The classical roots of wave mechanics: Schrödinger’s transformations of the optical-mechanical analogy✩ Christian Joasa , Christoph Lehnera a Max Planck Institute for the History of Science, Boltzmannstr. 22, 14195 Berlin, Germany Abstract In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat’s principle) and corpuscular mechanics (Maupertuis’s principle). Almost a century later, this optical-mechanical analogy played a central role in the development of wave mechanics. Schrödinger was well acquainted with Hamilton’s analogy through earlier studies. From Schrödinger’s research notebooks, we show how he used the analogy as a heuristic tool to develop de Broglie’s ideas about matter waves and how the role of the analogy in his thinking changed from a heuristic tool into a formal constraint on possible wave equations. We argue that Schrödinger only understood the full impact of the optical-mechanical analogy during the preparation of his second communication on wave mechanics: Classical mechanics is an approximation to the new undulatory mechanics, just as ray optics is an approximation to wave optics. This completion of the analogy convinced Schrödinger to stick to a realist interpretation of the wave function, in opposition to the emerging mainstream. The transformations in Schrödinger’s use of the opticalmechanical analogy can be traced in his research notebooks, which offer a much more complete picture of the development of wave mechanics than has been previously thought possible. Key words: optical-mechanical analogy, Hamilton, W. R., Schrödinger, E., quantum mechanics, wave mechanics 1. The roots of wave mechanics The genesis of wave mechanics has been treated by many authors. In a first stage, these studies relied mostly on Erwin Schrödinger’s published works and reminiscences of his colleagues (Klein, 1964; Gerber, 1969; Kubli, 1970). These accounts have been substantially revised by ✩ The authors are members of the Project on the History and Foundations of Quantum Physics, a collaboration of the Max Planck Institute for the History of Science and the Fritz Haber Institute of the Max Planck Society in Berlin. This paper grew out of a collaboration with Jürgen Renn on the roots of wave mechanics, whom we thank for numerous discussions and significant advice. The authors would also like to thank Massimiliano Badino, Jed Buchwald, and Jesper Lützen for their help; Michel Janssen for detailed comments on the draft of this paper; and especially Ruth and Arnulf Braunizer for permission to quote from the unpublished writings of Erwin Schrödinger and for their hospitality. Christian Joas acknowledges support by a grant-in-aid from the Friends of the Center for History of Physics, American Institute of Physics. Email addresses: cjoas@mpiwg-berlin.mpg.de (Christian Joas), lehner@mpiwg-berlin.mpg.de (Christoph Lehner) Preprint submitted to Studies in History and Philosophy of Modern Physics June 16, 2009 historians who also considered the existing correspondence (Raman and Forman, 1969; Hanle, 1971, 1977, 1979; Wessels, 1979). The various authors have drawn different conclusions about the roots, the trajectory, and the goals of Schrödinger’s project on wave mechanics, which we will discuss in more detail below. However, in the years from 1925–1927, Schrödinger wrote, besides his well-known four communications on wave mechanics (Schrödinger, 1926b,c,e,f) and several other relevant publications (e. g., Schrödinger, 1926d) dozens of notebooks comprising hundreds of pages. These notebooks are an obvious source for a more detailed understanding of his work and his ambitions in the years of the creation of wave mechanics. Schrödinger’s notebooks have been discussed by Kragh (1982, 1984) and Mehra and Rechenberg (1987a,b). Both found various tantalizing passages from the notebooks relevant for the discussion. However, both limit themselves to the study of a small set of notebooks with immediate relevance to the roots of the wave equation. Mehra and Rechenberg give extensive summaries of these notebooks but do not attempt to use them for a coherent picture of the genesis and development of wave mechanics. We have studied a larger set of 27 notebooks contained in the AHQP that we identified as possibly stemming from the period of the development of wave mechanics, as well as five earlier notebooks and manuscripts important for the prehistory of Schrödinger’s program.1 In Fig. 1, we present a list of all studied notebooks.2 Analyzing this larger set of notebooks, we arrive at a rather coherent picture of Schrödinger’s motivations and thought development through the creation of wave mechanics. Therefore, we do not agree with the assessment that “Schrödinger has left few traces of how his ideas evolved as he worked towards wave mechanics” (Wessels, 1979), which is frequently found in the older literature on the subject. In the existing literature, one can roughly distinguish three characterizations of the roots of Schrödinger’s wave equation. Already Klein (1964), and later Hanle (1971), pointed out that an important root of the development of wave mechanics is Schrödinger’s interest in gas statistics in 1924–1925. It was in this context that Schrödinger encountered Albert Einstein’s paper “Quantum theory of the monatomic ideal gas” (Einstein, 1924) which used Bose statistics to derive a state function of the ideal gas. Einstein mentioned that Louis de Broglie’s idea of matter waves could help to understand the physical content of the Bose-Einstein statistics. In the fall of 1925, instigated by this remark, Schrödinger studied de Broglie’s thesis. In his paper “On Einstein’s gas theory” (Schrödinger, 1926a) he pointed out that the Bose-Einstein counting procedure, which seems rather ad hoc as a counting method for particles, can be understood as a straightforward Boltzmann counting method for standing wave modes (Schrödinger called this ”natural statistics”). This was the starting point in Schrödinger’s development of a wave equation for matter waves. Raman and Forman (1969) favor an alternative explanation for Schrödinger’s interest in de Broglie’s idea of matter waves. They point to a paper from 1922, “On a remarkable property of the quantum orbits of a single electron” (Schrödinger, 1922), which uses an argument similar to de Broglie’s to derive Bohr’s quantized orbits. Although Schrödinger’s argument did not arise 1 This was first pointed out by (Raman and Forman, 1969). These notes are excerpted in (Mehra and Rechenberg, 1987a,b). 2 The notebooks are reproduced in the Archive for the History of Quantum Physics (AHQP) available on microfilm in several institutions (original repository: American Philosophical Society, Philadelphia). They will be quoted by their AHQP reel and document numbers, as shown in Fig. 1. The AHQP dating is incorrect for some of the notebooks, especially for the early and later ones. However, since the evidence is rather ambiguous in many cases, we will not attempt to give dates for all the notebooks in the list, but will only discuss in the text the dating for the notebooks that we treat in more detail. 2 AHQP 39-3-001 39-3-002 39-3-003 39-3-004 39-3-005 40-5-001 40-5-002 40-5-003 40-6-001 40-6-002 40-7-001 40-7-002 40-7-003 40-8-001 40-9-001 41-1-001 41-1-002 41-1-003 41-1-004 41-2-001 41-2-002 41-2-003 41-2-004 41-2-005 41-4-001 41-4-002 41-4-003 41-4-004 41-4-005 41-4-006 41-5-001 41-5-002 Title Tensoranalytische Mechanik I Tensoranalytische Mechanik II Tensoranalytische Mechanik III & Optik inhomogener Medien Hertz’sche Mechanik und Einstein’sche Gravitationstheorie [untitled manuscript draft connected to the previous one] [unidentified notes on electromagnetic(?) waves] H-Atom — Eigenschwingungen Eigenwertproblem des Atoms. I. Starkeffekt. Eigenwertproblem des Atoms. II. (Allgemeine Theorie) (Starkeffekt) (Störungstheorie) Starkeffekt fortgesetzt Summen von Binomialen. Eigenwertproblem d. Atoms. III. Zeeman- und Relativität als Störungen [. . . ] Berechnung der Hermiteschen Orthogonalfunktionen [loose notes on intensities, i. e., transition probabilities] Intensitätsberechnung für den Starkeffekt Einstein-Planck-Statistik [crossed out] Dirac. Nebenbedingungen. [. . . ] Elektronendrall!!! Undulatorische Statistik I. Undulatorische Statistik II. Kugelmittelmethode auf Dirac- und Gordonglg angewendet Intensitäten. Parallele zu Heysenberg und Lanczos Die schwebenden Fragen. [. . . ] Zu unscharfe Spektren Streckenspektrum, Intensitäten 1.) Studien über Integralgleichungen und Kerne. [. . . ] Koppelung. Ganz alt. Allgemeine Dispersionstheorie, Koppelung II. [. . . ] Koppelung mit dem Strahlungsfeld [. . . ] 1.) Dispersion und Resonanz 2.) Berlin, Juni 1928 Berlin 1928 Funkenwahrscheinlichkeit Darstellungen Zur Abel’schen Integralglng Figure 1: Possibly relevant notebooks from Schrödinger’s nachlass. First column: AHQP numbering (reel-section-item); second column: title. in a context of matter waves but rather was inspired by his study of Weyl’s unified field theory, Raman and Forman argue that the formal parallel between de Broglie’s and Schrödinger’s own work made him receptive to de Broglie’s ideas. Even though Raman and Forman are able to support this argument by some elements from Schrödinger’s correspondence, there is no evidence for a continuing interest of Schrödinger’s along the lines of this paper in the years 1922–1925. Wessels (1979), Kragh (1982), as well as Mehra and Rechenberg (1987a,b) consider a third explanation for Schrödinger’s interest in de Broglie: De Broglie’s use of the formal analogy between Fermat’s principle for ray optics and Maupertuis’ principle for corpuscular mechanics appealed to Schrödinger because of his own explorations of Hamiltonian mechanics around 1920 which had led him to study Hamilton’s optical-mechanical analogy. Three notebooks entitled “Tensor-analytical mechanics”, tentatively dated to 1918–1922,3 deal with an extension of classical mechanics inspired by Einstein’s recently published theory of general relativity and 3 There is very little hard evidence for the dating of these notebooks. For the lower bound we use Schrödinger’s mention of (Weyl, 1918) in the second notebook, for the upper bound the assumption that the notebooks were written before (Schrödinger, 1922), which uses Weyl’s gauge theory that is not mentioned in the notebooks. 3 by Heinrich Hertz’s reformulation of classical mechanics in differential geometrical form, presented in (Hertz, 1894). In the second of these notebooks, Schrödinger explored Hamilton’s optical-mechanical analogy in a section entitled “Analogies to optics” (see Fig. 3).4 The opticalmechanical analogy was prominently used by Schrödinger in his second communication on wave mechanics as a heuristic justification, but there has been considerable debate how much of a role it played in the actual process of discovery. In this paper, we want to show that all these seemingly disparate roots of Schrödinger’s interest in wave mechanics are substantially interconnected in his thinking, extending Wessels’ observation that beyond the concrete questions of gas statistics and atomic theory, “Schrödinger’s work was shaped by his interest in foundational and conceptual matters and his concern over physical significance and physical models.” (Wessels, 1979, p. 313) From the study of Schrödinger’s notebooks, it becomes clear that the various roots all were different facets of his longstanding interest in extending classical mechanics, inspired equally by 19th-century analytical mechanics, Boltzmann’s statistical mechanics, Einstein’s General Relativity, and the problem of a physical understanding of the old quantum theory. The Hamiltonian analogy became the most fruitful piece in this complex network of theoretical speculation. During the development of wave mechanics, for Schrödinger both the content and the role of the analogy changed considerably. We will trace these developments through Schrödinger’s notebooks from the time of the First World War to 1926. In Section 2, we give a brief characterization of Hamilton’s analogy and its reception before wave mechanics. Section 3 studies Schrödinger’s exploration of the Hamiltonian analogy in the context of his work on analytical mechanics in the years 1918–1922. Section 4 discusses the heuristic role the analogy played in the first derivation of the wave equation. This also throws a new light on the old question why Schrödinger did not refer to the Hamiltonian analogy in his first communication on wave mechanics even though it played a role in his thinking. The transformation of the optical-mechanical analogy from heuristic tool to formal constraint in the context of Schrödinger’s search for a relativistic wave equation is treated in Section 5. We show from the notebooks that Schrödinger only reached his mature understanding of the optical-mechanical analogy—implying that classical mechanics is only a limiting case of a more general wave mechanics—in the course of preparing the second communication. The role of the optical-mechanical analogy for Schrödinger’s interpretation of quantum mechanics is discussed in Section 6. There is an intriguing similarity in the development of Schrödinger’s use of the Hamiltonian analogy to the complex history of Einstein’s use of the principle of equivalence in General Relativity. Also for Einstein, the thought that inertia and gravity are intimately connected played a heuristic role at first: it led him to accept the necessity of using a curved space-time for the relativistic theory of gravitation (Stachel, 1989). Only once he had the basic formal structure of Riemannian geometry in place he could use the principle of equivalence as a formal constraint on possible field equations (Renn and Sauer, 2007).5 4 The notebooks are also mentioned by Raman and Forman (1969), but not considered as very important in their argument. 5 See M. Janssen, ‘No success like failure...’: Einstein’s quest for general relativity, 1907–1920. In M. Janssen and C. Lehner (Eds.), The Cambridge Companion to Einstein, forthcoming. Here, also Einstein’s later use of the equivalence principle as an interpretational device is discussed. 4 2. Hamilton’s optical-mechanical analogy The optical-mechanical analogy goes back to William Rowan Hamilton (Hamilton, 1833).6 It was mostly ignored during the 19th century but reached considerable prominence with the development of wave mechanics. In this section, we will consider its original formulation and the history of its reception through the 19th century. Hamilton’s early work was mostly devoted to ray optics. In the 1820s, neither the corpuscular theory of light nor the wave theory were unanimously accepted. Hamilton himself became a vigorous defender of wave optics, yet in his work on geometrical optics he was not concerned with questions about the nature of light but treated ray optics as a purely geometrical problem. Hamilton’s treatment starts from a generalization of Malus’s theorem: for any bundle of light rays emitted from a point, there will be a family of surfaces so that all light rays are orthogonal to these surfaces. Hamilton shows that Malus’s theorem holds in full generality also for inhomogeneous and anisotropic media, and that the family of surfaces can be described by a potential function, the characteristic function (i. e., the surfaces are the surfaces of constant value of the characteristic function). This function can be found through solving a partial differential equation, which will be of central importance for our story. We will call it here, with Schrödinger, Hamilton’s partial differential equation. The characteristic function gives a complete description of an optical system of rays, such as in an optical apparatus. Thus, Hamilton is able to formulate a theory of ray optics that is as general as the Lagrange theory of mechanics: Those who have meditated on the beauty and utility, in theoretical mechanics, of the general method of Lagrange [. . . ] must feel that mathematical optics can only then attain a coordinate rank with mathematical mechanics, or with dynamical astronomy, in beauty, power, and harmony, when it shall possess an appropriate method, and become the unfolding of a central idea. (Hamilton, 1833, p. 315) As one of the chief merits of his method Hamilton saw the fact that the same formal theory applies to the geometry of light rays irrespectively of whether one considers light to consist of particles (obeying a principle of least action) or of waves (obeying Huygens’ envelope theory).7 While the consistency of ray optics with the emission theory of light is immediately plausible (light rays are simply the paths of the light particles), Hamilton showed that his theory of the characteristic function can also be understood from a wave theory of light (Hamilton, 1837). In the wave theory, the fundamental physical entities are not the rays but the surfaces of constant value of the characteristic function: they define the wave fronts of the light wave. Hamilton showed how Huygens’ construction of successive wave fronts also in this case leads to Hamilton’s partial differential equation, thus demonstrating the validity of his approach also in a wave theory of light. This, however, is only approximately true: At about the same time, Augustin Jean Fresnel began to argue that certain phenomena which could be described by ray optics only with the help of cumbersome additional assumptions were explainable much more straightforwardly 6 A detailed account of Hamilton’s derivation of the analogy can be found in Hankins’ biography of Hamilton (Hankins, 1980), which also treats the impact of the analogy on later physics and especially the development of wave mechanics. 7 “Whether we adopt the Newtonian or the Huygenian, or any other physical theory, for the explanation of the laws that regulate the lines of luminous or visual communication, we may regard these laws themselves, and the properties and relations of these linear paths of light, as an important separate study, and as constituting a separate science, called often mathematical optics.”(Hamilton, 1833, p. 314) 5 in terms of wave optics. This implies that wave optics is more general than ray optics and that the two are not equivalent.8 Hamilton does not seem to comment on this issue and it is not clear to us whether he was aware of this limitation to his “general method.” In the following decades, the wave theory of light quickly became universally accepted and Hamilton’s geometrical optics faded into oblivion even though it was the most general formulation of geometrical optics. In 1895, Heinrich Bruns rederived independently important parts of Hamilton’s optics in his theory of the eikonal (Bruns, 1895). He pointed out the close formal analogy between the eikonal in geometrical optics and the Hamiltonian action integral in mechanics without realizing that Hamilton had arrived at Hamiltonian mechanics through this analogy. A clear presentation of the Hamiltonian formulation of ray optics as the limiting case of a wave equation was given by Sommerfeld and Runge (1911, p. 290), who credit Peter Debye with the idea for the derivation. Only with Einstein’s light quantum paper (Einstein, 1905) did the corpuscular theory of light reemerge, and only at about the time of the genesis of wave mechanics did the question of the nature of light again become as open as in the early 19th century. What does all this have to do with mechanics? In direct continuation of his work on optics, Hamilton announced in 1833 that Lagrangian mechanics itself could be formulated in a way that is mathematically equivalent to his theory of ray optics. Hamilton was able to show that both optics and mechanics obey the same variational principle for a characteristic function, which in both cases can be determined by solving Hamilton’s partial differential equation. In mechanics, Hamilton’s partial differential equation nowadays is more commonly known as the HamiltonJacobi equation.9 It is a partial differential equation for the characteristic function S (qk , E), a function of the coordinates qk of the endpoint of the trajectory and the total energy E: ! ∂S = E. H qk , ∂qk Here, H is the Hamiltonian function (the energy expressed as a function of positions qk and momenta pk ) where for the momentum pk = ∂S /∂qk has been substituted. Hamilton’s characteristic function is a generalization of the integral that Maupertuis had called ‘action’ and that had to be minimized in the least action principle. Therefore, Fermat’s principle of the shortest time and the mechanical principle of least action are just specific formulations of Hamilton’s more general principle (see Fig.2). The only fundamental difference between optics and mechanics in this scheme lies in the interpretation of the integrand of the characteristic function: In mechanics, it represents the particle momentum while in optics, it represents the inverse of the phase velocity. Thus, in both cases, the action integral can be found by solving Hamilton’s partial differential equation, which therefore offers a most general method of solving dynamical problems. This constitutes the optical-mechanical analogy. The optical-mechanical analogy led Hamilton to his formulation of mechanics nowadays known as Hamiltonian Mechanics. The dynamics of a system can be entirely derived from the knowledge of a single characteristic function: By this view the research of the most complicated orbits, in lunar, planetary, and sidereal astronomy, is reduced to the study of the properties of a single function V; 8 For the history of the dispute about wave optics in the early 19th century see (Buchwald, 1989; Buchwald and Smith, 2001). 9 It is not to be confused with Hamilton’s equations ∂H = − ṗ, ∂H = q̇. See, e. g., (Goldstein, 1950) for a stan∂q ∂p dard treatment of Hamiltonian mechanics. Hamilton’s original formulation consisted of a set of two partial differential equations, one for the starting point and one for the endpoint of the trajectory (Hankins, 1980, p. 196). 6 Optics: Mechanics: Characteristic function is time of propagation T : Characteristic function is action intergral S: T= Z n ds = 0 c S = n refractive index, c light velocity Integrand is inverse phase velocity 1/u: Z p 2m(E − U)ds = 0 m mass, E − U kinetic energy Integrand is particle momentum p: p p = 2m(E − U) 1 n = u c Fermat’s principle: Maupertuis’s principle of least action: δT = 0 δS = 0 This implies: Light rays are orthogonal to surfaces of equal time T (wave fronts). This implies: Particle trajectories are orthogonal to surfaces of equal action S . Figure 2: Hamilton’s optical-mechanical analogy: The two different interpretations of Hamilton’s characteristic function in optics and mechanics. which is analogous to my optical function, and represents the action of the system from one position to another. (Hamilton, 1833, p. 332) The unifying potential of Hamiltonian Mechanics was acknowledged and fueled widely-held beliefs that all of science was eventually to be based on variational principles such as the principle of least action. However, it only had limited practical impact on 19th-century physics.10 Hamiltonian mechanics was extended by the mathematician Carl Gustav Jacob Jacobi and became in this form an important tool for celestial mechanics. The root of Hamiltonian mechanics in his optics, on the other hand, was mostly forgotten outside of Great Britain. One of the few exceptions was Felix Klein who already had discussed Hamilton’s work in optics in 1891 (Klein, 1892). When, as mentioned above, Heinrich Bruns noticed the similarity between his theory of the eikonal and Hamiltonian mechanics, Klein (1901) pointed out that this similarity reflected the historical roots of Hamilton’s theory. Klein’s own lecture notes on the derivation of Jacobian theory from optical considerations were only available in manuscript at the Göttingen library (Hankins, 1980, p. 204). One of the few people that took notice of them was Klein’s assistant Arnold Sommerfeld who wrote the above-mentioned paper with Iris Runge (Sommerfeld and Runge, 1911) and years later, in 1926, would point out the history of the 10 For the history of the reception of the optical-mechanical analogy in the 19th and early 20th century, see (Hankins, 1980, Chapter 14) 7 optical-mechanical analogy to Erwin Schrödinger who acknowledged Klein’s role in a footnote to his second communication on wave mechanics (Schrödinger, 1926c, p.490). Another thread in the reception of Hamiltonian mechanics in the 19th century, which also would become important for Schrödinger’s development, is the tradition of the geometrization of mechanics, starting with Jacobi’s elimination of time from the action integral which rendered the problem purely geometrical. The further history is intimately connected with differential geometry of variously curved spaces as developed by Gauss and Riemann.11 Already Gauss had shown that a statement analogous to Malus’s theorem (of the existence of surfaces orthogonal to light rays) holds for geodesics in a curved space. Beginning with Liouville, Lipschitz, and Darboux, French mathematicians realized that Hamilton-Jacobi mechanics could generally be understood as a theory of geodesics in a higher-dimensional space with variable curvature given by the kinetic energy. This approach was one of the roots of Heinrich Hertz’s analytical mechanics. However, Hertz went further and eliminated the concept of force altogether by introducing hidden masses and mechanical constraints (Bindungen) which connected the visible matter to these hidden masses. Therefore, any mechanical motion is a free motion in the manifold defined by the mechanical constraints. Hertzian mechanics can thus be seen as an attempt to geometrize all forces in the way that General Relativity would later geometrize the gravitational force. 3. Schrödinger’s work on analytical mechanics Schrödinger, in two unpublished manuscripts,12 attempted to develop the Hertzian idea of a geometrization of mechanics in a hope to connect it to Einstein’s general theory of relativity. In the first chapter of the manuscript, Schrödinger gives a programmatic introduction in which he poses the problem to find an intuitive understanding of general relativity and proposes to search for it by studying the relation between General Relativity and Hertzian mechanics. He writes: Given these manifest internal connections between Hertzian mechanics and Einstein’s General Relativity, it is hard for me not to attribute a deeper meaning to the circumstance that in both cases the “forces” make their entrance in the same mathematical garb, namely as the Riemann-Christoffel three-indices symbols of a quadratic form of the coordinate differentials.13 However, the manuscript breaks off and it does not become clear where Schrödinger hoped to find the deeper connection between Hertzian mechanics and General Relativity. In the same context Schrödinger wrote three notebooks entitled “Tensor-analytical mechanics” in which he explored the Hertzian formulation of mechanics.14 The notebooks contain 11 This story is treated in detail by Jesper Lützen’s book on Hertzian mechanics (Lützen, 2005). two manuscripts really form two parts of a continuing whole. Manuscript AHQP 39-3-004 seems to predate manuscript AHQP 39-3-005 but was later marked as chapter two. Manuscript 39-3-005 comprises chapters 1 and 3. Manuscript 39-3-004 is entitled “Hertzian mechanics and Einstein’s theory of gravitation,” but that title is crossed out, presumably when Schrödinger assigned it as chapter two for the extended manuscript. The manuscripts are discussed in (Mehra and Rechenberg, 1987a,b). 13 Schrödinger, AHQP 39-3-005, emphasis in the original (underlined). Both in General Relativity and in Hertzian mechanics, the Christoffel symbols are determined by the noneuclidean metric, and they in turn determine the actual trajectories. 14 AHQP 39-3-001, 39-3-002, 39-3-003. Mehra and Rechenberg (1987b, p. 529) assume that the two manuscripts (AHQP 39-3-004, 39-3-005) predate the three notebooks and that the notebooks are an “immediate continuation” of the manuscript. However, it is just as plausible to speculate that the manuscripts grew out of the more general considerations of the notebooks. 12 The 8 various speculations about a representation of mechanics in differential geometrical form. Schrödinger uses elements of Hertzian mechanics, Liouville’s geometric representation of mechanics, and General Relativity. He explores various connections between these ideas, without coming to clear conclusions. One line of thought, for instance, is the attempt to explain the rest mass of matter as the kinetic energy of a hidden motion in additional dimensions. Just as in the manuscripts, however, the fundamental motivation for Schrödinger’s explorations seems to be the search for a common root of general relativity and the noneuclidean representation of analytical mechanics, with the hope of a better physical understanding and possible extension of both. Especially noteworthy is that Schrödinger, in the first notebook, tried to explain quantum conditions as constraints (Bindungen) in the sense of Hertzian mechanics. In the Hertzian picture, as mentioned before, forces between particles are derived from constraints in a higher-dimensional space. These constraints determine the noneuclidean configuration space of the particles in which the actual trajectories are geodesics. Thus, forces have been completely eliminated. Since quantum conditions can also be seen as constraints in phase space, it seems that Schrödinger’s hope was that he could reformulate the old quantum theory in such a way that both forces and quantum conditions would be represented by constraints in a higher-dimensional space. This would have led to a unification of the concept of force and the concept of a quantum condition and thus to a physical understanding of the nonmechanical quantum conditions of the old quantum theory. In the second notebook (AHQP 39-3-002), Schrödinger starts by analyzing the concept of curvature in a Riemannian space. In this context, he summarizes the interpretation of curvature in terms of parallel displacement, referring to (Weyl, 1918). Schrödinger continues with an attempt to replace motion under influence of a force in flat space-time by constrained (geodesic) motion in a curved higher-dimensional space. He asks himself how the metric tensor has to be defined to this end. Schrödinger’s hope is that he can calculate the metric as a function of a scalar potential V and a total energy E so that the actual trajectories follow from a principle of least constraint. He realizes that he cannot directly calculate the metric since it depends on the motion itself. Schrödinger writes: The metric tensor for the (q, t)-space can not be given for the general case. It depends on the motion itself. One has to solve Hamilton’s partial differential equation first, in a certain sense as a field equation, one has to pick a suitable solution—and only then can one say: The trajectory of the system is an extremum of this field.15 Especially noteworthy in the light of what was to come is the idea expressed here that Hamilton’s partial differential equation can be interpreted as a field equation. Thus the solution of Hamilton’s partial differential equation determines an “action field” that in turn determines—through an extremum principle—the trajectories that solve the mechanical problem. This means that the “field” expressed by the wave fronts is ontologically prior to the trajectories whose shape is determined by this field. These considerations led Schrödinger to a more extensive inquiry into the geometric interpretation of Hamilton’s partial differential equation and, especially, Hamilton’s optical-mechanical analogy in the third notebook (AHQP 39-3-003). The second part of the notebook is entitled “Analogies to optics. Huygens’ principle and Hamilton’s partial differential equation” (see Fig. 3). After juxtaposing the mechanical principle of least action and the optical principle of least time, Schrödinger explores how Huygens’ principle is used to construct surfaces of equal 15 ibid., emphasis in the original (underlined). 9 Figure 3: Part of a page from Schrödinger’s third notebook on tensor-analytical mechanics (AHQP 39-3-003), tentatively dated to 1918–1922, with a discussion of Hamilton’s optical-mechanical analogy. The page shows Schrödinger’s parallel treatment of the variational principles of mechanics (left column) and optics (right column). time in an optical medium and how rays are constructed as orthogonals of such surfaces of equal time. The rays constructed in this way fulfill an extremum principle, the principle of shortest time. After some further explorations of geometrical optics follows a section entitled “Direct transfer to mechanics.” Here, Schrödinger applies the construction to the case of mechanics by replacing the time integral along the optical ray with the action integral along the trajectory. He obtains Hamilton’s partial differential equation and the Lagrangian equations of motion. This section does not contain any attempts at an extension of classical mechanics. However, it proves Schrödinger’s acquaintance with—and constructive application of—Hamilton’s optical-mechanical analogy as early as 1918–1922. The study of Schrödinger’s attempts to generalize mechanics sheds new light on the “Remarkable property” paper (Schrödinger, 1922). Here, as in the remark on Weyl in the notebooks quoted above, Schrödinger is concerned with the parallel displacement of a vector in curved 10 space, and as in the notebooks he is looking for a physical justification for quantum conditions. The paper is written in the context of Weyl’s gauge theory whereas the notebooks only rely on Weyl’s representation of General Relativity.16 Nevertheless, the similarity of intentions and outlook in (Schrödinger, 1922) and in the earlier notebooks is obvious. Therefore, the “Remarkable property” paper can be seen as an offspring of Schrödinger’s much more general program of extending mechanics and not just, as Raman and Forman (1969) see it, as a singular piece of evidence for Schrödinger’s developing interest in atomic physics in the early 1920s. As we will argue in the next section, it is this more general program that later would make Schrödinger receptive to de Broglie’s ideas and that eventually would be transformed into his research program in wave mechanics. 4. The genesis of wave mechanics: Hamilton’s analogy as a heuristic tool When de Broglie proposed the idea of matter waves, he explicitly used the analogy between ray optics and classical mechanics to justify his proposal.17 However, he seems unaware of Hamilton’s formulation of the optical-mechanical analogy. De Broglie represented the analogy as the formal equivalence of Fermat’s principle of the shortest path and Maupertuis’s principle of least action. It is remarkable that de Broglie had already arrived, from relativistic considerations, at the conclusion that the phase velocity of the matter wave is inversely proportional to the velocity of the corresponding particle, which exactly matches the relation in the optical-mechanical analogy (see Fig. 2). Schrödinger learned from Einstein about de Broglie’s idea in the context of gas statistics.18 Schrödinger’s interest in this subject grew out of his devotion to a Boltzmannian program of statistical mechanics. In 1924, Schrödinger investigated Planck’s derivation of the Sackur-Tetrode quantum theory of the ideal gas which had been criticized by various authors, foremost Paul Ehrenfest, as being ad hoc. The essential point of Ehrenfest’s criticism was Planck’s division of the state function by a factor N! in order to make entropy an extensive quantity. Planck justified this division with the observation that, without this division, permutations of identical particles would be counted separately. Ehrenfest countered that Planck’s counting procedure was mathematically incorrect and that entropy need not be an extensive quantity in any case. Schrödinger was already involved in this discussion when Einstein published his paper “Quantum theory of the monatomic ideal gas” (Einstein, 1924) which used Bose statistics to derive a state function of the ideal gas. Einstein had derived the density fluctuations of an ideal gas in their theory and shown that they displayed the same duality of a “wave term” and a “particle term” as the density fluctuations of blackbody radiation.19 He quoted de Broglie’s dissertation as a promising approach to understand this mysterious duality (Broglie, 1924). In a paper dated September 4, 1925, Schrödinger weighed the various definitions of entropy (Schrödinger, 1925). He argued that the Bose-Einstein statistics is the correct way to achieve what Planck had tried to, i. e., taking into account the identity of particles. But he was unsatisfied with the fact that Bose’s counting of cases seemed physically unnatural.20 Only when 16 Weyl’s gauge theory was not included in (Weyl, 1918) but only in the third edition of 1919. Schrödinger (1922) cites the fourth edition of 1921. It is not clear which edition the reference in the notebook AHQP 39-3-002 is to. 17 For a historical account of de Broglie’s theory see e. g. (Jammer, 1966; Kubli, 1970; Darrigol, 1986). 18 For Schrödinger’s work on gas statistics and its role for the development of wave mechanics, see especially (Hanle, 1975, 1977). 19 For an account of Einstein’s treatment of the wave-particle duality see (Klein, 1964). 20 See (Howard, 1990) for the debate about the statistical correlations in Bose-Einstein statistics. 11 Schrödinger read de Broglie’s thesis in the fall of 1925 did the idea of matter waves help him to see light in the quantum theory of the ideal gas: On December 15, 1925, he submitted his paper “On Einstein’s gas theory.” In this paper, Schrödinger wrote: A natural feeling rightfully objects against considering this new [Bose-Einstein] statistics as something primary, not amenable to further explanation. (Schrödinger, 1926a, p. 95) Schrödinger argued that matter waves offer a natural way to give a physical understanding to the otherwise mysterious Bose counting procedure that Einstein had applied to the ideal gas. Understanding particles as wave modes explains the indistinguishability of particles that is characteristic for the Bose-Einstein statistics. Unlike his contemporaries who were willing to accept the existence of a statistics sui generis for microscopic particles, Schrödinger saw the fact that the wave picture would lead back to a classical Boltzmann statistics as a strong indication for the correctness of the wave picture of matter. Therefore, the success of Einstein’s gas theory was for him a strong argument to get serious about the de Broglie-Einstein undulatory theory of the moving particle, according to which the latter is nothing but a kind of ‘crest’ on a wave radiation forming the substratum of the world. (Schrödinger, 1926a, p. 95) Schrödinger’s search for a “natural statistics” reflects his descriptive approach to statistical mechanics, which was strongly influenced by Boltzmann’s work (Scott, 1967). Wessels (1983) argues that Schrödinger’s commitment to intuitive pictures of reality does not presuppose a strong commitment to ontological realism. However, as the quote from “On Einstein’s gas theory” above shows, in the case of statistical mechanics, Schrödinger seems committed to a realist interpretation: Statistical mechanics is not merely a phenomenological theory, which can be formulated quite independently of the nature of the underlying mechanism. Rather, statistical behavior reveals the true structure of the building blocks of the physical world. Given this approach to statistical mechanics, there is a close connection between Schrödinger’s statistical arguments from “On Einstein’s gas theory” and his interest in the foundations of mechanics. De Broglie had argued in his thesis that quantum conditions can be explained by the requirement that there be an integer number of periods of the pilot wave along an orbit. When reading de Broglie’s thesis, Schrödinger realized the similarity between de Broglie’s argument and his own argument from 1922 that quantum conditions can be replaced by the requirement that the parallel displacement of Weyl’s gauge factor along the orbit be a multiple of a constant number.21 However, much more fundamental than this formal similarity is that de Broglie’s explicit use of the optical-mechanical analogy as a heuristic argument for the wave representation of matter must have resonated strongly with Schrödinger. After all, he had pondered the opticalmechanical analogy with a similar motivation of extending classical mechanics himself. But as we will show, the role of the optical-mechanical analogy was not limited to making Schrödinger receptive to de Broglie’s ideas, it also played an important role in his development of these ideas. Schrödinger quickly “got serious” about de Broglie’s idea and, probably during his Christmas vacation in 1925, tried to formulate a wave equation to give an exact description of de Broglie’s 21 Schrödinger remarked on this connection in his letter to Albert Einstein dated November 3, 1925 (see Moore, 1989, p. 191–192). The letter dates Schrödinger’s reading of de Broglie quite precisely: He states that he read the dissertation “a few days ago.” 12 matter waves.22 He quickly realized that this wave equation made good on the program already stated in his notebooks on tensor-analytical mechanics: it offered a physical explanation for quantization conditions. The discrete quantum levels could now be explained as the discrete solutions resulting from the eigenvalue problem of the wave equation, just as a vibrating body can only oscillate in specific discrete modes. Thus, the discrete orbits of the Bohr atom could be understood as discrete wave modes of the electron waves. A later reminiscence attributes the idea to derive a wave equation to a remark made by Debye when Schrödinger presented de Broglie’s dissertation in a Zürich colloquium (Bloch, 1976). Such a hint, however, seems unnecessary given Schrödinger’s firm acquaintance with classical wave theory evident from the lecture notes he took as a student in Vienna (AHQP, reel 39). Schrödinger himself prominently used the optical-mechanical analogy as a heuristic argument for the derivation of a wave equation for matter in his second communication on wave mechanics (Schrödinger, 1926c). Does this imply that it was the Hamiltonian analogy that led him to the wave equation? Then, however, it is striking that the first communication (Schrödinger, 1926b) does not mention the optical-mechanical analogy and derives the wave equation from an ad hoc variational principle: Schrödinger starts from Hamilton’s partial differential equation (Eq. (1))23 ! ∂S =E H q, ∂q and, for the action, substitutes (Eq. (2)) S = K log ψ. K is a constant with the dimension of an action which Schrödinger in the course of the paper shows to be equal to h/2π. The variable ψ, for the modern reader easily identifiable with the wave function, is introduced without any physical interpretation.24 With this substitution, Eq. (1) would be a differential equation for ψ. Instead of solving the differential equation H − E = 0, Schrödinger postulates without further justification to replace it by the variational principle (Eq. (3)) Z 0 = δ d3 r (H − E)  ! !2 !2 !  Z 2  ∂ψ 2  e 2m ∂ψ ∂ψ 3  = δ d r  + + − 2 E+ ψ2  . ∂x ∂y ∂z r K This leads to the differential equation (Eq. (5)) ! e2 2m ψ=0 ∆ψ + 2 E + r K 22 There is some debate in the literature about the exact date of Schrödinger’s first attempt, which is recapitulated in (Kragh, 1982), but as Kragh shows, the possible time frame is rather short, between Schrödinger’s letter to Einstein from November 3 (see above) and Schrödinger’s letter to Wien on December 27, see below. 23 The equation numbers are the numbers from (Schrödinger, 1926b). 24 Only at the end of the paper, Schrödinger writes: “Obviously, it seems very natural to relate the function ψ to an oscillatory process in the atom to which reality can be attributed to a higher degree than to the electron orbits whose reality is nowadays frequently doubted.” (Schrödinger, 1926b, p. 372) However, he states that he chose to abstain from an intuitive argument in this paper for the sake of bringing out the essence of his argument: the fact that the discreteness postulated by the quantum conditions can be explained as the discreteness of the solutions of a differential equation. 13 which today is known as the Schrödinger equation. The absence of any reference to the Hamiltonian analogy in this derivation led to the assumption that Schrödinger might not even have known about it at the time he wrote his first communication.25 As discussed in Section 3 and first pointed out by Raman and Forman (1969), Schrödinger was certainly well aware of the analogy since the late 1910s. This makes the omission of a reference to the optical-mechanical analogy in the first communication quite surprising. An extended discussion has arisen in the literature whether the optical-mechanical analogy played a role in the discovery of the wave equation or whether the succession of the communications reflects the fact that Schrödinger discovered the wave equation independently from the Hamiltonian analogy. Even without using the evidence from the earlier notebooks, Wessels (1979) doubts that the analogy played no role in Schrödinger’s thinking and distinguishes a heuristic role that the Hamiltonian analogy played for the discovery of the wave equation from the constructive role it played in the second communication. Kragh (1982) uses the evidence from the early notebooks to argue for the importance of the analogy in Schrödinger’s thinking and points out that Schrödinger mentions the “old Hamiltonian analogy” already in the notebook written for the first communication (AHQP 40-5-003) even though it doesn’t appear in the published text. On the other hand, he also recognizes that the earliest preserved notes on wave mechanics by Schrödinger (AHQP 40-5-002) do not arrive at the wave equation from the Hamiltonian analogy. It is of course not certain that these notes, entitled “Hydrogen atom. Eigenvibrations,” really represent Schrödinger’s first attempt to derive a wave equation. However, we do think that it is highly plausible for internal reasons: Schrödinger writes in a letter to Arnold Sommerfeld from Jan. 29, 1926, (Sommerfeld, 2004, pp. 236-238) that he recognized the coefficients from Sommerfeld’s solution of Hamilton’s partial differential equation for the relativistic Kepler problem in his own early calculations and that this gave him confidence that he was on the right track. AHQP 40-5-002 contains Sommerfeld’s coefficients, and therefore it is plausible that these are the calculations that Schrödinger refers to in the letter. Also Mehra and Rechenberg (1987b) study these notes in detail and come to the conclusion that they constitute Schrödinger’s first attempt at deriving a wave equation. They show that Schrödinger, without actually solving the relativistic wave equation he had found, was able to see that it would lead to noninteger quantum numbers different from Sommerfeld’s result. That was the reason that Schrödinger himself gave why he abandoned his first relativistic wave equation,26 today known as the Klein-Gordon equation. These circumstances make it seem very probable that AHQP 40-5-002 really was Schrödinger’s first attempt at a wave equation. But in these notes, Schrödinger makes no reference to the Hamiltonian analogy or a variational principle: he obtains the wave equation simply from inserting the known velocity of the de Broglie phase wave into a generic relativistic wave equation. Thus, the Hamiltonian analogy was not the way through which Schrödinger arrived at the wave equation originally, he rather used a very simple and straightforward abduction from de Broglie’s determination of the matter wave velocity. Nevertheless, Schrödinger turned to the optical-mechanical analogy soon after these notes. 25 Letter from Erwin Fues to Thomas S. Kuhn, Oct. 31, 1963, quoted in (Wessels, 1979). See also, e. g., (Hermann, 1977). 26 Letter to W. Yourgrau, February 29, 1956, quoted in (Yourgrau and Mandelstam, 1968, p. 113-114). At the time, Sommerfeld’s theory seemed well confirmed by the empirical facts, since it explained the observed fine structure of hydrogen. This agreement turned out to be coincidental since Sommerfeld, just like Schrödinger, had not taken into account electron spin. Sommerfeld’s theory, however, also used a classical expression for total angular momentum, instead of the quantum mechanical value. This canceled the error from the neglect of spin, while in Schrödinger’s case the problem became visible. See (Granovskii, 2004) for a modern account. 14 He mentions the analogy in AHQP 40-5-003, which was certainly written before the first communication, as Kragh (1982) points out. An even more substantial point about the role of the Hamiltonian analogy can be made from a thorough study of this notebook: We argue that the abstract variational principle presented in the first communication was found by Schrödinger through his occupation with the Hamiltonian analogy. The seemingly ad hoc variational principle is a stand-in for a more complex thought process motivated by the optical-mechanical analogy. AHQP 40-5-003 begins with the treatment of the nonrelativistic case, or more exactly with an approximation for particle velocities small compared to the velocity of light. Therefore it is plausible to assume that this notebook was written in direct continuation of the failed relativistic derivation of AHQP 40-5-002, trying out what would happen in the nonrelativistic limit. Schrödinger mentioned results from this section of the notebook in a letter to Wilhelm Wien dated Dec. 27, 1925 which implies that he must have started the notebook by that time. Among various rather preliminary calculations for the Stark and Zeeman effects, Schrödinger spends a considerable effort on finding a variational principle that would yield the wave equation, just as a variational principle in elasticity theory yields the corresponding wave equation. After several unsuccessful attempts at guessing the variational principle, there appears in the notebook a program that contains as its second item “the old Hamiltonian analogy between optics and mechanics”. The text under this heading exactly lays out the beginning of Schrödinger’s first communication that we summarized above and that mystified later commentators: Schrödinger starts from Hamilton’s partial differential equation and reinterprets it as a variational principle which indeed leads him to the (nonrelativistic) wave equation. The heading of the notebook entry shows that the invocation of the variational principle is not an alternative to an argument involving the Hamiltonian analogy but, quite to the contrary, is motivated by the analogy. This connection between the variational principle and the Hamiltonian analogy becomes clear if one considers Schrödinger’s previous work on tensor-analytical mechanics: As our quote (Section 3, p. 9) from his second notebook on tensor-analytical mechanics shows, Schrödinger had already thought about the interpretation of Hamilton’s partial differential equation as a field equation: Schrödinger had interpreted S as an “action field” which is given by Hamilton’s partial differential equation and in turn determines the particle trajectories through a variational principle δS = 0. This intuition became fruitful when he now tried to find a variational principle that would yield the wave equation for ψ. Because of the substitution S = K log ψ, the function ψ now stood for the action field of his earlier consideration. To find a variational principle that yields the wave equation for ψ, he replaced the field equation H − E = 0 given by Hamilton’s partial differential equation ∂ψ ∂x !2 + ∂ψ ∂y !2 + ∂ψ ∂z !2 − ! e2 2 2m ψ =0 E + r K2 R with the variational principle δ d3 r (H − E) = 0 or δ Z  ! !2 !2 !  2   ∂ψ 2 e 2m ∂ψ ∂ψ + + − 2 E+ ψ2  = 0. d r  ∂x ∂y ∂z r K 3 This means that he simply transposed the two steps of solving the field equation for S and solving the variational problem δS = 0 from his earlier construction. Of course, the permutation of the 15 two steps is in fact quite ad hoc:27 The variational principle does not follow in any strict sense from the Hamiltonian analogy and Schrödinger’s use of the analogy thus is purely heuristic. But in this heuristic role, the optical-mechanical analogy is of great importance to Schrödinger’s argument: it motivates the reinterpretation of Hamilton’s partial differential equation as the wave equation for a theory of matter waves. AHQP 40-5-003 displays how the interplay between heuristic ideas and formal analogies led Schrödinger to the formulation of a new theory out of the elements of Hamiltonian mechanics.28 This is how the notebook resolves the mystery of the introduction of the variational principle in Schrödinger’s first communication: Behind this seemingly ad hoc step stands Schrödinger’s exploration of Hamilton’s optical-mechanical analogy. 5. The search for a relativistic wave equation: Hamilton’s analogy as a formal constraint Already in the introduction entitled “The Hamiltonian analogy between mechanics and optics” to his second communication on wave mechanics, submitted less than a month after the first, Schrödinger discards the substitution S = K log ψ (Eq. (2) of the first communication) and writes: For the time being, we had described this connection [between the wave equation and Hamilton’s partial differential equation] only briefly in terms of its outward analytical structure by means of the per se unintelligible transformation (2) and by means of the likewise unintelligible transition from setting an expression to zero to the requirement that the space integral of that expression be stationary. (Schrödinger, 1926c, p. 489) He then moves on to present a new derivation that draws heavily on the Hamiltonian analogy. What made Schrödinger return to the optical-mechanical analogy? We argue in this section that the new derivation of the Schrödinger equation in his second communicaton arose in the context of unsuccessful attempts to find a new relativistic wave equation that would replace the one he had abandoned before the first communication. Schrödinger hoped that the relativistic equation would free him from the limitations of the nonrelativistic equation that he had presented in his first communication and would allow him to take into account, in a natural way, the interaction of the wave function and the electromagnetic field. He then would be able to explain the Zeeman effect and calculate the coefficients of emission and aborption of electromagnetic radiation within the wave-mechanical picture, thus going well beyond the old quantum theory and the rival matrix mechanics. The new derivation of wave mechanics presented in the second communication was the result of a fundamental advance in Schrödinger’s understanding and use of the optical-mechanical analogy. The crucial progress was the completion of Hamilton’s optical-mechanical analogy: Hamilton’s analogy consists of a formal relation between corpuscular mechanics and ray, not wave, optics. Thus, the wave equation for matter cannot be the formal analog of Hamilton’s 27 Nevertheless, the connection between the classical variational principle and Schrödinger’s variational principle for the wave function can be clarified formally. It can be shown that the classical principle is a limiting case to Schrödinger’s. See (Gray, Karl, and Novikov, 1999). 28 Renn and Sauer (2007) have discussed this interplay for the case of Einstein’s development of the theory of General Relativity and argue that it is a mechanism in the creation of new theories out of the elements of existing knowledge. 16 wave optics Schrödinger’s completion limit of short wavelength ray optics wave mechanics limit of short wavelength Hamilton’s analogy mechanics Figure 4: Schrödinger’s completion of the Hamiltonian analogy between optics and mechanics. partial differential equation. Rather, it has to transcend Hamilton’s partial differential equation in the same way as wave optics transcends ray optics, the latter being merely a limiting case of the former (see Fig. 4). Schrödinger is very explicit on this point in his second communication: Maybe our classical mechanics is the full analog of geometrical optics, and, as such, wrong, not in agreement with reality. It fails as soon as the radii of curvature and the dimensions of the trajectory are not large anymore compared to a certain wavelength, to which one can attribute a certain reality in q-space. In that case, one has to search for an “undulatory mechanics”—and the obvious way to this end is the wavetheoretical extension of Hamilton’s picture. (Schrödinger, 1926c, pp. 25) Until the present study of his notebooks, it has remained unclear whether Schrödinger possessed this knowledge already when he first derived a wave equation. Surprisingly, an examination of the notebooks shows that this insight only occurred to him after the completion of his first communication. Kragh (1982) only mentions the notebooks written for the preparation of the second communication in passing. Mehra and Rechenberg (1987b) present the notebooks in detail but do not treat the fundamental shift in Schrödinger’s thinking documented in the notebooks. The first notebook in which Schrödinger explicitly utilized the results of his first communication is entitled “Eigenvalue problem of the atom II” (AHQP 40-6-001). It begins with a section entitled “For the second communication” which lays out the Hamiltonian analogy. Mehra and Rechenberg (1987b, p. 549-550) claim that this notebook is the basis of Schrödinger’s treatment of the Hamiltonian analogy in the second communication on wave mechanics. Quite to the contrary, Schrödinger’s notes do not correspond at all to the content of the second communication. Rather, the notebook recapitulates the argument discussed in Section 4 that led him to the variational principle from the first communication. Schrödinger first sets out to “clear up the rather obscure relation between Hamilton’s partial differential equation (1) and the wave equation (5).”29 He claims again that the Schrödinger 29 The notebook contains references to the equations of the first communication using the exact same numbers as used in the published paper (see our summary of the argument on p. 13). In addition, a note added in proof to the first communication can be found verbatim in this notebook. Therefore, it is not doubtful, as Mehra and Rechenberg (1987b, p. 543) claim, that AHQP 40-6-001 was written after the first communication. 17 equation (5) simply is the wave equation that gives the wave fronts satisfying Hamilton’s partial differential equation. This is, however, only approximately true, as Schrödinger would state very cleary in the quote from the second communication above. The interesting observation is that Schrödinger does not yet realize this decisive restriction in the notebook: As in AHQP 40-5-003, he searches for a wave equation that will be exactly equivalent (and not just in the appropriate ‘ray optical’ limit) to Hamilton’s partial differential equation. Citing Whittaker,30 Schrödinger recapitulates Hamilton’s formal analogy between mechanics and optics and Hamilton’s construction of the surfaces of constant action through Huygens’ principle. He states again, even more explicitly, that it is obvious that the Schrödinger equation will give the desired surfaces of constant action: The construction of the family of surfaces required by the differential equation [i. e., Hamilton’s partial differential equation] is identical to Huygen’s principle. The orthogonal trajectories of the family are the extremum trajectories of Hamilton’s variational principle (in the time-free form of Maupertuis), they are the possible orbits of the mass point for the energy E. [. . . ] This, too, is clear, that ∆ψ + const (E − V) ψ = 0 (5) really describes a wave process satisfying the demands [i. e., that the wave fronts be surfaces of constant action] where ψ is the wave function and the equation is already independent of the time variable. Like Hamilton a century earlier, Schrödinger does not notice the difference between the construction of surfaces of constant phase according to Hamilton’s partial differential equation and a full wave process with varying amplitudes. Therefore he thinks it is “clear” that the wave equation (5) defines the classical surfaces of constant action. The following 19 pages contain several attempts to use the substitution S = K log ψ to derive a relativistic wave equation. Schrödinger’s first attempt starts from the form of Hamilton’s partial differential equation that contains time explicitly. He observes that it is nonsymmetrical in time and space coordinates. Therefore, he turns to the relativistic version of Hamilton’s partial differential equation, introducing canoncial momenta and the four-potential. He observes that now, instead of the gradient of the action, the four-gradient of the action minus the four-potential enters the geometrical construction of wavefronts in four-dimensional space and tries to gain an intuitive understanding of this fact by comparing it to a wave in a medium with a flow. Schrödinger tries to construct the wavefronts in the relativistic case, again through an application of Huygens’ principle. He is not able to show that this leads to Hamilton’s partial differential equation and breaks off. Instead he tries to directly solve the wave equation in a flowing medium, hoping that “maybe the correction will come out by itself.” However, the resulting wave equation is too complicated, and again he cannot show that it leads to Hamilton’s partial differential equation. Once more, Schrödinger turns to a reconsideration of the extremum principle of his first communication under the heading “Connection between the wave equation and Hamilton’s equation” (see Fig. 5). This passage clearly illustrates that Schrödinger still holds on to the substitution 30 Presumably (Whittaker, 1904), the German edition of which Schrödinger refers to in the second communication. 18 Figure 5: The top of p. 14 of Schrödinger’s notebook AHQP 40-6-001, with the title “Connection between the wave equation and Hamilton’s [partial differential] equation.” S = K log ψ: As in the first communication, he starts from Hamilton’s partial differential equation (Eqs. (1) and (2) in Fig. 5) and through the substitution (here Eq. (3)), arrives at the equation (here Eq. (4)) that was the basis of the variational principle leading to the nonrelativistic wave equation in his first communication. He tries to justify the variational principle of the first communication by showing that a solution of Eq. (4) is also a solution of the variational problem. This attempt had to fail. It breaks off with the remark “I’m not moving ahead!” In a footnote to the introduction of his second communication, Schrödinger explicitly states why: ψ does not really relate to the action function associated with a given motion in the way postulated by Eq. (2) [the substitution] of the first communication. (Schrödinger, 1926c, p. 489) After another attempt at direct construction of the wave equation (this time through an odd nonLorentzian transformation of the wave equation), he gives up his foundational explorations and uses the rest of the notebook for studying perturbation theory and the Stark effect. The common feature of all these attempts to find a relativistic wave equation is that they are based on an exact functional dependence of the Hamiltonian action S and the wave function ψ. There is another notebook, entitled “Eigenvalue problem of the atom III” (AHQP 40-7-001), that was written before the second communication on wave mechanics. It appears to be the direct continuation of the notebook entitled “Eigenvalue problem of the atom II” (AHQP 40-6-001) discussed above. It is in this notebook, again in a section entitled “For the second communication,” that Schrödinger takes a crucial step forward that leads him to the completion of Hamilton’s optical-mechanical analogy. He writes: For the second communication: 19 Naturally, one must not expect that any explicit function of S itself satisfies the wave equation (for instance cos S , or the like). For S is only the phase of the wave, say, the time at which a certain wave front reaches the point P in configuration space, expressed as a function of the coordinates of P. Thus, cos S only is the wavefunction without the amplitude. The latter naturally also has to be assumed to vary as soon as the rays diverge or converge if a wave equation is to hold. And since this diverging and converging of the wave normal is not equal in all points of a wavefront, the amplitude does not vary along all rays in a proportional way. Therefore, one cannot give a general function of S alone that satisfies the wave equation. The transition from Hamilton’s partial differential equation to the wave equation thus signifies in mechanics the exact same thing as in optics the transition from ray optics, which is generated merely by Fermat’s principle, to wave optics proper. One can speak of an undulatory theory of mechanics. Symptomatically, one encounters exactly the same indeterminacy as back then in optics: Initially, one only knows the speed of propagation and does not know how this speed is to be assigned to a specific elasticity and density [of the medium] respectively. (AHQP 40-7-001)31 What had escaped Schrödinger so far was the full impact of the fact that the connection established by Hamilton between wave optics and ray optics does not entail a complete equivalence of the two optical theories, but that ray optics is only a limiting case of wave optics.32 Schrödinger had not understood what this meant for the relation between classical and wave mechanics at the time he wrote his first communication: He did not realize that wave mechanics does not just impose additional constraints on classical motions (i.e. that it leads to quantum conditions), but that it means that the motions predicted by classical mechanics are only an approximation to the evolution of a full wave field. When Schrödinger tried to construct the wavefronts in the relativistic case, he failed in AHQP 40-6-001 because he was misled into believing that the full wave function would result from the relation S = k log ψ. This insight is reflected in the quote above: In AHQP 40-7-001, Schrödinger realized that S would only give him the phase of a more general wavefunction and that he needed to go beyond the picture of wave fronts determined by the classical action in order to also recover the amplitude of ψ. Thus, AHQP 40-7-001 contains Schrödinger’s decisive new idea: Wave mechanics is more general than particle mechanics in the same sense as wave optics is more general than geometrical optics.33 Hamilton’s analogy is only one axis in a four-way correspondence (see Fig. 4). It establishes a connection between ray optics and classical mechanics, but it cannot be used to 31 Emphasis in the original (underlined). Mehra and Rechenberg (1987b, p. 560-561) quote parts of this passage but do not realize that it constitutes a fundamental break with Schrödinger’s previous approach. 32 To our knowledge, (Sommerfeld and Runge, 1911) is the first paper that shows explicitly that Hamiltonian ray optics is a limiting case of wave optics. If Schrödinger had known this work at the time of writing AHQP 40-6-001, it would be hard to imagine that he should have tried to derive the wave function by the direct substitution S = K log ψ. In a footnote to his second communication (Schrödinger, 1926c, p. 490), Schrödinger refers to (Sommerfeld and Runge, 1911). However, from a letter to Wilhelm Wien dated March 4, 1926, Mehra and Rechenberg (1987b, p. 555) infer that Schrödinger only learned about this paper from Sommerfeld after he had submitted his second communication. This rules out that (Sommerfeld and Runge, 1911) was an inspiration for either of the two first communications, as Kragh (1982, p. 160) believes. 33 Klein (1970, p. 161) quotes a notebook of Paul Ehrenfest from the spring of 1904, in which Ehrenfest has a premonition of this insight: “The Hamilton-Jacobi equation of Lagrangian mechanics corresponds to a diffractionless optics. What is the super-Lagangrian mechanics whose Hamilton-Jacobi equation is adequate for describing the diffracted wave?” We thank Michel Janssen for this reference. 20 establish a direct connection between ray optics and the new wave mechanics. Rather, the direct connection is between the more general wave optics and wave mechanics. This is the completion of Hamilton’s optical-mechanical analogy. The ”old” Hamiltonian analogy, by its completion, changed its role from a heuristic tool (as employed by Schrödinger when writing his first communication) to a formal constraint: wave mechanics has to reduce to classical mechanics in the limit of wavelengths that are short compared to the curvature of the classical trajectories, just as wave optics reduces to ray optics in the appropriate limit. This is an extension and justification of Bohr’s correspondence principle, and Schrödinger hoped to use this constraint for the construction of a new mechanics—wave mechanics—that would be more general and powerful than both the Bohr-Sommerfeld quantum theory and the Göttingen matrix mechanics. Through the analysis of the notebooks, it is clear that this insight occurred to Schrödinger only after his completion of the first communication and that it resulted from his unsuccessful attempts at generalizing the non-relativistic wave equation for the atom. As we can see from the notebooks, Schrödinger had originally intended to present a fullfledged relativistic theory in his second communication. This is obvious from his early drafts for the second communication in AHQP 40-6-001 and only natural, given the fact that Schrödinger’s point of departure was de Broglie’s relativistic treatment of matter waves. However, also the new understanding reached in AHQP 40-7-001 did not lead to a different relativistic wave equation, and Schrödinger’s attempts to find a complete theory of the coupling of the wave function and the electromagnetic field got bogged down in further complications that we will discuss in the next section. Therefore, the relativistic project was not even mentioned, and Schrödinger limited himself to treat further successful applications of the non-relativistic equation. Nonetheless, the new view of the optical-mechanical analogy was also useful as an intuitive justification of wave mechanics. This is how Schrödinger presented it prominently in the second communication. At the same time, the completion of the optical-mechanical analogy had a deep impact on Schrödinger’s interpretation of the wave function: What the analogy came to mean for Schrödinger is that matter really consists of waves, just as Fresnel had found that light really consists of waves. 6. The “pending questions”: Hamilton’s analogy as an interpretational device In the following months, Schrödinger kept publishing at a rapid speed. He continued his presentation of nonrelativistic wave mechanics with two more communications that demonstrated the predictive power of the nonrelativistic wave equation, e. g. through the perturbative results for the Stark effect (Schrödinger, 1926e,f). He also established a link between wave mechanics and matrix mechanics in his paper “On the relation of the Heisenberg-Born-Jordan quantum mechanics to mine” (Schrödinger, 1926d), giving the first published account of a formal connection between the two.34 However, Schrödinger was also fully aware of the limitations of what he had achieved so far. Not only did he not yet have a relativistic wave equation or a systematic account of electromagnetic interaction, an even more basic question still had to be answered: The wave function was not defined in 3-dimensional space, as one should expect if one was to think of it as a kind of physical wave. Rather it was defined in configuration space, which has 3N dimensions if N is the number of involved particles. There is no straightforward relation between a function on 3N dimensions and one on three dimensions, or even N different functions 34 Also for this paper a preparatory notebook is preserved, AHQP 41-2-001, tentatively dated in AHQP to 1927, but clearly written before the publication of (Schrödinger, 1926d). 21 on three dimensions.35 To recover a classical picture, Schrödinger therefore had to search for some three-dimensional functional of the full wave function representing the physical effects of the wave function (for example, representing the charge-current density as the source of the electromagnetic field). Hence, the problem of finding a physical interpretation of the wave function was intimately connected with the previously mentioned problem of the coupling of matter wave and electromagnetic field, and so also with the question of the relativistic wave equation. In the spring of 1926, Schrödinger’s private and public research programs began to diverge: While his published work focused on successful applications of the nonrelativistic wave equation, Schrödinger’s notebooks show him working intensively on the “pending questions”36 of his more general program, which he only rarely alluded to in his published work. Especially noteworthy for this is the notebook entitled “Coupling. Very old” (AHQP 41-4-001), in which he explores various expressions for the charge-current-density connected with the wave function. Therefore, it is most probably from the time before he settled on ψ̄ψ as the expression for the charge density, which he discussed in (Schrödinger, 1926f, p. 134-139). This latter expression is explored in the presumably following notebook, AHQP 41-4-002. In this notebook, an interesting tension in Schrödinger’s research strategy can be observed: At various points, he still starts with Hamilton’s partial differential equation and tries to generalize it into a wave equation. But mostly he tries to tackle the problem more directly, presumably in an attempt to find an alternative to the relativistic wave equation he still believes he has to abandon. In those places, he tries various expressions for a charge-current density derived from the wave function and uses those to derive a wave equation with a coupling term to the electromagnetic field. This amounts to abandoning the idea of finding the wave function through a generalization of the classical Hamiltonian approach, where the “action field” is an abstract entity that cannot carry a physical charge. Schrödinger replaced this approach with straightforward model-building in a much more traditional way. Why did Schrödinger abandon his promising constructive principle so quickly without being able to reap the fruits of his advance? Presumably, the unambiguity with which the Hamiltonian analogy led him back to the Klein-Gordon equation made Schrödinger despair of the prospects of the constructive principle and turn to basic physical intuitions about waves in classical field theory. Because that was what he believed he had learned from his struggle with the optical-mechanical analogy: the failure of classical mechanics means that matter really consists of waves, just like in the 19th century the failure of ray optics meant that light really consists of waves. Hence, the struggle for a replacement for his relativistic wave equation did not only lead Schrödinger to his completion of the optical-mechanical analogy, it also induced him to abandon the use of the analogy as a constructive tool and replace it with the use as an interpretational device.37 Schrödinger’s commitment to a physical interpretation of matter waves became decisive when, in the summer of 1926, Max Born proposed the probabilistic interpretation of the wave function. Schrödinger objected vehemently—not surprisingly, since for him this meant a return to the obsolete particle picture and its quantum jumps. In opposition to the emerging mainstream, Schrödinger insisted on a field interpretation of the wave function, guided by his literal reading of the optical-mechanical analogy. This is not the place to follow the ensuing debate between 35 Schrödinger notes this problem in a letter to Arnold Sommerfeld (February 20, 1926), mentions it in (Schrödinger, 1926c, p. 526) and discusses it more extensively in (Schrödinger, 1926f, p. 135). The problem also is noted by Hendrik Antoon Lorentz (letter to Schrödinger from May 27, 1926) 36 This is the title of the notebook AHQP 41-2-002. 37 The philosopher of science may notice that Schrödinger’s struggle to find a relativistic wave equation matching the fine-structure splitting of Sommerfeld’s account is a striking example of putting too much weight into empirical falsification (see also footnote 26). 22 Schrödinger on one side and Bohr and the Göttingen school on the other.38 But the story told here should help to understand Schrödinger’s deep commitment to the field interpretation for the rest of his life, which persisted even after he stopped opposing the probabilistic interpretation publicly after the Solvay meeting of 1927. Schrödinger was never satisfied with the Copenhagen orthodoxy, and would eventually resume his critique in the thirties and turn to unified field theory in a search for answers to his “pending questions.” 7. Conclusion Hamilton’s optical-mechanical analogy played a central role in Schrödinger’s development of wave mechanics. The changes in his use of the analogy, already noticed by Wessels (1979), can be traced in Schrödinger’s research notebooks. The notebooks document every step in the development and offer a more extensive and coherent picture of Schrödinger’s thinking and aspirations than previously thought possible. Schrödinger first explored the optical-mechanical analogy in the context of an ambitious program of generalizing mechanics around the end of World War I. When Schrödinger encountered the analogy again in de Broglie’s dissertation, it attracted his attention and became a preliminary heuristic tool for the development of his ideas. This heuristic role of the analogy is not recognizable from Schrödinger’s first communication on wave mechanics, but is displayed in the notebooks he wrote in its preparation. His struggle to overcome the limitations of the nonrelativistic wave equation led to a deeper understanding of the analogy, reflected in the second communication. The notebooks show that he only understood the full impact of his wave mechanics during the preparation of the second communication: Classical mechanics is just an approximation to the ‘undulatory mechanics’ Schrödinger was proposing. The optical-mechanical analogy, thus completed, became a formal constraint for his continuing search for a relativistic wave equation, and at the same time an intuitive justification for his program. The success in the struggle for a reinterpretation of the optical-mechanical analogy reinforced his commitment to a physical wave interpretation of the quantum mechanical wave function and led to the first interpretation debate in quantum mechanics. This picture of the classical roots of wave mechanics serves as a corrective to an overly simplistic reading of Thomas S. 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