Angel "Java" Lopez en Blog

Publicado el 20 de Septiembre, 2014, 18:56

Estoy explorando una deducción de la ecuación de Schrödinger en:

La Ecuación de Schrödinger

pero no es el camino que siguió el propio Schrödinger. Este tomó una analogía entre óptica y mecánica, debida originalmente a Hamilton, para extender las ideas de de Broglie.

Encuentro hoy el excelente

The classical roots of wave mechanics: Schrödinger"s transformation of the optical-mechanical analogy

donde leo el resumen:

The optical-mechanical analogy played a central role in Schrödinger's reception of de Broglie's ideas and development of wave mechanics. He was well acquainted with it through earlier studies, and it served him as a heuristic model to develop de Broglie"s idea of a matter wave. Schrödinger"s struggle for a deeper understanding of the analogy in the search for a relativistic wave equation led to a fundamental transformation of the role of the analogy in his thinking into a formal constraint on possible wave equations. This development strongly influenced Schrödinger"s interpretation of the wave function and helps to understand his commitment to a wave interpretation in opposition to the emerging mainstream. The changes in Schrödinger's use of the optical-mechanical analogy can be traced in his research notebooks, which offer a much more complete picture of the development of wave mechanics than has been generally assumed. The notebooks document every step in the development and give us a picture of Schrödinger's thinking and aspirations that is more extensive and more coherent than previously thought possible.

No conocía la existencia de esas "notebooks", les recomiendo la lectura de este "paper".

Busqué hoy una relación que recordaba vagamente, entre algunas ideas de Klein, pasadas por Sommerfeld a Schrödinger sobre esa analogía entre óptica y mecánica. Parece que no tuvo tanta influencia como pensaba: Schrödinger había ya encontrado por otros medios parte de esa analogía, reflejada en un segundo "paper" que presentó. Sommerfeld, al leer el texto, todavía no publicado, le recordó la similitud entre una ecuación y la ecuación de la ecoinal. Buscando en Google Books, encontré este fragmento del excelente e interminable The Historical Development of Quantum Theory de Jagdish Mehra, Helmut Rechenberg (varios tomos), pp 553 en adelante:

As this point we may insert a remark which is perhaps essentially superfluous here but nevertheless may help to avoid a wrong impression that one possibly obtains from reading the published paper. In a footnote to p. 490 of the latter, Schrödinger, just after stating the connection between Hamilton's dynamical principle and Huyguens' optical principle, added the remark: 'Felix Klein has since 1891 repeatedly developed the theory of Jacobi [i.e, Hamilton's theory in the pure mechanical formulation of Jacobi] from quasi-optical considerations in non-Euclidean higher spaces in his lectures on mechanics. Cf. F. Klein, Jahresber. d. Deutsch. Math. Ver. 1, 1891 [Klein, 1892] and Zeit. f. Math. u. Phys. 46, 1901 [Klein, 1901] (Ges.-Abh, II, pp. 601 and 603). In the second note, Klein remarks reproachfully that his discourse at Halle ten years previously, in which he had discussed this correspondence and emphasized the great significance of Hamilton's optical works, had "not obtained the general attention, which he had expected." For this allusion to F. Klein, I am indebted to a friendly communication from Prof. Sommerfeld. See also Atombau, 4th ed., p. 803' (Schrödinger, 1926d, p. 490, footnote 1; English translation, footnote 3 on pp. 13-14)

Back in the fall of 1891, at the Versammlung Deutscher Naturforscher und Arzte in Halle, the Göttingen mathematician Felix Klein had spoken -as we have already pointed out in the previous section- on some recent English papers on mechanics, and had a particular called attention to the close relation of Hamilton's dynamical theory of light rays an his -Klein's- own consequences from this relation, namely, 'that one, moreover, by proceeding to higher-dimensional spaces, is able to reduce every mechanical problem to the determination of the path of ray propagating in a suitable optical medium' (Klein, 1892, p. 35). Klein had never published any detailed results of that sort in a journal; however, he reporter later in his Vorlesungen uber die Entwicklung der Mathematik im 19, Jahrhundert which appeared posthumously in 1926 (after Schrödinger first publications on wave mechanics): 'I have especially indulged, in the summer of 1891, in the pleasure of treating all mechanics, in the footsteps of Hamilton, as a kind of optics in the n-dimensional space; and I have included Jacobi's extension of the theory...; the elaboration of these lectures has been available for twenty years in the Göttingen Reading Room' (Klein, 1926, p. 198).

Vean que menciona "in the footsteps of Hamilton". Tengo que encontrar la referencia, pero todos los comentadores señalan que Hamilton ya se había dado cuenta de la relación entre su trabajo en óptica geométrica y la mecánica.

As in his paper of 1901, Klein complained here about the total neglect that physicist had accorded to the powerful ideas suggested by him. Perhaps one reason for this particular neglect -at least in Germany- may be seen in the fact that physicists in general paid hardly any attention to the sketchy hints in the mathematical journals, nor did they have access to the Reading Room in the Mathematical Institute at the University of Göttingen where the elaborated lectures of Klein lay...

Acá los autores insertan dos notas, que no quiero saltear:

267: The only exception, to which Klein referred, was the mathematician Eduard Study; the latter had, in a paper 'Uber Hamiltons geometrische Optik und deren Beziehung zur Theorie der Beruhrungstransformations' ('On Hamilton's Geometrical Optics and Its Relation to the Theory of Contact Transformations,' Study, 1905): 'worked out from the correct point of view a new form' (Klein, 1926, p. 198). In a footnote, which he had obviously added much latter -the manuscript of the lectures of mathematics in the nineteenth century must have been written in the years after 1910- Klein quoted two further papers by another mathematician, Georg Prange, entitled 'W.R.Hamiltons Bedeutung fur die geomestrische Optik' ('W.R.Hamilton's Importance for Geometrical Optics,' Prange, 1921) and 'W.R.Hamiltons Arbeiten zur Strahlenoptik und analytischen Mechanik' ('W.R.Hamilton's Papers on Ray Optics and Analytical Mechanics,' Prange, 1923).

268: As we have mentioned and discussed in Section 4, the British mathematician Edmund Whittaker discussed, in his textbook on analytical mechanics (Whittaker, 1904, and later editions), the optical analogue to Hamilton's dynamics in some detail; he also presented (in Sections 125 and 126 of his book) the connection between Huyguens' principle and contact transformations and he considered higher-dimensional spaces.

Volvamos al texto principal:

... However, there was one physicist, Arnold Sommerfeld, who had indeed seen the manuscript of the above-mentioned lectures when he was assistant for mathematics at Göttingen during 1894-1897. He had also referred to Klein's two notes -quoted above- in his book Atombau und Spektralinien...

Ese libro instruyó a toda una generación de físicos cuánticos, sobre el espectro atómico.

..., for example, in the fourth edition in Appendix 7. There he wrote:

Finally, several remarks on the origin of Hamilton's theory. We base our statements on two notes of F. Klein ([1892, 1901]). Hamilton was an astronomer who studied the propagation of light rays in optical instruments. He had available the ray-optics (Newton's emissive optics), which describes the path of light particles in a section-wise homogeneous, or more generally, an inhomogeneous medium by means of integrable ('totale') differential equations. Hamilton attempted to incorporate this method into the then developed wave optics, which describes the optical states by partial differential equations. The real differential equations of wave optics is the equation of vibration (I), from which one derives -for sufficiently small wavelengths- the differential equation of wave surface (II), i.e.,

Here u represents a vector, v a scalar, delta2 denotes the so-called "second differential parameter", and delta1 the "first differential parameter", that is, for a Euclidean line element,

Tengo que revisar el por qué de esas dos notaciones. Imagino que varían en espacios no euclideanos.

If one integrates Eq. (II), one obtains a family of wave surfaces or surfaces of equal phases by putting v = const. Their ortogonal trajectories are the orbits of the light particles, and hence constitute the light rays. Provided we know a complete solution of Eq. (II), then we can derive from it the trajectories of the light particles by mere differentiation... Of course, one must use, for the purpose of general dynamics, a non-Euclidean line element; and one must, for systems having more thant three degrees of freedom -as Klein realized- go over to a higher-dimensional space. (Sommerfeld, 1924d, pp. 803-804).

Es interesante mencionar una nota:

269 In his autobiographical sketch, published posthumously, Sommerfeld mentioned that he went to Göttingen in October 1893, where he soon came under the decisive influence of Felix Klein; Klein drew his attention to the problems of mathematical physics and tried to win him 'for his view of the problems which he had formulated in previous lectures' (Sommerfeld, 1968, p. 675). In 1894, Sommerfeld became 'Klein's assistant at the mathematical reading room and had, in this position, to work out his lectures' (Sommerfeld, 1968, p. 675).

Sigamos con el texto principal. Veamos la acción de Sommerfeld sobre Schrödinger.

Evidently, Eqs. (I) and (II) correspond to Eq. (209) and the eiconal equation (208a), respectively, as Sommerfeld and Runge had shown in 1911 in a paper quoted by Sommerfeld in Atombau. Sommerfeld also reported in his book that in the extension of Hamilton's theory by Jacobi the optical aspect had been lost, and that later Bruns rediscovery it in his theory of the eiconal. When he (Sommerfeld) then say, in February 1926, the manuscript of Schrödinger's second communication on Quantisierung als Eigenwertproblem, he wrote to Zurich and reminded Schrödinger of the ideas of Klein and of those in his own paper. Obviously, Schrödinger did not known of Klein's work, and had also overlooked the above-quoted passage in the fourth edition of Atombau, a book, which he had otherwise study quite carefully. In any case, he immediately dispatched a letter to Wein. He wrote:

Please do excuse me for having to bother you with a request to insert the enclosed two remarks in my manuscript "Quantisierung als Eigenwertproblem, Zweite Mitteilung" [for the] Annalen der Physik... I owe both hints to a friendly postcard from Professor Sommerfeld, which he still wrote to me just before his departure for England. It would not only be very embarrasing for me to omit the citations of Klein and Sommerfeld, but I am also extremely happy to point to none other than Felix Klein as that person, who 35 years ago, and unfortunately in vain, had hinted at the importance of these connections. Had Klein's lectures been distributed in a form other than the valuable and rare manuscript editions, then probably the connection with quantum theory would have been discovered some time ago. It is indeed so evident. From time immemorial one writes the first term of the action function as: minus the energy multiplied by time. Therefore, as soon as one interprets the action function as the phase of the wave, i.e., forms a periodic function of it, one must note that the frequency of this wave is proportional to the mechanical energy constant (Schrödinger to Wein, 4 March 1926)

No se encontró la "postcard" de Sommerfeld, pero debe datar de principios de marzo. La relación entre energía y frecuencia es uno de los grandes descubrimientos de la física: una relación inesperada, que permea todos los desarrollos posteriores.

Concluyen los autores:

Thus the references to Felix Klein's work as well as to the paper by Sommerfeld and Runge -and to Appendix 7 in Atombau- were added to the second communication only after the author had completed the manuscript; they did not influence his results and development at all.

Interesante tema. Si quieren los detalles matemáticos y físicos, les recomiendo el excelente Fundamentos de Mecánica Cuántica, de Borowitz, donde el autor sigue el camino de Schrodinger para llegar a sus resultados, explorando entonces las relaciones entre óptica geométrica y mecánica, y luego pasado de óptica geométrica a ondas, para conseguir unir la mecánica con las ondas, el germen de la mecánica cuántica de entonces.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia