From Matrix Mechanics and Wave Mechanics to Unified Quantum Mechanics B

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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 twoT 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 motion 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

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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 approach 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. tial singularities, depending on t like a sine func- In reality, however, S/K is not large generally, so tion with frequency ν = E/h, can exist in accor- one has to stick to (5) and to use the mathematics dance with (3). of Wave Theory to integrate this equation. So one has to replace in (3) by a product ¯ Next comes my own contribution, namely the con- of a new function (x, y, z) depending only on nection with the Göttingen Mechanics. For the sake position with the factor of simplicity I shall consider a one-dimensional problem and use Cartesian coordinates (in the three-di- 2πiνt 2πi(E/h)t e = e mensional case and with arbitrary coordinates every- thus obtaining thing goes just so, also if gyroscopic terms are added). So let the wave-equation be given as = ¯ e2πi(E/h)t d2 2m + [E Epot (x)] =0 then dx2 K2 − ∂2 4π 2 = E2 ∂t2 − h2 (compare (5), the bars are omitted). Now let E1,E2,...,En,... be the eigenvalues, and one obtains 1, 2,...,n,...a complete set of eigenfunctions. For these we have 2 2 4 [E Epot (x, y, z)] m c ¯ − − 0 ¯ + 0 for n = m (4) + 2 2 =0, ∞ c K nmdx = 6 Z ( 1 for n = m putting,∆ as Schrödinger does, K = h/2π. −∞ This is an eigenvalue problem for the possible values of E = hν. These ν are enormously The first equation (orthogonality) follows from large, because in E the energy of the electron at Green’s formula, the second means a normalization rest is included. The Frequency Condition now of the multiplicative constants in the n. Any arbi- says that the light waves can formally be contrary function of x can be expanded in a series with

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respect to the n. Now one considers in particular seem to be connected with the idea that the or- the expansion of xn dinary light waves are different oscillations (beats) of de Broglie’s radiation. The fact that in xn = xnmm(x); (I) and (II) no indefinite phases occur is due to Xm the trivial reason that in passing from (3) to (4) (I) + ∞ the periodic factor depending on time has been xnm = xnmdx suppressed. If this factor is taken into account, Z−∞ one obtains in xnm and (px)nm besides exp[(2πi/h)(En Em)t] a phase factor One also puts − exp[2πi(δn δm)] in which δm and δn are the + − ∞ ∂n phases of eigenvibrations belonging to E and (px)nm = iK mdx; m ∂x E . In principle, in the Göttingen theory as well (II) Z−∞ n ∂n as in de Broglie’s statement of the quantum iK = (px)nmm(x) problem, no description of the motion of the ∂x m X electron in the atom in space and time is given. In the latter theory this is clear from the fact that (i = imaginary unit, K = h/2π). Now xnm = xmn is outside the domain of validity of Geometrical Op- real, (px)nm = (px)mn purely imaginary. It can be − tics it is impossible to construct “rays” in de shown without difficulty, that the matrices for x Broglie’s Wave System that can be considered as and px thus defined satisfy the equations of the orbits of particles. The problem of the asymp- Göttingen Mechanics. Namely totic linkage with the usual pictures in space and pxx px = iK, time for the limiting case of large quantum num- − x − bers remains unsolved. Yet it is a definite 1 2 p + Epot (x)=E (Diagonal matrix) 2m x progress to be able to see the problems from two different sides. It seems one also sees now, how From the rule of multiplication it follows that the ma- from the point of view of Quantum Mechanics trix belonging to any function F(x) of x is just given the contradistinction between “point” and “set by of waves” fades away in favour of something + ∞ more general. Fnm = F(x)nmdx. Cordial greetings for you and the other peo- Z −∞ ple at Göttingen (especially Born, in case he is I shall not write out the calculations in detail; you back from America; please show him this letter). will be able to verify the assertion easily. Yours, (carbon copy signed) W. Pauli I have calculated the oscillator and rotator according to Schrödinger. Further the Hönl-Kronig- COMMENTS ON PAULI’S LETTER formulae for the intensity of the Zeeman components Pauli’s wave equation (1) is called in the let- are easy consequences of the properties of the spher- ical harmonics. Perturbation theory can be carried ter “the wave equation of de Broglie’s radiation over completely into the new theory, and the same field.” It is not given in de Broglie’s thesis, but thing holds for the transformation to principal axes, it is very easy to derive it from the given ex- which in general is necessary if degeneracies (mul- pressions for ν and λ. It is valid for plane waves, tiple eigenvalues) are cancelled by external fields of i.e. for a free election. force. At the moment I am occupying myself with the Equation (3) is essentially the Klein-Gordon calculation of transition probabilities in hydrogen equation. It is true that the magnetic terms are from the eigenfunctions calculated by Schrödinger. missing, but Pauli expressly says in the course For the Balmer lines finite rational expressions seem of his letter that “everything goes just so if gy- to come out. For the continuous spectrum the situ- roscopic terms are added,” which shows that ation is more complicated: the exact mathematical Pauli, who was at that time thinking very hard formulation is not yet quite clear to me. about the anomalous Zeeman effect, knew per- As regards Lanczos, my considerations have only fectly well how to handle magnetic fields. He very few points of contact with his ideas. He considers omitted the magnetic terms only “for the sake a problem for which the eigenvalues are the recip- of simplicity.” rocal energy values, whereas here the eigenvalues are We know from Schrödinger’s letters that he just the energy values. In his exposition certain func- also tried the Klein-Gordon equation, but he tions depending, like Green’s function, on two points, gave it up because it did not yield the right fine- play an essential role; such functions are not used structure of the hydrogen atom. here. On the whole I feel that Lanczos’ approach has The Klein-Gordon equation was discovered in- not much value. dependently by Schrödinger, by Pauli, by Klein About the physical significance of the expres- and Gordon, and by at least two other people.1 sions (I) and (II) I do not know much. In any case they To Pauli’s orthogonality relations

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+ ∞ 0 for n = m ready contains the valuable insight that nmdx = 6 Heisenberg’s atomic dynamics is capable ( 1 for n = m Z−∞ of a continuous interpretation. For the rest, the paper of Lanczos has less points we may remark that in the one-dimensional case of contact with mine than one might think the eigenfunctions are single and real, so that at first sight. complex conjugate factors are not needed. n∗ The determination of the system of for- The paper of Lanczos, to which Pauli refers mulae, which Lanczos leaves quite unde- at the end of his letter, was published in termined, cannot be found in the direc- Zeltschrift für Physik 35 (received 22 December tion of identifying the symmetrical kernel 1925). I feel it has more value than the con- K(s,σ) with Green’s function of our wave temporaries suspected. Let us use our hindsight equation… For this function of Green, if and start with Schrödinger’s equation, which I it exists, has as its eigenvalues the quan- shall write as tum levels themselves.

(6) + V = E This is an error of Schrödinger, for which I have − no explanation. We have just seen that the eigen- leaving out all numerical∆ factors. Lanczos con- values of Green’s kernel K(P,Q) are 1/E and not E. siders a finite domain in q-space, so let us en- Schrödinger continues: “On the contrary, the kernel of Lanczos is required to have as its eigenvalues close our atom in a large sphere of radius R. As the reciprocal quantum levels.” a boundary condition we may assume =0on Schrödinger just missed the point. the boundary. Since the zero-point on the energy If Lanczos’ kernel K(P,Q) is identified with the scale is quite arbitrary, we may suppose that it Green’s function of Schrödinger’s differential equa- lies below the lowest energy value. It follows that tion, its eigenfunctions φ ,φ ,... are Schrödinger’s zero is not an eigenvalue. 1 2 eigenfunctions. Under these assumptions, the boundary value Besides the integral operator K defined by the ker- problem nel K(s,σ): + V = u, − = 0 on the boundary K(s)= K(s,σ)(σ )dσ, ∆ Z can be solved by means of Green’s function Lanczos introduces two more integral operators p K(P,Q) as follows; and q:

(7) (P)= K(P,Q)u(Q)dQ. p(s)= p(s,σ)(σ )dσ, Z Z Replacing u(Q) by E(Q), and dividing by E, one q(s)= q(s,σ)(σ )dσ obtains Z in such a way that 1 (8) K(P,Q)(Q)dQ = (P). 1 Z E (9) pq qp = 1. − 2πi This integral equation is equivalent to Schrödinger’s equation (6). Its eigenvalues are Since p and q are supposed to be integral operators, just 1/E, the reciprocal energy values. the unit operator 1 must also be an integral opera- Now this is just the kind of integral equation tor Lanczos considers. He does not specify what kind of function K(P,Q) is, but he does say that 1(s)= E(s,σ)(σ )dσ. the eigenvalues of his integral equation are the Z reciprocal energy values. This implies, as Lanczos says, that E(s,σ) is zero for Now let us hear what Schrödinger says about a σ = s, and that 6 the paper of Lanczos. In a footnote on p. 754 of + his “equivalence” paper he writes: ∞ E(s,σ)dσ Similar ideas are exposed in an in- Z−∞ teresting paper of Lanczos, which al- is equal to 1. Hence, Lanczos’ function E(s,σ) is just Dirac’s function δ(s σ ). 1Jagdish Mehra has informed me that the other peo- − ple were V. Fock (Z. Phys. 38. 242 (1926), 39. 226 Lanczos concludes that the functions p(s,σ) and (1926)); H. van Dungen and Th. de Donder (Compt. q(s,σ) cannot be everywhere finite. In fact, if one Rend. Acad. Sci. Paris, July 1926); and J. Kudar (Ann wants to reach complete agreement between Lanc- Physik 81. 632 (1926)). zos, Schrödinger, and Pauli, one has to assume

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q(s,σ)=s δ(s σ ), Lanczos: You are absolutely right. You rehabili- · − tated my work. Pauli was a vicious man, as everybody h p(s,σ)= δ0(s σ ). knows. Anything which didn’t agree with his ideas −2πi − was wrong, and anything was right only if he made it, if he discovered it, which is all right for such a great Next, Lanczos defines the matrices corresponding to man. He could allow himself such viciousness, but I the operators p and q: am very grateful to you for pointing out what you have. I certainly was aware of these connections, but as you pointed out, the weakness was that I didn’t spec- pik = p(s,σ)φi(s)φk(σ )dsdσ Z ify the kernel function and the special functions. At that time, you know, after the matrix mechanics it qik = q(s,σ)φi(s)φk(σ )dsdσ looked as if you couldn’t do anything with the con- Z tinuum, and one would have to operate with the dis- and proves that the matrices p and q satisfy the Born- continuum. Everything is discontinuous in the matri- Jordan condition ces. Now I was very much interested at that time in integral equations, and actually the integral equation which I used was not the Schrödinger equation, but h pq qp = 1. the inverse equation, because only a differential equa- − 2πi tion can be changed to an integral equation, as you pointed out. And that is the reason why the energies From this analysis we see that Lanczos’ approach actually come in with a reciprocal, so that the Green’s had more points of contact with the ideas of function which I had is basically the Schrödinger equa- Schrödinger and Pauli than these two suspected. His tion with the source of a delta function. If you have a weakness was that he was not able to specify his func- source, and that source happens to be a delta functions K(s,σ),p(s,σ) and q(s,σ). tion, then this function does have a physical signifi- Let us now return to Pauli’s letter. Pauli says at cance, whereas it looks after Pauli’s criticism that it the end: “The problem of the asymptotic linkage has no physical significance. But it does have a phys- with the usual pictures in space and time for the lim- ical significance. iting case of large quantum numbers remains un- van der Waerden: Yes, but after you read Schrödinger’s paper, did you realize that this was the solved.” case? More light on this problem was shed by the study Lanczos: Afterwards, it was too trivial. I mean, it of the behaviour of wave packets. A most interest- was no longer of interest, because Schrödinger came ing contribution was a little-known paper of Ehren- along and he did it. As it often happens, it is the sec- fest, in which he proved that the centre of gravity of ond man who hits the nail on the head and not the first a wave packet moves according to the classical law: one. force = acceleration mass, provided the force ex- 2. Cornelius Lanczos emigrated from Hungary to × erted upon the electron by the electromagnetic field the U.S. in 1931 to take a position at Purdue Univer- is calculated by integrating the Lorentz force over sity, and joined the AMS in 1934. In 1952 (while Schrödinger was acting director) Lanczos moved to the the charge density e . Another important con- − ∗ Dublin Institute for Advanced Study where he re- tribution was, of course, Heisenberg’s Uncertainty mained (including serving as director for a time) until Principle, which was also derived from the study of his death in 1974 at athe age of 81. the behaviour of wave packets in q-space and p- W. Moore’s biography Schrödinger’s Life and space. Thought (Cambridge University Press, 1989) indicates A quite new point of view was Born’s interpreta- that Schrödinger and Lanczos had considerable social as well as scientific contact during the years they were tion of ∗ as a probability density, proposed in connection with his study of collisions. Dirac ex- both in Dublin. He also writes (p. 450) “Erwin was tended Born’s probability interpretation to much quite fond of him (Lanczos) but this feeling was not reciprocated.” more general measurements. However, in this lecture 3. Independently of Schrödinger or Pauli, Carl I wanted to restrict myself to what happened in Eckart, (then a young American at Caltech) established March and April of 1926, so I shall stop here. the relationship between the Heisenberg and Editor’s notes: 1. Unknown to van der Waerden, Cor- Schrödinger formulations of quantum mechanics in a nelius Lanczos, whom he had never met, was in the audi- paper published in Phys. Rev. 28, (1926) 711–26. See K. Sopka Quantum Physics in America, vol. 10, The His- ence. When the moderator introduced them, van der Waer- tory of Modern Physics (American Institute of Physics, den was visibly pleased and exclaimed, “Oh, This is 1988). marvelous. I didn’t know that you were here at this symposium or that you would come to this lecture.” Later in the discussion period, Lanczos made a remark about Einstein’s approach to quantization. van der Waer- den then took the opportunity to initiate the following ex- change: van der Waerden (to Lanczos): Did you know all this to which I have referred in my paper? Were you aware of these connections?

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