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In footnotes or endnotes please cite AIP interviews like this:
Interview of Peter Debye by Thomas S. Kuhn and George Uhlenbeck on 1962 May 3,
Niels Bohr Library & Archives, American Institute of Physics,
College Park, MD USA,
For multiple citations, "AIP" is the preferred abbreviation for the location.
Part of the Archives for the History of Quantum Physics collection, which includes tapes and transcripts of oral history interviews conducted with circa 100 atomic and quantum physicists. Subjects discuss their family backgrounds, how they became interested in physics, their educations, people who influenced them, their careers including social influences on the conditions of research, and the state of atomic, nuclear, and quantum physics during the period in which they worked. Discussions of scientific matters relate to work that was done between approximately 1900 and 1930, with an emphasis on the discovery and interpretations of quantum mechanics in the 1920s. Also prominently mentioned are: Niels Henrik David Bohr, Max Born, Léon Brillouin, Louis de Broglie, Hendrik Brugt Gerhard Casimir, Paul Drude, Paul Ehrenfest, Albert Einstein, Walter Friedrich, Dirk ter Haar, Fritz Hasenöhrl, Werner Heisenberg, David Hilbert, Heinz Hopf, Herman Julius, Felix Klein, A. Kleiner, Paul Langevin, Max Theodor Felix von Laue, Fritz London, Gustav Mie, Walther Nernst, A. A. Noyes, Max Planck, W.C. Röntgen, Paul Scherrer, Erwin Schrödinger, Arnold Sommerfeld, Pierre Weiss, Max Wien, Wilhelm Wien, Zangger, Zermelo; Eidgenössische Technische Hochschule Zurich, Massachusetts Institute of Technology, Universität Leipzig, and Universität München.
Why didn’t you study in Aachen?
Well, that was just a question of money. You see Maastricht is about 20, miles, 31 kilometers, from Aachen. And that time it was very clear that I wanted to go to Delft because I wanted to become an engineer, an electrical engineer. The only choice would be Delft or Aachen or Ypres in Belgium, because that was also very near. Then the question was, what does it cost, and when it turned out that Aachen would be a lot cheaper than Delft at that time. So I went to Aachen. I lived there with another boy from Maastricht in one room. He was also going to study their. He gave it up later and became a dentist, I went home every Saturday, was in Maastricht Sundays, and then on Monday morning at 1 o’clock the train went to Aachen. You know there was one hour time difference, and so I had to go off rather early in order to get the eight o’clock lectures. So then I was just an ordinary student with the other boys of Maastricht. In order to give you a feeling for what it was, you could live for less than 6o marks a month, about $l5 now.
And was Sommerfeld already there? He was not in engineering?
He was already there. You see, he had to give a course in technical mechanics. I didn’t know his background at that time, but later I knew. He had been an assistant of Felix Klein. And at that time Felix Klein had this feeling that there should be more connection between universities and technical institutions, Technische Hochschule. And so he complained always that engineers wanted their mechanics taught by engineers, and he did not like that. And he said, well he wanted to bring in more university into the technical part. At the same time he wanted to bring technical things into the university. He was responsible for making what became the aeronautical institute, where he brought Prandtl. That was originally not such an aeronautical institute, it was just mechanics. And he had another institute which was more electrotechnic, in connection with physics. And. Klein had this book about the spinning top. Sommerfeld worked with him on that. He brought Sommerfeld originally to Clausthal, and he was very, very efficient in bringing him to Aachen, just pushing the engineers out and bringing a university man in.
What did that do? Was that already reflected in the curriculum?
Weil not in the curriculum, but in the manner in which things were taught. Now at that time experimental physics was given by a man by the name of Wöllner — his brother was a famous singer. Wöllner was a man of the old business. And he was for experimental physics. There was nobody for theoretical physics there, but there was what they call in German extraordinary professor; the ordinary professor was Wöllner, and the extraordinary was Max Wien at my time. The one of the Wien effect and electrolytes and so on. He was a nephew or something of Willy Wien. He was later in Jena. He was quite a good man but also an experimentalist. Now in about l904 or '05, I followed a course in Maxwell theory from (Wöllner). It was an extra course which I did not have to take, but I was interested... This was all the business of mechanical models of the aether, no real Maxwell theory at all. That was from the professor of experimental physics. He wanted to do something extra. Now the one man from which I got something was Max Wien, because I had to follow experimental courses in physics, and we really got friendly to each other. I was in the laboratory a lot of times. He was interested in self induction, in (currents), in hearing, and such things, He had a lot of instruments there vibrational things where you could change the frequency and were responsive to alternating (currents) and such things.
There’s something I don’t quite understand Wöllner, the professor of experimental physics, gives his special topic on the theoretical problems of the models of aether; whereas the extraordinary, who presumably is to balance off, is really the experimentalist.
Well the extraordinary was at that time also an experimental physicist. You see there was no theoretical physics. Even Sommerfeld was there for mechanics, and so his course was the course in mechanics. What you learned in this course was general mechanics, mostly applied to solids, elasticity theory and such things. This was when Sommerfeld wrote the papers on the friction. I was involved in that too, I can also tell you about my mathematics. The one good thing about mathematics which I got was from the mathematics professor who gave an excellent course on Fourier series. He was talking about the old business heat conduction and so, on, and all these transformations. This was, I think, the second year. There were extra courses, you see, which you had not to take. There was a general scheme which was written down for somebody who wanted to have electrical engineering. And there you see you have to make one examination after two years. And after you have done that you have still two years or so to get your final examination. I made my final examination and for that I constructed a 200 kilowatt turbo generator. So I went through the whole business.
What sort of courses went into the curriculum?
Now there were experimental physics, for instance. No chemistry course. That was a thing which I did for myself, I followed a course in chemistry, but it wasn’t necessary. Then some courses in mathematics, then in what you call analytical geometry, and descriptive geometry, making a lot of pictures, and oh a lot of these things. We had to make a lot of drawings. Then construction of machines and such things, where we also had to make a lot of drawings. And then the mathematics courses. I never in my secondary school in Maastricht had anything about differential or integral calculus so .I learned them first in Aachen. Differential and integral calculus was the main thing. And there was one course for instance about vector analysis and such things. But that was approximately all
But for the physics requirement would it have been just ‘the course in experimental physics?
Yes. And laboratory of course. ‘That was allAnd for engineering you had to know a lot more of these practical courses - construction of machines and electro-technique and. such things. Now, when I became connected. With Sommerfeld — — and very soon he wanted me to become his assistant, so I became his assistant before I was an engineer. And then what I had to do was make up the problems.... Well, he had in his course also one, it may have been two hours where the boys could handle their problems. They came there and delivered their problems and they were discussed. It was not like it is here, you see, that you take so much care about this. Then he had some ideas. Well first of all, one of the things which I had to do was this business of vibrations and the connection of vibrations, like the sympathetic pendulum. And he had this one case where you have a spring, and you can make this rotation at the same frequency as the vertical vibrations so we had to make springs with the mechanician, and then we made Lissajous pictures of that. It is published somewhere.
Then he was interested in friction, and this is well-known. Well, I had to make this thing, and we never aid much with it then because we left. But then I was going to measure the temperature at different places in this eccentric with turbo-elements which I put in, but I never did that. Then he went on to the Euler problem. You know you have a thing like this and you put a weight on, and. then from this weight it falls over like that, it is unstable. And Sommerfeld connected it then with the frequency. You had this thing there, and then you let it go like this. Then for a certain weight the frequency was zero you see. Buckling yes, that’s what it’s called. I say that because there is a big thing in the neighborhood of Aachen for making bridges and so on. They had a case where if you put a thing on here [at blackboard again], then it buckles out here in the middle like this. And the question was, how could you calculate that? So he went out to this place, looked at these things, and made some experiments there.
He never did anything final about it. But you see Sommerfeld was really a mathematician. He was also interested in electro-dynamics, because at that time he wrote a few articles. I have looked at them some time ago. He wrote a few articles even at the time when he was still in Göttingen, about electrons moving with velocities larger than that of light. And he calculated this much business of it. Did you see that? It was a year or two before Einstein, and it was approximately the same time that Lorentz published his transformation. Sommerfeld wanted this thing to be published in the Amsterdam Academy. So I got the manuscript and. had to translate it into Dutch, and it is my Dutch translation that has been published in the Amsterdam Academy and was presented by Lorentz. But that was a year or two before Einstein. And then when Einstein came everybody was saying that is no good and everybody said it is nonsense, and laughed about it — until Cherenkov....
How is it at that time in Aachen? Did Sommerfeld and. you study the relativity theory?
Study? Well you see Sommerfeld had a special way of doing things. At that time he had no connection. There was nobody who was thinking the same way, so he tried to make a connection with some students. The two students with which he made a connection were I and. another man of the name of Rogowski, who was later at Siemens. What he did was, he invited us to come to his house. We came to his house in the evening at 8 o’clock, had the evening meal, supper. And then you sit in his room. And in his room he began to talk. He talked about the things he was interested in, and you sat there as a kind of an audience. He asked you about it, although you did not know anything about it. He tried it out, so to say. And in this way I learned a lot. But you sat there until 11 or 12 o’clock in the evening and you talked and he talked. These sessions were perhaps twice a week or so. It was not regular. You might meet him and he would say, “Oh come up, I have to talk about something.” That was the way he did it.
Do you remember the early discussions of relativity?
Relativity came much later. What I am talking about is before relativity, you see. And then later on there was discussion about relativity, but what he liked most of it was the way Minkowski put it. Sommerfeld is an original, a real mathematician.
But the spcia1 theory comes out still in the Aachen period.
Yes of course, of course. But you see Sommerfeld had the feeling that you have mathematical equations, like Maxwell’s equations. Then he said, “Well, Mxwel1’s equations are no good if you are going to talk about” — I can still hear him say that — “about general solutions.” He always said, .1 have learned from Fourier that you have to make a solution and develop it so far that you can really put numbers in it’. And apply it, you see. That was his main purpose, not fundamental equations themselves, but after you have equations to make a mathematical solution which is really practical. That was really the way he was, you see. But now the relativity theory of course not yet so far. So he was very much interested in it, and accepted it right away, you see, but that was all, so to say.... When the paper came out we had discussions about it, but that was just to say, “well this is all right.” There was nothing to it.
Nothing about the paradoxes and so forth?
No, no. Well the few paradoxes we went over that very easily. This was not like the paradoxes later with Einstein and quantum theory.
No, no but I mean some of the simultaneity paradoxes.
Yes, yes, well we accepted that rather easily. It was not hard to accept it. After all we had Lorentz’s transformation, and then it was just a. question of philosophical interpretation of the Lorentz transformation. That was all
Was Sommerfeld pretty well convinced already that there was no need for a mechanical aether or anything of this sort?
Oh yes, oh yes, right from the beginning. We didn’t bother about that. I had enough of that in this course of Wöllner, you know. No no, that was done, approximately, as if the equations came from heaven. There were two principles induction principle; and you had only to add the displacement current; then you are through, you see.
When you speak then in your own work occasionally then of aether, this has no physical significance? .
No physical significance. No you see, if you look at the literature. I just remembered that because I looked it up several months ago. There is a talk which is given by Lord Kelvin in one of these Friday evening lectures at the Royal Institution. This is just in 1900. And there he talks about the clouds on the heaven of physics, you see. And one cloud is the equipartition and the other cloud is the aether. Anyway, the aether was nothing mechanical for us.
What about the range of problems that are just then emerging as quantum mechanics?
Oh,no quantum mechanics was not there at allI can tell you the impression we had of these early things of Planck. We said maybe there is something to it but it is absolutely illogical. It is also illogical if you look at it. You have a vibrator which you put in the radiation, and you calculate what the energy this thing is going to get is. There is no question about quantum or anything there. Later on you get the same thing and put it in a hot place and talk about the statistical energy it is going to have .. And you find you should do that the same way, and you find kT, of course. And then later on you say, well I am going to give it energies, epsilon. And thinking of the fact that Wien’s law exists, you have to put this epsilon equal to hY. So to everybody that was absolutely illogical. So they said, well this is an interpolation formula and perhaps there is something to it, but not from the derivation. I must say I like Einstein much better in the way which he said essentially: veil now that is a thing which we have — this quantum. And let us apply it in different places and see whether the rule applies. But all that came later. That was not at the time in Aachen.
Were there even discussions of Planck’s work in Aachen?
Yes, yes. But you see it is superficial like this. It might be something, and there is a formula which seems to represent the radiation business, but whether the quantum is now something fundamental or not, we don t know.
I mean if you speak about these discussions,that must have been such an evening discussion between you three, because that’s the only three people who were interested.
Yes, yes, yes. There was nobody else. But Sommerfeld in this period was not yet trying to take this problem up himself. He even did not know those problems very well. He got into these problems when he got to Munich. Then he said, “Well now I have to study in order to give my courses.” And we studied together in order that he could give his courses..
That about statistical mechanics?
Oh, that came much later. That is a thing which he did not like very much. He liked more the other things. Well, you can see that.
Well, I have wondered about this, because you in particular use it very early.
After all, we were free. There was not a question that we were going to do just what he asked us.
But did you learn it after the move to Munich?
Yes, yes, yes. And I never learned anything of that kind from Sommerfeld. I can tell you how that was really. From Sommerfeld I learned mathematics. You see I had never a course for instance in complex variables. But all I know about it I got both because it was applied, and in talking to him. I got that. But physics, really fundamental questions in physics, you could never get from him. He was much more: The laws are there, I am going to translate it into a mathematical formula. Now, you know how we got to Munich? In Munich they had this thing, and they asked Lorentz first to come, and Lorentz decided no, I don’t want to. Then they asked the second one on the list, Wiechert. You know Wiechert in Göttingen, who had published some papers about electrodynamics? And he did not want to go, because he got his earthquake place there. And then the third man on the list was Sommerfeld. Now of course he had been waiting for that. He went right away. Then he asked me, did I want to go with him? I hesitated a little bit, because as an assistant in Aachen I got 150 marks a month, and going to Munich I was going to get l02/2 Well these were practical and important questions at that time.
But in Munich Sommerfeld was a theoretical physicist really?
Sommerfeld was a theoretical physicist. And he also inherited a small laboratory with some instruments, and a mechanic. That was in the academy, the old academy in the (???) Strasse. Later on he had also his laboratory in building of the University. But at that time when he came there he had only to give his course at the University and he had his place on the second floor of the Academy on (???) Strasse. There was a mechanic,(Seine) was his name. And a shop. And a number of physical instruments. Then you see Sommerfeld told me, “Now we are going to have one day a week, one afternoon a week, and in this afternoon we are going to have a physical practical, and you are going to try to teach it His fingers were no good that way. But that was the place where Hof and those people worked for the doctors’ degree. Oh Sommerfeld t that time — and let us not forget it — had been interested in turbulence, and so Hopf had to do something about turbulence. He had to place a thing which came down like this in order to make a two dimensional thing,he had to let water flow over it. I still remember how I brought him a whole sack-full of sugar so he could make a sugar solution.
Was it unusual for the man brought in as a theoretical physicist to have his laboratory?
Now, you see, that was there. Boltzmann had been there, and they had had his laboratory. Sommerfeld inherited that. And he did not know quite what to do with it, you see. So he was also thinking of some problems which could be handled there. He put Ewald on a problem there and Ewald still talks about it. Well, he put him on a problem - you know these currents which you can induce in vacuum?... He says the best thing that happened to him was when he smashed the -
Why did it take quite so long after Boltzmann left Munich to the new appoint- ment? Do you know?
I think it is because they had these three men. First of all they are slow in Munich. Secondly there were three men on the list. They first tried to get Lorentz, and that takes a certain time; and then they tried to get Wiechert — Well, I don’ t know. I don’ t know anything about that because then I was not there.... Well, you see they had, all along, a course in theoretical physics. What was the name of this fellow? This is the one man who thought he found out that hydrogen peroxide gives off rays like X-rays. Weil you see what he had done, he had taken his peroxide and a photographic plate and put a sheet of aluminum, and still it got black, you see. And he had forgotten that aluminum has holes in it, so when he put two sheets about each other he had no effect anymore.
You speak of Sommerfeld saying, “Now I have to learn physics.”
Well you see there are four courses theoretical mechanics, thermodynamics, electromagnetism, and optics. These were the four fundamental courses. And he had never given them, so he had to do something about it. And at the same time be began also to talk about radiation. And then Planck comes in and so on. But he had to study that, and. he studied that the way he had in Aachen, you see. He talked to me. Then I had to sit there and make some remarks. At the same time I had to understand in order to know something. So he did that in order to be able to give the courses he had to give.
Was it at that point then that you did your own statistical mechanics
Oh yes, yes, yes. You see I had not much to do. I had this room there. And what I had to do was nothing else than give this one or two hour seminar for the problems to be handled by the students. I did not even handle the problems. Sommerfeld wanted to do that. I had only to make the problems. That was all I had to do for the whole week. There were not so many students. At that time there was Ewald and there was Hopf and who else? A few were there. And since he wanted to use this laboratory which he had, there, he put some of them on some more physical problems, not mathematical problems. Till: In these courses now, these four courses, do you remember approximately how many students listened?
Oh you might have had something like 30, 40. They were all experimentalists or preparing for the high school teaching exams. You see I made my Dr.’s degree at Munich And in order to get the Dr.’s degree we had to write a thesis. Well, Sommerfeld said, “Why don’t you calculate all the light pressure on spheres.” So you see I had to develop Bessel functions and so on. Then he said, “A fine problem would be” this is characteristic for Sommerfeld “A fine problem would be if now you could show that this formula, which are these products of Bessel functions and spherical harmonics, if the wavelength gets small, go over into the optic formulae we know for the ray optics.” My work on the Bessel functions came from this special problem. If you look at these theories, you see you have more and more terms, and then you begin to end up at the place where the index of the Bessel functions becomes comparable to the argument. And so there were no formulae for that. So I needed them in order to sum up those things. And in order to do that I used the representation of the Bessel functions which really came from Sommerfeld — this integral in the complex plane. Then I looked for the saddle point. So this way it was connected. With this physical problem.
What would Sommerfeld have said if one had said to him, “What is the point of this sort of problem?”
Oh he would have said, “That’s the old Fourier point.” You have your mathematics but the mathematics is no use as long as you have not translated it in such a way that you can apply it. You see he was not really so much interested except later he got interested in the fundamentals — in the physical problem itself. He was much more interested in the mathematical problems. How after you have the differential equations you are going to do something with it....
Well now it would be very interesting if you could recall a little bit how the discussions of these modern problems in quantum theory and relativity came about.
Well they came about in connection with his courses all with his courses.
Was he already at Aachen good as a teacher?
Oh yes, oh yes; he was very good as a teacher.
And he was concerned to be a good teacher?
Was he concerned? It was more for his own sake you see. As a teacher he liked to put things in such a way that everybody was interested. But then for his own sake he wanted to talk privately to people whom he could talk to.
And who responded And. these meetings in the evening, they continued in Munich?
No, they did not continue there. ‘There it was more — there he was — well, distances were larger and so on, There he had enough time for doing these things. So in the day he just came in and said, “Well, let’s sit down and talk about, these things.” Then he'd talk the whole afternoon or so. There was not just evenings.
Did the concern with quantum mechanics then you think develop naturally out of his learning physics for the sake of the courses?
Yes, yes, yes. When I left, that was really the time when he got more and more interested in it. I got my Dr.’s degree first, and then I had to wait two years before I could make this Habilitation in 1908 or 1909. My Habilitation thesis was no good — something about electron conductivity in metals. Just a chewing up of the old business, nothing new in it. Then I could give a course. Now the first course I gave was radiation theory. Then made this whole business of Planck and so on. What was the name of this Russian who was later the professor for theoretical physics in Pasadena? Epstein. Nell Epstein was sitting in this course.... Epstein came then from Russia. Then he was going to live in Munich. During the time he lived in Munich he was, so to say, a private person who was very much interested the Same way in mathematical things as Sommerfeld. That is why he was there and tried to do things. He did things about ellipsoids in an electromagnetic field and diffraction theory’. He was the same type as Sommerfeld.
Really then you gave the course in radiation theory in 1910.
Something like that yes. You see I only gave one course and then I got away. Then I went to Zurich. And you can see from that what time it was.
It is striking that until 1911 nothing on quantum mechanics comes from Sommerfeld.
Yes, I know, I know. I learned about that because I was giving the course’. Later on he introduced this course with general courses.... Being around at that time, we had to learn things.
What about the Einstein papers?
Oh, we knew that they were around.
Oh, but I mean also photoelectric effect and so on.
Ah. That was coming up then you see. That is what I appreciate so mach in Einstein. That he says, ‘You see there is a thing. We don’t know whether it is right or not, but now we are going to find all the places where we can perhaps apply it, and then see whether it works.” That was his attitude.
Also the specific heat paper of Einstein?
Yes, yes, yes. You see the real thing in the specific heat paper was that I did not believe well, Nernst did not want Planck’s formula. He said this is really nothing else than an interpolation formula.... And at the same tine Nernst had this third law.... In connection with that, he was going to do low temperature measurements with F. A. Lindemann,... They got these deviations from Einstein. In order to represent it they introduced half quanta. So they had a formula with O per cent Ii and O per cent h /2. I did not stomach that. When he was talking about that, it was already the time when I went to Zurich....
In this course of yours on radiation theory, you talked to Sommerfeld about these problems? I take it that at this stage of the game you were perhaps even further into them than he was.
Oh yes, that is true. He had not paid very much attention to it, and I was interested in them. I was going to give this course and so I went into it • And then I talked to him.
How did he feel about the quantum in this period? About the energy discontinuities for instance?
Well you can see that from this one thing which I published. I looked at it yesterday in order to be prepared a little bit. You know this thing, quantization of the space? r whole business was, Planck is illogical; on the other hand it looks as if the whole thing is very good. Can we get rid of this illogical part? That was the real reason. And I read it yesterday and it is very clearly expressed there.
You knew Jeans already?
Ye, yes, yes. This enumeration of the eigen frequencies is really in Jeans.... I had used that in the course, too.
Did this 1910 paper come out of the course, so to say?
In connection with the course. Because my intention was to make it, for myself, less illogical than it was. UI-I: And that is why you refused to use this connection between radiation intensity and average energy of the oscillators?
Yes, yes, yes, yes, yes, yes. Well I said, Well then I don’t have to go into this whole business which is without quanta.
The thing which still struck me is this: This famous formula for average energy — Einstein simply derived it by writing down the sum • flow Now you didn’t’ do it that way. You still use, partially, the Planck formulation.
No, because you see you have to introduce the h v.... I say that you have the Hohlraum, you have a cavity. This cavity really represents an infinite number of vibrators. If you put energy in there you don’t know anything about the energy, because every vibration can have as little or as much as you want• So you have to add something which characterizes the temperature equilibrium. Now introducing this temperature equilibrium, I can come out with my quanta. This is the reasoning, you see. Without bringing in Maxwell equations again, you see, or waves. If you think of Planck, then he brings in the waves in order to get his resonator,.,. Now look here. How does Einstein come to that?
It is the Boltzmann distribution.
Yes, yes, yes. But then he has to ask, “How do I get to the Boltzrmann distribution?”
I see. That is what you wanted also to bring out.
You’ve got already in the Jeans paper the du formula. I’m curious as to why you don’t just take this formula.
You see, I don’t want to take a formula from somebody else. You always want to do it your own way. That is the only thing. That is the thing. I also really learned, from Sommerfeld, never take anything from anybody else. Really start with the things from the beginning. And he always insisted that you should. also put these things in the publication. He always wanted publications to be much longer. He wanted everything in the manuscript.
We automatically now look back at the Jeans paper and say these are aether’ oscillators. But they’re not necessarily. They’re standing wave solutions of Maxwell equations. And what you do is to show that they can be treated as oscillators. Was that new?
Oh, I don’t t know whether it was new. I don’t think it was new. I think you will find that in everybody else. No, there’s nothing, and my feeling now is I would say there’s nothing else but that I say then, "All right, I understand it now from the beginning, from Maxwell’s equations, you see. I could also read it in the literature perhaps.
And then of course your specific heat was an immediate consequence of this.
Yes, of course. No, you see with the specific heat I started with the lattice, and I wrote out equations and I saw and I talked. Well at that time in Zurich there was a mathematician of the name of (Haar). You may have heard of him. He was a Hungarian. He was a very good mathematician. He came also from Göttingen originally. And he was taking care of the courses for Zermelo. Zermelo was the other mathematician at that time. He had to go to the heights , because he had something with his lungs. There had to be somebody to replace him, and this was this Haar. And we had always lunch together, he and I and a relative of Einstein. At that time I tried to get out of Haar a good method for finding out the vibrations of a lattice. And then in desperation, because that was too difficult, I made this approximation. [Uhenbeck leaves for rest of morning]
In the 1910 paper, when you do the Planck derivation for the energy distribution of the oscillators, you don’t say that you do it differently from Planck, but it is different.... In Planck’s derivation I see no way to interpret that for1a for W as a probability, and therefore to justify taking about s as k times the logarithm of it.... Your treatment is immune to this criticism.
Yes, I know, I know. Bu you read mine — I read it yesterday. Read mine, and the first line is so interesting. I say there because Planck is so illogical, I want to put something in which is immune to that.... I don’t say that especially. But you see this whole thing came really up as a result of this course, and in the course you have to explain things to people. So how much of that was in this direction and how much in the other direction I don’t know.... That was one of the main puzzles. That was a main puzzle. This may have been part in answer to that, but I don’t know details.... You see, what I really wanted, at least what I feel at the moment that I wanted, is the only thing I can do with it, is that I really wanted something that I could talk about the probability of distributing some things over some other things.... I was very much content about it. As a matter of fact Langevin was very content about it. The only reaction I ever got was when I met people later. You see we were very alone at that time. We had no contacts with anybody. And later on when I me then Langevin, he talked about it. He talked about it. And. that was the way he could understand it.
Were there other reactions of this sort from other people?
No reaction at all, no reaction at all at that time. You see even now there was not a reaction, because people which put together things did not put it in.
I think you have to have looked at a number of the derivations of the Planck formula to realize the sense in which this is different, which is this making it a real probabilistic derivation. You don’t mention at all that there is
Well, but I did it’.... As a young man you don’t want to invite criticism. I thought it was already a big criticism, that dare to write down against Planck, you see, and say that he is illogical.
Do you remember talking about this paper with Sommerfeld?
Oh yes. Well, he accepted it. He had not talked about it. And he was not a man to talk about probability and entropies and such. The thermodynamics was the last thing he was interested in. Thermodynamics was something which he did not like very much. You can see approximately what his attitude was by this paper which I published with him, I think even two years or three years later. He never liked it at all. And he was just pressed to publish that, because he wanted that to be published.
1910 is also the year in which two Hopf-Einstein papers appear. Now Hopf had not been doing this sort of thing at all with Sommerfeld.
No, but Hopf was interested in Einstein as such and so he went to Berne and visited Einstein. And Hopf was quite independent. His father was a very rich hop merchant.
Did Hopf also play a role in stimulating this interest?
Oh no, Hopf was just hanging on. He was the one, you know, who did these experiments about the Reynolds numbers.
What about von Laue?
Oh, von Laue’. This is a very nice story. I don’t know whether you know that. One day during the time I was assistant of Sommerfeld, be got a letter from Planck. And Planck said, “1 have a young man here in Berlin and he is very nervous. He cannot stand the whole way of living in Berlin. But he is very good, and he would be good if he could umhabilitieren. He was habilitiert at Berlin, you see. If he could unhabilitiert himself. Would you be willing to accept him in Munich?” And I saw this letter at the time because Sommerfeld after he got it and Sommerfe1d said yes, yes that’s all right. And that is the way Laue came to Munich you see. And then Sommerfe1d wanted to have some recompensation for that, so to say. Now Sommerfeld was the Editor of the Enzykloapedie der Mathematischen Wissenschaften.
He had also impressed me already to write an article, and he always tried to find people to write articles. So he got Laue and said, “Well you should write me an article about interference.” And that is really the beginning of the whole business, you see So Laue got interested. He had only been interested in radiation, and the entropy of radiation, and so on, you know. He only began to talk about interference because he had to write this article. And I still remember - I was not there when he made his discovery, I was not there anymore. But I still remember. You see we came always together. The way we lived in Munich was this, you see. We had a table in the Hofgarten. The Hofgarten is a place — there were three cafes there, like Viennese cafes in the open and then there was the residence.
We all went to lunch about 12 o’clock. I always went to lunch with Peter Paul Koch, who was later a professor in Hamburg and was one of the assistants of Roentgen’ s. And he was always very (hungry) at 12 o’clock, because he was a man who always started his day at 5 o’clock in the morning. So we then together went down to the (Rose) Strasse, had lunch, and then after lunch we came to one of these cafes. And there was a table and. a lot of people came together there — several people, a medical man and so on. At this table also Laue came, because that was the place where we met, We sat there from I to 2, something like that. Then like the Viennese cafes we got the newspapers and we talked about God knows what’. And also about practical things, about physical things. And there I remember Lane was saying, “Well now I have t calculate the interference of linear grating and then the gratings in plane, and then experiments. I should also put in gratings in space’.”
And this is before the whole idea?
Oh yes, yes. It was so to say a kind of a preparation for it, as it looked afterwards. And at this same table there were several people who were working on X-rays in the Roentgen laboratory. These were this man Knipping and the other one, Friedrich....
How clear were people in Munich that X-rays are electro-magnetic rays?
Not clear at allRoentgen did not like to accept it at all himself, and Sommererfeld. had. some discussions with him after the experiments there, you see. Sommerfeld thought they were definitely electromagnetic radiation. There was already this whole business of Haga and Wind.... So that was all accepted by the younger people. But not so much by Roentgen, you see.
The younger people as a group were pretty convinced?
As a group they were convinced that this was radiation, yes.
So that Bragg’s theory - —
Had no influence, no.
You were no longer there even when Laue first got the idea?
No, no. I don’t think that he would have talked about that if he had got the ideas, because he was very secretive really. I heard from others that he talked to Friedrich. Because Friedrich was doing these experiments, you see. And Knipping in the laboratory always called him the watchmaker, because he was the man who always wanted to do all the little details of the instruments.... So Laue got Friedrich to put a crystal in, you see, and to let X-rays go through. And then they got this diagram with the spots, these big spots. And the first thing Laue did was to put this in an envelope and to give it to Sommmerfeld so that he could give it to the Bavarian Academy. It should be deposited at the Bavarian Academy until a later time, you see. It was a sealed envelope so that he could show that it had been done at the moment when he was ready to publish it.
How did you feel when you heard about that one?
Oh I thought it was nice, very nice. A very nice idea. I had no idea about doing something like that, you see. But I am certain that my feeling was that it had been spoken in the air for a longer time, and that Laue had to wait until Friedrich was good enough to try it. It was not a thing then which was under such pressure that they did it right away.... It cannot be too much time, but not so that he told Friedrich about it and the next day Friedrich tried to do it.
Coming back now to the quantum problem itself and the question of the quantum of action as against the energy quantum. Sommerfeld was very insistent on that formulation, wasn’t he?
Yes, and he liked that very much, because after all the product of energy times time, the product, we’ d better say, of the coordinate times the corresponding momentum, has always the same dimensions, ... independent of your definition of the coordinate. And that was the thing which fascinated Sommerfeld.
It’s somewhat strange taking this as fundamentally a quantum action — which may come naturally out of the mathematics — but in physical terms it makes no particular sense at the time.
Well now you see, Planck had introduced energy quanta, but then you had to ask, how is energy quanta dependent on the frequency? Then he has the displacement formula of Wien. Then it is decided that the energy quantum must be proportional to otherwise thermodynamics would not be good. So from a physical point of view it has to be introduced that way. Then for Sommerfeld this mathematical thing was the most important. Then you find out that this thing which you call the quantum of action is really the thing which already appears in ordinary mechanics, and has always, independent of the choice of the coordinates, the dimension energy times time. This is a queer thing. That was the one thing which decided Sommerfeld that this is something now fundamental
But in the early days it does nothing very useful.
Oh no, but you see, for Sommerfeld such a mathematical connection was absolutely everything.
This is an idea that he begins to make a good deal of in 1911. Planck had already done something of this sort.
Well what Planck has done is first make this diagram with the ellipses. And then the question is, are you going to take these ellipses as such, or are you going to take the whole surface? You see that’s where this zero point energy comes in. I never understood why he did. that. I think it is also a formal thing originally, and. it is so very important in that,
You think with Sommerfeld it was a formal ...?
Yes, yes, yes.
It’s ultimately terribly important. One is impressed with the fact that at the beginning it mostly makes trouble.
Well it makes trouble. But on the other hand. it is trouble which you have to go through, Otherwise you cannot quantize more complicated systems. You have no principle.
But people are not yet really quantizing more complex systems, are they?
Well, as a matter of fact they are trying. They are trying. You see that was the feeling. If it is important, then it has to be formulated in such a way that you can apply it to any system. Now they did not succeed with this direction in which they were going.
And in which people are going already now before the Bohr atom?
Oh yes, oh yes. That was long before the Bohr atom. The characteristic thing of the Bohr atom is that Bohr was with Rutherford, and that he had a picture of the Atom, you see. As soon as he had the idea,’ This is picture and I am going to apply to this picture,” that was all right. But before that there was always the question, “How can I get in a spectrum an accumulation of lines which are finite frequencies? How can I stop those lines? I remember, at those tines there was a lot of t]k about — what was the name of this fellow? There was a man who had died then later in Switzerland, and he had calculated frequencies of plates with a queer distribution of masses on them, in order to represent the Paschen-Runge formula. Oh that was a big thing! That was the question, you see. That was the question, “How to stop the series at a finite frequency.” And he went at it in a very complicated way, you see, for queer potential energies.
Did Sommerfeld worry about that problem?
Oh, Sommerfeld worried — I have one anecdote’. This was still at the time when I was at Aachen. Sommerfeld and I went at Easter vacation. We went by bicycle to the Mosel Valley. And one evening we were at a place where a wine merchant wanted to sell some wine. And. he wrote into a book there, this guest book you see. He wrote, “As soon as I know how to explain the Balmer formula, I am going to try your wine.” This was something which was in the minds of the people.
Well now did Sommerfeld go right on being interested?
Yes, yes, yes.
How did he feel about the Thomson atom?
Well, the Thomson atom was discussed in Munich for a long time. I think that even Ewald made some model there, if I remember, with magnets. Now you know Hasenohrl, he must have been a very good fellow, because he really had the feeling that energy and mass are connected. You see once he calculated, if you have a cavity, and you have radiation in it, and you want to accelerate that, then you have to put an extra force on because you have radiation in the interior, because you have this pressure there. So in a sense he is the "Vor1aüfer" of Einstein in this respect.
You said before you didn’t like the photo-electric paper, the Debye-Sommerfeld paper.
Oh that’s no good, that’s no good. That is a kind of a frustration, you see. And I would never have published it if Sommerfeld had not insisted.
How did that start?
It started that way, like everything at that time, “Is it possible perhaps to explain the quantum starting with classical things?” That was the whole problem of Einstein, his whole life. There was a big effort to try that, and to see what you could do, and it is not worth publishing. That was the background.
Does this start with Sommerfeld himself?
Yes, yes. That started with Sommerfeld himself... We worked on that, and I was-not—I did not like it at allSommerfeld thought it was a possibility. I was convinced it was no possibility and I tried to convince him we should not publish it but he wanted to.
You liked the Einstein photon notion better than he did, didn’t you?
Well with this notion the rest is trouble. You cannot have the notion alone. I don’t know which paper of Einstein’s it is, but he talks once about the fluctuations. You find, that the fluctuations are not represented by quanta, are not represented by interference, but you have to run both things at the same time.
That’s a later paper though.The 1907 fluctuation paper, which goes right on into the photo-electric effect is just fluctuation of photons. ... . In your inaugural lecture at Zurich you talked about quanta of energy and the atomistic structure of energy.
I don’t know. I could not get that paper. I have not read it for forty years, so I don’t know what is in it.
Although you did not like the photo-electric paper, was it taken very seriously by many people?
Yes, yes, yes, but I think mistakenly. Because it was, you see,the old business. If you have to do something quite new, nobody really wants to do that you see. And if there is a kind of possibility of an escape, which is not taking new things, then everybody likes that.
But it made the problem of the time required for low- intensity radiation very sharp.
It really convinced him that it was no good to try that, although he wanted it published. It really convinced him.
He was really pretty well convinced already by the time it appeared?
Oh yes, yes.
He was definitely not happy at all about the photon notion You were much more prepared to take that seriously?
Yes, yes, I was much more prepared, but he was not happy about it. But you see, the older people were, the less happy they were. And Einstein, he, certainly didn’t want to accept it. He had no place for the quantum in his whole relativity. And that was the main thing for him. And so he always tried to find new tricks which were contradictions.
Still in this same general area, and during the Munich period, did the Solvay Congress have anything very much to do with stimulating interest in the problem of the quantum?
I don’t know, because I was not at the first Solvay Congress.... I don’t think that the Solvay Congress, as such, had much to do with it. It was the general atmosphere. Because there was so much frustration, you see. And a lot of people even if they didn’t’publish it tried to do something, and they never, never succeeded. I remember at that time we always came together in the southern part of Bavaria, skiing. Willie Wien, and Sommerfeld, and Gusta Mie, and some of these people. We came together there and we sat together and there was a lot of discussion about all of these things. What I feel now from these is this frustration.
Can you remember what seemed most frustrating? Was it all these problems that are not responding. Is it the fundamental problem of the discreteness of energy? Is it the difficulty of solving particular problems like this?
Not that, no, not that. It is the question, you have a certain scheme of things, mechanics, electro-dynamics, and so on. There is no place for the quantum in it • Then you are going to say, well, maybe you have overlooked something. That was the general attitude you see. Maybe if I take a special structure or something like that, then I might see how I can get to something like the quanta.
When you say a special structure, you mean structure of phase space?
No, a special structure of special mechanical systems, with special properties. Like this man with the plates, these are special. Although you know it is not a plate. There was this trying to find out whether there is a connection with the whole thing. You see the thing goes so deep. You have to abandon everything. And it was clear even at that time that you have to abandon Maxwell’s equations.
That time is now when?
Oh that was the time when we talked to each other in Munich. You cannot even use them anymore. They have also to be quantized’. But that was far away. I remember later, and that is not much later, I was in Leipzig talking to Heisenberg, asking whether he could calculate for me how the scattering of light by light would be. Which means that the Maxwell equations are not linear anymore. We found out that it was so little that we could not do an experiment. But that is characteristic for this type of thing. Most people also had the feeling, when they talked about it, that it is not so good to have a cavity and put the radiation in it. This radiation is never going to get into a thermal equilibrium. As long as they believe in Maxwell it will not’. You have always to have this black little thing of dirt. In order to make this radiation black radiation, you have to put in a little particle. And that would mean that the quantum is not general.... This was the fundamental thing.
This is the notion that the quantum may be associated with the insertion of the little black particle, the dirt, so that you may try to isolate the quantum in the little extra piece of material.
In the material, yes.
When do you suppose did Sommerfeld come to the conviction that the quantum is fundamental? He surely had not got it when he was at Aachen?
No, not at Aachen, no question about it. But I think that must be approximately the same time when he also was insisting on this integral pdq.
By 1910 or '11, something of this sort. Here is Sommerfeld now coming to grips with a fundamental problem, which is just the sort of thing you say earlier he had avoided.
Yes, yes, yes.
How did he feel about this?
Well that I dont know, that I don’t know so well, because at that time we were not in daily contact any more....
How did you work together on the photo-electric effect paper?
By correspondence. We started to talk about it, and then we tried to do something in different places. I was not really good.
We really now have got you to Zurich. I’d be glad to have you go on and talk about Zurich, or we could stop for a while.
Well, what did you want to know about Zurich? I had no real contact with Einstein, except the evening when he went with his family to Prague. Sommerfeld and I met him at a place, one of the places in the neighborhood of the (???) to have sonic beer.
Because you were there to replace Einstein.
Well, that was later. At that time I did not know, you see. He was just on his way to Prague, with his wife and some children. Well, I met him after that several times, but that was the first time I met him. And so my relation with Einstein had nothing to do with Zurich as such. In Zurich, you see, at the University, you know there are now two things in Zurich, the University and the Eidgenossische Technische Hochschule. The University is supported by the cantons. Now there was a physicist by the name of Kleiner. He was an experimental physicist and there was no theoretical physicist. And. then Einstein made his doctor’s degree with this Kleiner. Then because he had his family already and he had the money, he went to Berne. And was at the patent office there which is where he published his things, you see. Kleiner was convinced that that was very good, although he was not a mathematician at all.
He was at the same time a member of the Cantonsrat. So he pushed through the Cantonsrat a professorship of theoretical physics, created with the idea to give it to Einstein. And that was done. But then it was of course not much money which they gave. And so as soon as Prague was offered to Einstein, he went there. I still remember when we were talking in Munich about that he said, “Well, they are queer in Austria.” It was still Austria-Hungary you see. “They want to have a religion.” You see, you could not become a professor without having a religion. So he said, “Now I have returned in der Schoss meiner Voreltern,” You see in Zurich he was a Swiss citizen, You know he was born in Ulm. And as a Swiss citizen he had given no religion. You could do that in Switzerland. But in going to Prague he had to have a religion. When I came to Zurich then, of course, they were interested in relativity. And so one of the first courses I had to give, except for the general courses, was the course in relativity. So I gave the course in relativity, the first thing. People were all right. The one man who was really all right, who was also very important in the life of Einstein, was the man of the name of Zangger.... He was the forensic medical man. But he was very much interested in all these general things. He was talking about relativity and so on too. He was interested more from the philosophical point of view. But the main thing at the laboratory there was Kleiner. Kleiner had always been doing experiments for dielectrics. And at the same time at the Technical Hochschule there was Pierre Weiss, who was a specialist in magnetism.
You’re one of the first people, I think, to try to use the quantum on the problem of magnetism....
Yes, it comes from the contact in Zurich. Just as my polar things come from this contact. You see what it was. In the University everything was going on about dielectrics, but more on a technical level. And then the contact with Pierre Weiss, and Langevin theory tells you, you have to do something of the same type in dielectrics.
Now people in Germany do not seem to be much worried by the magnetism problem. The French are worried by it.
Yes. Yes. That was because it was really a ‘French problem. It started with Curie. When Curie made his experiment about magnetism, and showing that the paramagnetism of oxygen was proportional to 1 over T. So that was the thing which had to be understood. And that was when Langevin made this calculation, this statistical calculation, if you want. And then what Pierre Weiss did to it. You see nobody understood, at that time, paramagnetism, until Pierre Weiss came with this field. This extra field, which is the same idea as the Lorentz-Lorentz business. Only that he had to have a coefficient which is much bigger. And that was the trouble then.
You were then the only theoretical physicist at the University? When you were there.
At the University, yes.
You didn’t stay so long.
Well then they came from Holland. I am a Dutchman. They had come before. They asked me to go to Tubingen. I just declined that. Then they came from Holland. I could not do very much there. I’m a Dutchman, I thought I had to. And I really regretted that, I did not like to go away from Zurich. And Kleiner also impressed me much that I should stay, and so on. But Holland of course. Now in Holland I had bad experience. In Holland you see, at Utrecht, the professorship was really Wind. This man of Haga and. Wind, He was the theoretical physicist. Now it was very clear at Utrecht. There a theoretical physicist is a theoretical physicist and can have nothing to do with the laboratory. Now I was spoiled by Zurich, because in the laboratory of Kleiner I went into the laboratory and did everything I wanted to.
So I had my contact with the experimental part. But that was impossible at Utrecht. It was even so that when I went to this Julius, who was the experimental physicist, and asked him, “Could I have a room so that I could do some experiments about dielectrics with the polar molecule.” He said, “No theoretical physicist in my laboratory.” So I was absolutely insulated. I had to be a theoretical physicist with paper and pencil, but nothing else. And that was at the same time that they offered me this business in Göttingen. In Göttingen they had two laboratories. One laboratory for the theoretical physicists and one for the experimental physicists. The one for the theoretical physicists was in the hands of Woldemar Voigt, which he used for crystal physics. And then he felt that he did not want a laboratory, and that he only wanted to give the courses. They offered me that. Well of course that’ s what I did.
How early had you been aware of the specific heat problem? Did you deal with that at all in your course on radiation theory?
Not in radiation theory, but in my course in Zurich. 1 think I say somewhere that I had first presented this thing iii my course in thermodynamics and then published it.
How did you feel about the Nernst-Lindemann work, the difficulties in getting exact agreements or more exact agreements?
Well you see the point was that I was so convinced that if there is a quantum in the sense of Planck, then his formula is the only thing it can be. And you cannot start fooling around with it. And now they were doing that for the specific heat. So it had to be explained or understood. With Planck’s formulation, can I really get the representation of the experimental facts? That was the problem. And I felt that it could be done if you did it (real), if you said an atom is not just a vibrator. So that is why then I tried to calculate these vibrations in a lattice. And for a linear lattice it is easy enough. What Born and Karman did is too complicated. And so I said to myself, really, if I have this spectrum approximately, it is good enough, because then I will have the right thing for low temperatures, because they are at the end of the spectrum so it isn’t important. And for high temperatures the spectrum isn’t at all important. I have only to know how many lines there are in it. So it is good on both sides, you can get deviations in the middle which cannot be very important. That was the idea.
After this period you go very much away from this whole line, don’t you?
I still did things like that in Zurich and Göttingen.... Well, I had the feeling that Sommerfeld did not like it at all, that I was doing things on quantum theory at the time when I was in Göttingen.... I was talking about Zeeman effect and such things. He did not like that. He wanted to have that for himself. So I decided well all right, I won’t do that anymore.
Were your two papers on the Zeeman effect independent of each other?
Oh yes, yes. So when I talked to him I really felt that it was not right. So I had problems enough, so why do anything against him; I really owed him so much.
These problems were so big and there were already so many people working by then on them.
Yes, yes. But you see there was not very much difference in the way I was doing them and the way he was doing them. It was not so that I had. a very novel idea, you see. And so if you are working in the same direction and using the same principles, well then it is much better if one yields.
This was the Zeeman effect paper in particular, was it then?
One of the things I’m curious about, to stay away from the papers now for a moment, there is such a lot of scientific activity on these basic problems in Germany through the first war.
Well of course there were scientists who were Germans first. Then if they could do something for the war then they would do it rather than else. But there are a lot of Germans which are not that way, which think these philosophical ideas are much more important than any human things like war. One of the men who felt that way was certainly Hilbert, though he is not a physicist. What Hilbert’s idea was this. When I came to Göttingen he said, “Now I don’t know anything about physics. You have now your electron.” At that time it was the electron there. “Well I want to have a mathematical theory from which the electron comes out.” The same thing we want to do nowadays with all the other particles. At that time he said, “The electro-dynamics must be formulated in such a way that it explains the existence of an electron. And that is really my aim. And so why don’t you make a seminar, and then I will come to this seminar at the Rosengarten, in order to learn physics. Maybe then I can get an idea that way.” This was when I first came to Gottingen. And so I made this, I think it still exists — Seminar about the Structure of Matter. That was every week once, and Hubert always came.
When you turned to things like the molecular problem, did you go on being concerned with any of the quantum mechanics?
Well, you can say I was concerned, for instance, when I talked about magnetism and low temperatures, because there is this contradiction. If Langevin’s formula rules, then you can never get entropy zero at zero. And that was the real reason why I worried about that. That was much later. So I had always these kinds of problems in mind, but, I did not do anything mathematical about it. You can always say what I wrote down later about the Compton effect. That was also something which was absolutely independent of Compton.....
In that paper on the Compton effect you speak of having had ideas of this sort for some time.
Yes, well that means the following thing. You see, we were together. You know I brought Scherrer to Zurich.... I knew Scherrer in Gottingen. When I came to Gottingen there was a young man who had come over from Zurich and had married. Now, as a matter of fact, Scherrer was we1l-to-do, but he had married the daughter of a man who was considered the most wealthy man in Switzerland. Her father had not liked that very much, and so the young pair went to Germany and not knowing too much, they had to go to a place for philosophy. And so they went to Königsberg. Well, very soon they saw that that was not good for physics, and so they went to Göttingen. And when I came to Göttingen Scherrer was there working with Voigt, and doing experiments about the Zeeman effect and absorption, This was the type of thing Voigt was doing.
Then I started to do myself some experiments in the laboratory about the scattering of X-rays, using old equipment there. Then he came in and got interested in that. And so we got in touch with each other. He got his doctor’s degree and so on. And when this thing at the technical school in Zurich was offered to me, I had the idea that for engineers we should have different kinds, of physics. Some kind of physics for mechanical engineers; some kind of physics for chemical engineers and so on. So I could take over one. Picard, the one who always went up in the balloons, was there, left over from Pierre Weiss. And then I wanted one special course for chemists. And so I talked to the people there and I created a new professorship and I got Scherrer this professorship. Being a Swiss himself it was easy, and he was a good man. And so we were together in the laboratory. We always understood each other very well.And so I talked to him, oh a lot of times, about this Compton effect, this so-called Compton effect. I calculated, there should be this change in frequency. We should do something about it someday if it is really right. And we never came to it. This is what I mean.
What led you to think about photon collision computations?
There is this business. I thought at that time that I might find in this way a way out of the dilemma which goes back to Barkla that the scattering coefficient divided by the density is 0.2 for X-rays. And this is old electro-dynamics. And we know, experimentally, that it is not true for high frequencies. And so I thought that if I made the quantum come in there I might find an explanation for that. But that is why I began to say, “Well, we don’t know anything, so let’s use just the general laws, conservation of energy, conservation of momentum, and see what that says.” That was the idea.
But this would still be thinking of light as particles?
Well, not necessarily as particles. You make no commitment. And you use only the general laws, general conservation laws, and see what they would say. That does not mean that you are talking about particles.