it is a constant charge density, and that does not make radiation. Only if it is partly in the lower, partly in the higher state, that is, if I have interference, only then radiation is emitted." In so far, we had a definite interpretation which was not complete in that sense that we did not give interpretations for everything — we only gave interpretations for some cases. But in these cases the interpretation was quite definite, in line with the old Einstein interpretation, and it was different from Schrodinger's interpretation.

Kuhn:

I didn't, of course, mean to say that you didn't say how to interpret the formalism to solve certain sorts of problems. But there are certain sorts of statements that Schrodinger makes, such as that the electron is a wave, for which you have really no equivalent sort of statement that tells you that much about what the world is like. Schrodinger asks you now to understand why the world behaves in accordance with the wave equation. You never suggest at this point what an electron must be like — what forces between particles must be like — what it is like to think of mechanical quantities as matrices, even though what comes out of the matrices are the sorts of things one goes around measuring with a meter stick. That sort of larger interpretation. Now by '27, you're doing that. You must have been talking about it almost from the beginning.

Heisenberg:

Yes, there's no doubt about that. We always spoke about what it really means, but in the first stage of the development our attitude was that we had learned that this old way of talking about orbits and so on was not the correct way. The whole idea of classical orbits must be meant most "symbolically". But what the word "symbolically" means — that we don't know yet. So the only thing we know is that we can predict some things, for instance amplitudes. And there we are pretty certain that we do predict it the correct way. We believe that this is right. What it means in terms of classical language — that we don't know, and that, of course, is a problem which we must find out later. We did not say that we have an interpretation. We believed that an interpretation must come out of this mathematical scheme. In so far, our interpretation goes no further than saying that these are the stationary states, these are the amplitudes, the Einstein probabilities and so on. So we used the word "interpretation" in a slightly different sense from what you did. We did not mean by "interpretation" to produce a picture of what nature really is. We doubted whether nature is really anything. We felt that what we can say is just, "Doing that experiment, this is what will come out." So the word "interpretation"changes its meaning. That's perhaps an important thing.

Kuhn:

That is the point I felt you were perhaps coming to, but I wanted to make it explicit.

Heisenberg:

Yes, and the point is that my disappointment about Schrodinger's first paper came from this very point that he thought he could give an interpretation in the old sense. I thought that such an interpretation in the old sense does not exist. That was my very deep conviction. So I could not put it into quite rational terms. That could only be done a year later, but I felt that if a man tries to make such a direct interpretation — that he says that an electron is a wave, and so on — then he must certainly be wrong. That was the only thing I was convinced of. I would put it this way: We had seen that we could, possibly, understand nature, but at the same time we had to learn that the word "understand" means something different from what we had believed it would mean in the earlier times. So it is a whole change of the attitude about what we can do and cannot do. "Understanding" means suddenly just predicting experiments. "Understanding" does not mean something like in the old classical physics. That change took place already in the very first stage of quantum mechanics, I would say in '24 already. We didn't want to go back to the old line, and that was a disappointment with Schrodinger. Schrodinger apparently tried to push us back into a language in which we had to describe nature by "anschauliche Methoden". That I couldn't believe. Therefore, I was so upset about the Schrodinger development in spite of its enormous successes. I felt, "Now Schrodinger puts us back into a state of mind which we have already overcome, and which has certainly to be forgotten."

Kuhn:

When you speak of this now you sound almost as though the idea that the electron was a wave was, after all, an old idea that had been tried out and rejected.

Heisenberg:

Well, that was perhaps not fair to say. At least, it was not older than de Broglie's and Einstein's paper.

Kuhn:

Now who had known about de Broglie's paper?

Heisenberg:

Some people knew about it, at least in Gottingen they had spoken about I and I told you about this conversation with Elsasser and Franck.

Kuhn:

Well, you can see that this is one of the reasons I asked. When you told me about that conversation you said, "This was the first time that I realized that there might be some reason to believe in a wave." The Elsasser paper was in mid 1925, but really before matrix mechanics. This implied that somebody had been trying to persuade you to believe in the electron as a wave already; de Broglie would be the only person and it's not likely that you knew of de Broglie earlier than that.

Heisenberg:

Well, I have probably not known earlier than this discussion with Elsasser. This discussion definitely made an impression on me, but only in the sense to say, "Well, yes, this funny dualism which we know from the light quanta and which I like mostly to interpret along the lines of the Duane paper, this funny dualism seems to be a very fundamental thing and so it might also be useful to talk in those terms about the electron." But I would always have the electron in my mind as a small ball, as a sphere. Only I would say, "It may also be useful sometimes to call it a wave, but only as a kind of talking, not as a reality." And if Schrodinger would have expressed his paper in that way, by saying that it may have some meaning to consider in some respect the electron as a wave, I probably wouldn't have objected to it. I objected only when he really tried to say, "I want to get rid of the quantum jump." That was the very moment that I said, "No, it can't be."

Kuhn:

Is it possible that in this discussion with Born and Franck about the Elsasser paper what may have impressed you was less the feeling "Maybe we have to think of the electron as a wave," than the fact that this actually reflected back on the photon problem? I was impressed that in your paper with Kramers, there is very early the remark, "Of course; the photon hypothesis is useful because it gives us a microscopic account of the macroscopic conservation of energy, but that's no reason at all to believe there really is such a thing .as a photon." Now in 1924, this is a pretty cavalier attitude about the photon, coming from very high circles.

Heisenberg:

Yes, well, the trouble is that in talking about such things, it's very difficult to make any statement at all because after all, everybody knew that one had contradictions in the theory. The different senses which one used did contradict each other and if you have contradictions, you can say "almost anything", according to a famous statement of the mathematicians. That's just it: We knew that there were waves, that there were photons as particles and you had to combine the thing somehow. This "somehow" was just a vague term, "unklarer Quatsch," as Pauli would say, but we couldn't help it. When this "unklare Quatsch" was also applied to the electrons we said, "Well, all right, we are in a mess anyway, why not also take the electrons into this."

Kuhn:

Well, we probably have to let it go at that. I was very much impressed with your recollection of this conversation about the Elsasser paper being important, but I was very puzzled. The way you told it, it would have made good sense if this conversation had taken place after the Schrodinger equation and you'd been saying, "Now, I saw that maybe there was a sense in which we might have to say "waves". But this conversation is pretty clearly before the Schrodinger equation and therefore I was trying to reconstruct what the real importance was, this did open up for you against the background of prior resistance.

Heisenberg:

Yes, well, I think the main impression was this: The whole thing was in a mess already. Even the electron being a point — already that is doubtful. One certainly may have had to say, "Well, there are no such electronic orbits in the things." Everything was vague talk, we didn't mind. It probably was earlier than this discussion with Drude about the gamma ray microscope. Even then the ganra ray microscope was the other side of the problem.

Kuhn:

If it's the Elsasser paper, and you are pretty sure it was, and it would be likely that that would be Born and Franck because both of them were somehow involved with the paper, then it's pretty well got to be before you leave for Heligoland?

Heisenberg:

Yes, that's just as it was. I recall that it was in the summertime and it must have been just before I left for. Heligoland, yes.

Kuhn:

Good. Well, think that is where we should stop.