Letters on Wave Mechanics: Correspondence with H. A. Lorentz, Max Planck, and Erwin Schrodinger Read online

  Thank you very much for kindly sending me your lecture,3 which I had already read with the greatest interest several days earlier. I was especially captivated by the dramatic force with which you sketch the status of the theory of relativity and the quantum theory—in the third section—and with the way you pick out the key difficulty and make it comprehensible without formulas. Just this difficulty concerning the energy unfortunately still persists, quite unimpaired.

  If I did not answer your card, which gave me so much pleasure, at once, it was because I wanted to send along at least a little something that was new. Enclosed are the results for the Stark Effect in H. It seems that the intensities come out completely right. The assumption on which it is based is that the electrical charge density is given by the square of the wave function, and that the normalization integral has the same value for all the individual proper vibrations that belong to one coarse Balmer level. I cannot yet describe the numbers I am sending you as incontestable because the calculation is very involved and I have not yet checked everything again. In any case Epstein’s formula for the splitting comes out completely unaltered (as I already said at the end of my “Second Paper”); also the “Selection Rule for the azimuthal quantum number”. Moreover, the “exclusion of zero for the equatorial quantum number” also comes out quite automatically—there is no proper vibration that would correspond to the quantum orbit that collides with the nucleus. It is also very gratifying that although the three unobserved components at relative distances of 5, 6, and 8, are not actually “forbidden” theoretically, they receive an intensity that is 80 to 700 times smaller than that of the weakest observed component, so that their non-appearance becomes very understandable.

  I am now calculating Hα, Hβ, Hγ. The calculations are unfortunately terribly difficult to see through and I cannot manage to bring them into a simpler form.

  With best compliments and greetings I remain, dear Professor, always

  Yours faithfully,

  E. Schrödinger

  3. Planck to Schrödinger

  Berlin—Grunewald

  24 May 1926

  Dear Colleague,

  I have owed you my thanks for sometime for your kindly having sent me your last Annalen article on quantization. You can imagine the interest and enthusiasm with which I plunge into the study of these epoch making works, although I now make very slow headway penetrating into this peculiar train of thought. In connection with that I have high hopes of the beneficial influence of a certain amount of familiarity which in time facilitates the use of new concepts and ideas, as I have often found already. But what especially delights me, and the reason for my really writing you today, is the joyful hope that we may soon have the opportunity to hear you and to talk to you here. As my colleague Grüneisen4 tells me, your visit to a meeting of the Physical Society has not been cancelled but only somewhat postponed, and it may even still take place this semester. Let me tell you explicitly how much pleasure all of the physicists here would have in hearing you yourself present your new theory and in coming into contact with your ideas. And don’t be afraid that we will make too many demands on you and tire you out. I do not know if you are already familiar with Berlin. But I hope you will find that in certain respects life here is freer and more independent than in a smaller city where everyone checks on everyone else, and there is no possibility of completely withdrawing at some time without anybody noticing it.

  I should like to express just one little selfish request. In case you can come in July, please not before the 11th. Because at the beginning of July I have to go to Bonn for a few lectures and I would be sad if I missed your visit here as a result. Above all, however, I wish you the relaxation that you need after your demanding labors, and the complete recovery of your powers. I should be especially grateful if, at your convenience, you would send me a brief card with a word about your travel plans.

  In the meantime, with best regards,

  Yours sincerely,

  M. Planck

  4. Schrödinger to Planck

  Zürich

  31 May 1926

  My dear Professor,

  Thank you very much for your kind and extremely gracious letter of the 24th, which now has finally decided me to accept the attractive invitation for this semester, however things may go. I have just written to Mr. Grüneisen. It goes without saying that, so far as I am concerned, a date when you are absent from Berlin is out of the question. Now Mr. Grüneisen was kind enough to point out to me that it might also be possible to consider a slight postponement of the date of the meeting, and since a postponement of the July 9th meeting would surely come too near the end of the semester, as he himself thinks, I have allowed myself to suggest that perhaps the June 25th meeting could be put off until July 2nd. Would that still work out with your trip to Bonn? The 25th of June would not be acceptable to me because from the 21st to the 26th a number of foreign physicists (among them Sommerfeld, Langevin, Pauli, Stern, P. Weiss) are meeting here for lectures and discussions. Now the connections work out so badly that I would have to leave here on the afternoon of the 23rd at the latest, if I do not want to travel through the night directly before the Berlin meeting. And I should not like to do that because then I am often completely exhausted and may possibly speak very badly.

  I should be very grateful if you would give me some hints, in just a few words, as to how I should plan my lecture. What I mean is, should I think more of the fact that you and Einstein and Laue are in the audience—a thought without which I should feel uneasy—or should I direct myself more to those gentlemen who are further removed from theoretical work; which would of course have as an inevitable result that those named above (and a considerable number of others) will be very bored. In other words: should I recapitulate in a simplified way what has already been published or, passing over that lightly, talk more about perturbation theory, the Stark effect, and general intensity formulas? (Otherwise I could only mention these latter things briefly at the end, or else it would get to be too long; it takes about an hour for a general survey of the fundamentals, for the purpose of orientation and without much calculation, as I know from our colloquium here).

  Naturally I can also do both, if there is the opportunity, one in a general meeting and the other in a more restricted colloquium.

  Today I received a very kind and very interesting letter of 13 closely written pages from H. A. Lorentz5 which I still have to study in detail, of course. He raises a good many interesting questions; however, he does not reject it at all, on the whole, but still appears to be very critical. Lorentz sees one of the chief difficulties in reinterpreting classical mechanics as “wave mechanics” to lie in the fact that the “wave packet” which is to replace the “representative point” of classical mechanics in macroscopic problems, (possibly also in the motion of the electron on paths of slight curvature), that, I say, this wave packet will not remain together, but, on the contrary, will gradually spread into larger volumes by “diffraction”, according to general theorems of wave theory. I felt that to be a serious point at first—yet, strange to say, it seems not to be the case, at least not always. For the harmonic oscillator (which always remains the simplest typical example of a mechanical system which one can work with so easily and agreeably), I was able to produce a wave packet, by superposition of a large number of neighboring characteristic oscillations of high order (i.e. high quantum number), which is practically confined to a small spatial region, and which as a matter of fact revolves in precisely the harmonic ellipses described by classical mechanics for an arbitrarily long time without dispersing! I believe that it is only a question of computational skill to accomplish the same thing for the electron in the hydrogen atom. The transition from microscopic characteristic oscillations to the macroscopic “orbits” of classical mechanics will then be clearly visible, and valuable conclusions can be drawn about the phase relations of adjacent oscillations. For the present these phase relations and amplitude relations remain postulates, how
ever; they can naturally also be so arranged that for large quantum numbers a “revolving” mass point does not result: e.g. since the structure is linear it can also be arranged so that two wave groups, revolving independently of one another, result—perhaps the equations are only approximately linear.

  A second very delicate question that concerns Lorentz is the energy that is to be assigned to a characteristic oscillation. It is quite certain that the Balmer-Bohr energy value is not to be ascribed to the characteristic oscillation. In general one should not consider the individual characteristic oscillation as the equivalent of the individual Bohr orbit; that is a mistaken parallel, as the above construction shows. The concept “energy” is something that we have derived from macroscopic experience and really only from macroscopic experience. I do not believe that it can be taken over into micro-mechanics just like that, so that one may speak of the energy of a single partial oscillation. The energetic property of the individual partial oscillation is its frequency. Its amplitude must be determined in quite another way—I believe by normalizing the integral of the square of the total excitation to the value of the electronic charge.

  Mr. Grüneisen was kind enough to hold out to me the prospect that either you or Mr. von Laue would offer me hospitality. If it doesn’t cause too much trouble I am naturally very pleased about it, and in any case I am very grateful for your kind offer. I would strive to give as little inconvenience as possible, and ask that it be so arranged that you are disturbed as little as possible; naturally any improvised lodging you choose is completely adequate for me.

  Thank you once again for all the kindness that is always shown me by Berlin in general and by you especially, Professor Planck. With sincere respect, I remain

  Yours faithfully,

  E. Schrödinger

  5. Planck to Schrödinger

  Berlin—Grunewald

  4 June 1926

  Dear Colleague,

  I am extremely pleased that you could make up your mind to visit Berlin before the end of this semester, and I know for certain that the rest of the physicists here think the same way.

  My colleague Grüneisen informs me that he has some doubts with regard to July 2nd and suggests July 16th instead. I should just like to join him in this. The semester here lasts until the beginning of August so that things are still in full swing in the middle of July and we need not be afraid that many will have already gone away. Grüneisen himself is an exception, to be sure, but he has to set out so early that he would unfortunately miss your visit all the same. But the 16th of July would suit the rest of us very well, and the only question is whether it is suitable for you yourself.

  My wife and I would be especially happy if you would stay with us. We hope very much that we will be able to make you comfortable in our house. I shall take care above all that you remain master of your own actions to the greatest possible extent, and especially that, at those times over and above the “official” periods dedicated to the Physical Society, you have the opportunity to withdraw and to occupy yourself as you see fit. I know from experience how pleasant it often is to have a possibility of this kind. Moreover, my house stands at your disposal night and day for as long as you are inclined to stay.

  You also talk about the level at which your lecture should best be given, or rather at which it should begin. I would like to propose, in agreement with my colleagues, that you imagine your audience to be students in the upper classes who, therefore, have already had mechanics and geometrical optics, but who have not yet advanced into the higher realms; to whom, therefore, the Hamilton-Jacobi differential equation, if they are acquainted with it at all, signifies a difficult result of profound research, deserving of reverence, and not by any means something to be taken for granted. Under no circumstances, however, should you be afraid that any one of us will consider one sentence of yours to be superfluous. For even if the sentence should not be necessary for an understanding of your train of thought, it would always offer the particular interest of seeing what special paths your thought takes and which particular forms your perception favors. For all of us the main point of your lecture will be what you yourself in your letter designated as a general survey of the fundamentals for the purpose of orientation without much calculation and without many individual problems. Perhaps it would be easier and more natural for you to carry this out, if on the other day, Saturday morning the 17th of July, you were to give a second lecture in our Colloquium, aimed at more special matters with supplements and continuations of the lines of thought you will have described at the more general meeting. I hope that this seems suitable to you, since you already indicated such a possibility yourself. That can very easily be arranged, and I ask you only to let me know so that we can take care of matters.

  What a cross-fire of critical, enthusiastic, and questioning acclamations might now besiege you! But still, it is a thing with incredible prospects. I see that you have already energetically taken hold of the big question of whether and under what conditions a wave packet will remain intact. I have such a feeling that for closed systems it is the boundary conditions that take care of the conservation [of the wave packet], whereas a satisfactory solution for phenomena in an unbounded space seems to me to be possible only on the basis of new assumptions. That, however, is a cura posterior.

  In the meantime my cordial greetings and the friendly request that you write me the day and hour that you arrive here.

  Yours faithfully,

  Planck

  6. Schrödinger to Planck

  Zürich

  11 June 1926

  My dear Professor,

  Please do not be annoyed with me because I am just today answering your extremely kind letter of the 4th of June. I have written to Mr. Grüneisen in the meantime that I now finally accept for July 16th, and in fact it also suits me excellently because then I need to conclude my lectures a few days earlier, and besides, these last lectures are no longer worth much, since the men already have their heads full of the vacation. I am very sorry, however, not to be able to see or to become acquainted with Mr. Grüneisen himself, but unfortunately that can’t be helped.

  Now first and foremost my very hearty thanks for your kind invitation to stay with you which I of course accept with the utmost pleasure. The words with which you offer me your house as a “place of refuge from Berlin” express a boundless, thoughtful, concerned kindness that has truly touched me. You are quite correct that one is most often in want of just this possibility of being alone for a few hours in situations where everyone around is striving to be nice to one. I hope, however, that I will not need to make much use of this possibility in the present situation, despite my end-of-semester fatigue. Not only would I really like to give as much as I possibly can, both in and outside the “official” hours, to the gentlemen in Berlin who are so friendly as to be interested in my work; but also from a purely selfish standpoint I should like to make full and intensive use of the opportunity to discuss the things that have held me completely captured for months, with a number of the most distinguished scientists with the widest variety of research interests. If one still gets a little tired after a few days—the pleasure of the interesting dialogues would be sufficient compensation, to say nothing of the stimulation and the positive challenge.

  I will hold to your advice for which I am very grateful, concerning the general lecture, and will naturally be very happy if anyone still has the desire to listen to me on the following day in the more restricted group.

  By the way, during the last few days another heavy stone has been rolled away from my heart: I have the interaction of the atom with an incident light wave, thus the theory of dispersion. I had considerable anxiety over it because it was to be feared that the eigenfrequencies themselves would appear as the locations of the resonances in the case of a forced oscillation, and furthermore, that the forced vibrations would not depend on the existing nearby proper oscillations, i.e. not on the state in which the atom happens to be. And that would be n
onsense. But it all resolved itself with unheard of simplicity and unheard of beauty; it all came out exactly as one would have it, quite straightforwardly, quite by itself and without forcing. This is the way: what I called the “wave equation” up to now is really not the wave equation but rather the equation for the amplitude. It no longer contains the time at all, but instead of it, it already has an integration constant E, (see Eq. [18″] of my second paper.) The time dependence must be given by , or, what is the same thing, we must have

  One can eliminate E from this equation and equation (18″) and one thus obtains the true wave equation which is of fourth order in the coordinates, perhaps of the type of the vibrating plate.

  The main point is now this: one may now in a free and easy way also let the potential energy be an explicit function of the time in this true wave equation. The interaction energy with the incident wave can be added on as a perturbing term, and perturbation theory straightforwardly applied, which is quite simple. The result is essentially the so-called Kramers dispersion formula, with completely exact assertions about the phase and polarization of the secondary radiation, naturally assuming that the eigenfunctions and eigenvalues of the unperturbed atom are known.

  What is still missing from the whole picture is only the interaction with its own wave, i.e. what corresponds to radiation damping. I believe that can no longer be very hard.

  Naturally, perturbation theory can still be applied to many other questions too, e.g. the perturbation due to an α-particle or an electron flying past. I believe that it is a rather considerable step forward because the whole course of an event in time can now be exactly followed—at least in principle.