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From Matrix Mechanics and Wave Mechanics to Unified Quantum Mechanics B vanderwaerden.qxp 1/20/97 9:35 AM Page 323 From Matrix Mechanics and Wave Mechanics to Unified Quantum Mechanics B. L. van der Waerden Editor’s note: Like many mathematicians of in which the operator H is obtained from the his generation, van der Waerden was ex- classical Hamiltonian H(p, q) by replacing every tremely broad. In 1932 he published one of momentum p by the first books on group theory and quantum K ∂ h mechanics. In the following remarkable paper, ,K = . delivered in Trieste in September 1972 at a i ∂q 2π symposium celebrating Dirac’s seventieth C. The eigenvalues E are the energy values. birthday, he clarifies the connection between To these three hypotheses, Schrödinger added the Heisenberg and Schrödinger formulations Bohr’s postulate: of quantum mechanics and earlier work by D. Em En = hνmn. Cornelius Lanczos. − This theory was presented in Schrödinger’s first and second communications on “Quantisierung als Eigenwertproblem” in Annalen der Physik 79. The first communication was received on 27 January, he story I want to tell you begins in and the second on 23 February 1926. March 1926 and ends in April 1926. On the other hand, matrix mechanics was invented Early in March two separate theories by Heisenberg in June 1925, and presented in a fully existed: matrix mechanics and wave developed form in Dirac’s first paper on quantum me- mechanics. At the end of April these chanics (received 7 November 1925) and also in the Ttwo had merged into one theory, more power- famous “three-men’s paper” of Born, Heisenberg and ful than the two parents taken separately. Jordan (received 16 November 1925). This theory Wave mechanics was based upon three fun- was based upon four mechanical hypotheses and damental hypotheses: two radiation hypotheses. The mechanical hypothe- A. Stationary states are determined by complex- ses are: valued wave functions (q), which remain 1. The behaviour of a mechanical system is deter- finite everywhere in q-space. mined by the matrices p and q (one matrix q for B. The functions satisfy a differential equa- every coordinate q, and one p for every mo- tion mentum p). K H = E 2. pq qp =( )1 if p belongs to the same coordi- − i nate q, otherwise equal to 0. 3. H(p, q)=W = diagonal matrix, having diagonal el- Article reprinted from The Physicist’s Conception of Na- ements En, the energy values. ture, J. Mehra (ed.), 1973, pp. 276-293, D. Reidel Pub- 4. Equations of motion:∂H ∂H lishing Company, Dordrecht, Holland, with kind per- p˙ = , q˙ = mission from Kluwer Academic Publishers. − ∂q ∂p MARCH 1997 NOTICES OF THE AMS 323 vanderwaerden.qxp 1/20/97 9:35 AM Page 324 These hypotheses imply letter, but he signed the carbon copy (which is quite unusual), and he kept the letter in a plas- p = a e2πi(νm νn)t mn mn − tic cover until his death. I am indebted to his En = hνn. widow, Franca Pauli, for giving me her consent to publish this letter. The radiation hypotheses determine the frequency and intensity of the radiation emitted or absorbed: PAULI’S LETTER 5. Em En = hνmn. [This letter was probably written and typed at − 6. The transition probabilities are proportional to the Copenhagen] 2 amn . | | 12th April 1926 In his second communication, Schrödinger con- Dear Jordan, fesses that he did not succeed in finding a link be- Many thanks for your last letter and for your tween his own approach and Heisenberg’s. This was looking through the proof sheets. Today I want written in February 1926, but in March he found the to write neither about my Handbüch-Article nor link. In his paper “Über das Verhaltnis der Heisen- about multiple quanta; I will rather tell you the berg-Born-Jordanschen Quantenmechanik zu der results of some considerations of mine con- meinen”, received 18 March 1926, Schrödinger writes: nected with Schrödinger’s paper “Quantisierung “In what follows…the inner connection between als Eigenwertproblem” which just appeared in Heisenberg’s Quantum Mechanics and my own will the Annalen der Physik. I feel that this paper is be made clear. From the formal mathematical stand- to be counted among the most important recent point one may even say that the two theories are iden- publications. Please read it carefully and with de- tical.” votion. Now is this true? Are the two theories really equiv- Of course I have at once asked myself how his alent in the formal mathematical sense? results are connected with those of the Göttin- Equivalence (or identity, as Schrödinger says) gen Mechanics. I think I have now completely would mean clarified this connection. I have found that the A, B, C, D 1, 2, 3, 4, 5, 6. ⇐⇒ energy values resulting from Schrödinger’s ap- proach are always the same as those of the Göt- Now what Schrödinger actually proves is tingen Mechanics, and that from Schrödinger’s A, B, C = 1, 2, 3, functions , which describe the eigenvibrations, ⇒ one can in a quite simple and general way con- and of course struct matrices satisfying the equations of the D = 5. Göttingen Mechanics. Thus at the same time a ⇒ rather deep connection between the Göttingen Moreover, if time-dependent functions are al- Mechanics and the Einstein-de Broglie Radiation lowed, satisfying Schrödinger’s time-dependent dif- Field is established. ferential equation, one can prove 4. However, hy- To make this connection as clear as possible, pothesis 6 can in no way be derived from I shall first expose Schrödinger’s approach, styled Schrödinger’s set of hypotheses. a little differently. According to Einstein and de The converse = Schrödinger does not even at- ⇐ Broglie one can assign to any moving particle tempt to prove. Yet he refers to his proof as “Äquiv- with energy E and momentum G, taking care of alenz-Beweis”, and he asserts confidently: “Die Äquiv- the relativity terms, alenz bestecht wirklich, sie besteht auch in 2 umgekehrter Richtung.” m0v m0c G = 2 ,E = 2 From the formal logical point of view, one may 1 v 1 v − c2 − c2 even say that it is impossible to derive A, B, C from q q 1, 2, 3, 4, 5, 6, because in hypothesis A the notion normed in such a way that the energy at rest is 2 2 2 2 2 4 “stationary state” occurs, which does not occur in = m0c , hence E c G = m c an oscillation − 0 1, 2, 3, 4, 5, 6. with frequency ν = E/h and wave length After the publication of this paper, everybody ac- λ = h/ G . (This assignment is invariant with re- | | cepted Schrödinger’s conclusion that the two theo- spect to Lorentz transformations.) The phase ve- ries are “equivalent”. Everybody except Pauli. He locity V is knew better. E On April 12, just after the publication of V = λν = , G Schrödinger’s first communication, but before his | | “equivalence” paper came out, Pauli wrote a very re- hence the wave equation of de Broglie’s radia- markable letter to Jordan, in which he established tion field the connection between wave and matrix mechanics, 1 ∂2 in a logically irreproachable way, independent of =0 Schrödinger. He never published the contents of this − V 2 ∂t2 ∆ 324 NOTICES OF THE AMS VOLUME 44, NUMBER 3 vanderwaerden.qxp 1/20/97 9:35 AM Page 325 assumes the form sidered as difference-oscillations of the de Broglie- radiation. Planck’s constant enters the theory only G2 ∂2 at that point where one passes from the energy of (1) 2 2 =0. − E ∂t the states to the frequency of the radiation of de Taking care of the∆ relation Broglie. Neglecting relativistic corrections one obtains 2 2 2 2 4 2 ¯ E c G = m0c from (4) by putting E = m0c + E and expanding ac- − cording to powers of 1/c2: between energy and momentum, one obtains 2m0 2 2 4 2 ¯ ¯ ¯ E m c ∂ (5) + 2 (E Epot ) =0. (2) − 0 =0. K − − c2E2 ∂t2 This equation∆ is given in Schrödinger’s paper, and Now if we have∆ a mass point moving in a field he also shows how it can be derived from a Varia- of force and if Epot is its potential energy, the tion Principle. relation between energy and momentum be- Here is another remark for which I am indebted comes (taking care of the variability of the mass) to Mr. Klein. The difference between the general 2 2 2 2 4 Quantum Theory of periodic systems and (E Epot ) c G = m c − − 0 Schrödinger’s Quantum Mechanics based upon Equa- tion (5) is, from the point of view of the de Broglie- provided E is again normalized so that for the 2 Radiation, the same as the difference between Geo- mass point at rest E Epot = m0c . (For the hy- − metrical Optics and Wave Optics. Namely if the wave drogen atom with relativistic correction one ob- 2 length of the de Broglie-Radiation is small, one can viously has to put Epot = Ze /r). Substituting − put in (5), as is well-known this into (1) one obtains instead of (2) ¯ = ei(1/K)S . = If S/K is large, one now obtains from (5), according (3) 2 2 4 2 [E Epot (x, y, z)] m0c ∂ to Debye, the Hamilton-Jacobi differential equation ∆ − − =0. ¯ c2E2 ∂t2 for S. In this case becomes a univalued point function only if the moduli of periodicity of S/K are The phase velocity now depends on position. integer multiples of 2π. This leads to the usual con- Schrödinger’s approach is now as follows: A dition pdq = nh, which has been interpreted al- quantum state of the system with energy E is only ready byR de Broglie from the point of view of the possible if a standing de Broglie-Wave without spa- geometrical optics of his Radiation Field.
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