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Latin American journal of physics education

A brief history of the mathematical equivalence between the two quantum mechanics

Carlos Miguel Madrid Casado, Collado Villalba

Published Jan 1, 2008• 13 citations
OpenAlex

Abstract

The aim of this paper is to give a brief account of the development of the mathematical equivalence of quantum mechanics. In order to deal with atomic systems, Heisenberg developed matrix mechanics in 1925. Some time later, in the winter of 1926, Schrodinger established his wave mechanics. In the spring of 1926, quantum physicists had two theoretical models that allowed them to predict the same behaviour of the quantum systems, but both of them were very different. Schrodinger thought that the empirical equivalence could be explained by means of a proof of mathematical equivalence.

Related papers

Why were two theories (Matrix Mechanics and Wave Mechanics) deemed logically distinct, and yet equivalent, in Quantum Mechanics?

Slobodan Perović

A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger’s 1926 proof, a myth. Schrödinger supposedly failed to achieve the goal of proving isomorphism of the mathematical structures of the two theories, while only later developments in the early 1930s, especially the work of mathematician John von Neumman (1932) provided sound proof of equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to a large extent on this equivalence, was deemed a myth as well. If such analysis is correct, it provides considerable evidence that, in its critical moments, the foundations of scientific practice might not live up to the minimal standards of rigor, as such standards are established in the practice of logic, mathematics, and mathematical physics, thereby prompting one to question the rationality of the practice of physics. In response, I argue that Schrödinger’s proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr’s Model that was discovered and published by Schrödinger in his First and Second Communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics could be derived successfully from eigenfunctions as well (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of isomorphism of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr’s atom. And although the full-fledged mathematico-logical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr’s complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it.

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Abstract This paper re-evaluates the formative year of quantum mechanics—from Heisenberg’s first paper on matrix mechanics to Schrödinger’s equivalence proof—by focusing on the role of radiation in the emerging theory. We argue that the radiation problem played a key role in early quantum mechanics, a role that has not been taken into account in the standard histories. Radiation was perceived by the main protagonists of matrix and wave mechanics as a central lacuna in these emerging theories and continued to contribute to the theoretical development and conceptual clarification of quantum mechanics. Studying the interplay between quantum mechanics and radiation, the paper provides an account of (a) how quantum mechanics was able to connect to its empirical basis in spectroscopy and (b) how Schrödinger’s equivalence proof emerged from his explorative calculations on the emission of radiation.

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This chapter begins by understanding the classical view of an electromagnetic wave and proceed step-by-step in exploring the historical development of quantum mechanics. The frequency of the electromagnetic wave is the same as the frequency of mechanical oscillation of the charge, and the intensity of the electromagnetic wave is the square of the amplitude of the mechanical oscillation of the charge. The origin of quantum mechanics is linked to the discovery of then new laws of radiation by Max Planck in 1900. The explanation of the empirical facts of radiation by Planck using an extraordinarily powerful yet simple model lead to the discovery of the "quantum" nature of radiation. The work to develop a mathematical theory of quantum mechanics continued for a very long time. Two mathematical approaches for the formalism of quantum mechanics emerged. One was developed in 1925 by Werner Heisenberg, called matrix mechanics, and the other was developed by Erwin Schrodinger in 1926 called wave mechanics.

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Elements of quantum mechanics Quantum mechanics was discovered in 1925 through groping efforts to compromise two apparently contradictory pictures on the fundamental entities in nature. One was the wave picture for light which was later extended to matter by de Broglie, another was the corpuscular picture of matter which was later extended to light by Einstein. Schrödinger's wave equation came as a natural development of the first stream, while Heisenberg's matrix mechanics was presented as a unique proposal from the second stream. In spite of completely different appearances, the two theories proved, within a couple of years after their discoveries, to be equivalent. This is a most beautiful example that the physical reality exists independent of the mathematical framework formulated for its description. In this chapter, we will give a very brief review of the principles of quantum mechanics, 1–3 mainly with the harmonic oscillator as a model system for the following reasons. The first is historical: the electromagnetic wave, whose interaction with matter is the subject of this book, is a harmonic oscillator, a system which was for the first time subject to “quantization”, thus opening a way to the discovery of quantum mechanics. The second is technical: the harmonic oscillator is one of very few examples of analytically soluble problems in quantum mechanics. The third is pedagogical: the harmonic oscillator is a system best suited for realization of the equivalence of the two different pictures mentioned above and hence for a deeper understanding of the principles of quantum mechanics.

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Schrödinger's wave mechanics

Guido Bacciagaluppi, Antony Valentini

Schrödinger's work on wave mechanics in 1926 appears to have been driven by the idea that one could give a purely wave-theoretical description of matter. Key elements in this picture were the idea of particles as wave packets (Section 4.3) and the possible implications for the problem of radiation (Section 4.4). This pure wave theory, in contrast to de Broglie's work, did away with the idea of particle trajectories altogether (Section 4.5). The main conflict, however, was between Schrödinger and the proponents of quantum mechanics (in particular Heisenberg, Section 4.6), both in its form at the time of Schrödinger's papers and in its further developments as sketched in the previous chapter. For reference, we provide a brief chronology of Schrödinger's writings relating to wave mechanics up to the Solvay conference: Paper on Einstein's gas theory, submitted 15 December 1925, published 1 March 1926 (Schrödinger 1926a). First paper on quantisation, submitted 27 January 1926, addendum in proof 28 February 1926, published 13 March 1926 (Schrödinger 1926b). Second paper on quantisation, submitted 23 February 1926, published 6 April 1926 (Schrödinger 1926c). Paper on the relation between wave and matrix mechanics (‘equivalence paper’), submitted 18 March 1926, published 4 May 1926 (Schrödinger 1926d). Paper on micro- and macromechanics (coherent states for the harmonic oscillator), published 9 July 1926 (Schrödinger 1926e). […]

Cambridge University Press eBooks•Oct 22, 2009•0 citations
The aim of this paper is to give a brief account of the development of the mathematical equivalence of quantum mechanics. In order to deal with atomic systems, Heisenberg developed matrix mechanics in 1925. Some time later, in the winter of 1926, Schrodinger established his wave mechanics. In the spring of 1926, quantum physicists had two theoretical models that allowed them to predict the same behaviour of the quantum systems, but both of them were very different. Schrodinger thought that the empirical equivalence could be explained by means of a proof of mathematical equivalence.
A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger’s 1926 proof, a myth. Schrödinger supposedly failed to achieve the goal of proving isomorphism of the mathematical structures of the two theories, while only later developments in the early 1930s, especially the work of mathematician John von Neumman (1932) provided sound proof of equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to a large extent on this equivalence, was deemed a myth as well. If such analysis is correct, it provides considerable evidence that, in its critical moments, the foundations of scientific practice might not live up to the minimal standards of rigor, as such standards are established in the practice of logic, mathematics, and mathematical physics, thereby prompting one to question the rationality of the practice of physics. In response, I argue that Schrödinger’s proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr’s Model that was discovered and published by Schrödinger in his First and Second Communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics could be derived successfully from eigenfunctions as well (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of isomorphism of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr’s atom. And although the full-fledged mathematico-logical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr’s complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it.
Abstract This paper re-evaluates the formative year of quantum mechanics—from Heisenberg’s first paper on matrix mechanics to Schrödinger’s equivalence proof—by focusing on the role of radiation in the emerging theory. We argue that the radiation problem played a key role in early quantum mechanics, a role that has not been taken into account in the standard histories. Radiation was perceived by the main protagonists of matrix and wave mechanics as a central lacuna in these emerging theories and continued to contribute to the theoretical development and conceptual clarification of quantum mechanics. Studying the interplay between quantum mechanics and radiation, the paper provides an account of (a) how quantum mechanics was able to connect to its empirical basis in spectroscopy and (b) how Schrödinger’s equivalence proof emerged from his explorative calculations on the emission of radiation.
This chapter begins by understanding the classical view of an electromagnetic wave and proceed step-by-step in exploring the historical development of quantum mechanics. The frequency of the electromagnetic wave is the same as the frequency of mechanical oscillation of the charge, and the intensity of the electromagnetic wave is the square of the amplitude of the mechanical oscillation of the charge. The origin of quantum mechanics is linked to the discovery of then new laws of radiation by Max Planck in 1900. The explanation of the empirical facts of radiation by Planck using an extraordinarily powerful yet simple model lead to the discovery of the "quantum" nature of radiation. The work to develop a mathematical theory of quantum mechanics continued for a very long time. Two mathematical approaches for the formalism of quantum mechanics emerged. One was developed in 1925 by Werner Heisenberg, called matrix mechanics, and the other was developed by Erwin Schrodinger in 1926 called wave mechanics.
Elements of quantum mechanics Quantum mechanics was discovered in 1925 through groping efforts to compromise two apparently contradictory pictures on the fundamental entities in nature. One was the wave picture for light which was later extended to matter by de Broglie, another was the corpuscular picture of matter which was later extended to light by Einstein. Schrödinger's wave equation came as a natural development of the first stream, while Heisenberg's matrix mechanics was presented as a unique proposal from the second stream. In spite of completely different appearances, the two theories proved, within a couple of years after their discoveries, to be equivalent. This is a most beautiful example that the physical reality exists independent of the mathematical framework formulated for its description. In this chapter, we will give a very brief review of the principles of quantum mechanics, 1–3 mainly with the harmonic oscillator as a model system for the following reasons. The first is historical: the electromagnetic wave, whose interaction with matter is the subject of this book, is a harmonic oscillator, a system which was for the first time subject to “quantization”, thus opening a way to the discovery of quantum mechanics. The second is technical: the harmonic oscillator is one of very few examples of analytically soluble problems in quantum mechanics. The third is pedagogical: the harmonic oscillator is a system best suited for realization of the equivalence of the two different pictures mentioned above and hence for a deeper understanding of the principles of quantum mechanics.
Schrödinger's work on wave mechanics in 1926 appears to have been driven by the idea that one could give a purely wave-theoretical description of matter. Key elements in this picture were the idea of particles as wave packets (Section 4.3) and the possible implications for the problem of radiation (Section 4.4). This pure wave theory, in contrast to de Broglie's work, did away with the idea of particle trajectories altogether (Section 4.5). The main conflict, however, was between Schrödinger and the proponents of quantum mechanics (in particular Heisenberg, Section 4.6), both in its form at the time of Schrödinger's papers and in its further developments as sketched in the previous chapter. For reference, we provide a brief chronology of Schrödinger's writings relating to wave mechanics up to the Solvay conference: Paper on Einstein's gas theory, submitted 15 December 1925, published 1 March 1926 (Schrödinger 1926a). First paper on quantisation, submitted 27 January 1926, addendum in proof 28 February 1926, published 13 March 1926 (Schrödinger 1926b). Second paper on quantisation, submitted 23 February 1926, published 6 April 1926 (Schrödinger 1926c). Paper on the relation between wave and matrix mechanics (‘equivalence paper’), submitted 18 March 1926, published 4 May 1926 (Schrödinger 1926d). Paper on micro- and macromechanics (coherent states for the harmonic oscillator), published 9 July 1926 (Schrödinger 1926e). […]