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Heisenberg’sUmdeutungPaper

Anthony Duncan A, Michel Janssen B

Published Aug 17, 2023• 1 citation
Publisher

Abstract

Abstract This chapter covers the paper known as the Umdeutung (= reinterpretation) paper, published in July 1925, with which Heisenberg laid the foundations of matrix mechanics. It carefully examines Heisenberg’s work on intensities and his correspondence in the months leading up to his famous trip to Helgoland, where he aimed to recover from poor health. It was in Helgoland that he developed the outlines of the new framework that would eventually become matrix mechanics. The chapter then gives a detailed analysis of the Umdeutung paper itself, including the quantization condition, as well as a derivation of the quantized energies of the simple harmonic oscillator, including the zero-point energy, both of which would assume a central role in the paper.

DOI
10.1093/oso/9780198883906.003.0004

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Abstract This chapter covers the paper known as the Umdeutung (= reinterpretation) paper, published in July 1925, with which Heisenberg laid the foundations of matrix mechanics. It carefully examines Heisenberg’s work on intensities and his correspondence in the months leading up to his famous trip to Helgoland, where he aimed to recover from poor health. It was in Helgoland that he developed the outlines of the new framework that would eventually become matrix mechanics. The chapter then gives a detailed analysis of the Umdeutung paper itself, including the quantization condition, as well as a derivation of the quantized energies of the simple harmonic oscillator, including the zero-point energy, both of which would assume a central role in the paper.
Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.
In 1925, Heisenberg, Born, and Jordan developed matrix mechanics as a strategy to solve quantum-mechanical problems. While finite-sized matrix formulations are commonly taught in quantum instruction, following the logic and detailed steps of the original matrix mechanics has become a lost art. In preparation for the 100th anniversary of the discovery of quantum mechanics, we present a modernized discussion of how matrix mechanics is formulated, how it is used to solve quantum-mechanical problems, and how it can be employed as the starting point for a postulate-based formulation of quantum-mechanics instruction. We focus on the harmonic oscillator to describe how quantum mechanics advanced from the Bohr–Sommerfeld quantization condition, to matrix mechanics, to the current abstract ladder-operator approach. We also describe a number of different activities that can be included in the quantum mechanics classroom to celebrate this centennial.
Abstract This chapter covers the process of the consolidation of the ideas in Heisenberg’s Umdeutung paper into the Two-Man-Paper by Born and Jordan, the Three-Man-Paper (Dreimännerarbeit) by Born, Heisenberg, and Jordan, and a paper by Dirac. Both Born and Jordan recognized that Heisenberg’s peculiar non-commutative multiplication rule is recognized to be nothing but the standard multiplication rule for matrices. They also replaced Heisenberg’s quantization condition, the Thomas–Reiche–Kuhn sum rule, by the now-familiar commutation relations of position and momentum. The chapter then discusses how Dirac independently derived these commutation relations, relying on a profound analogy with the Poisson brackets of classical mechanics
Abstract This chapter covers the history of dispersion theory starting from the classical theories of the late nineteenth century, involving charged harmonic oscillators with resonance frequencies at the absorption frequencies of the dispersing materials to account for anomalous dispersion, the phenomenon that the index of refraction decreases rather than increases with frequency around the absorption frequencies. It then covers dispersion theories formulated in the context of the old quantum theory, in which the link between radiation frequencies and orbital frequencies is severed. Drawing on these efforts, especially by Ladenburg and Reiche in the early 1920s, and using Bohr’s correspondence principle and other ingredients of the old quantum theory, Kramers found a successful quantum formula in late 1923. He only published a full treatment of his new dispersion theory in a paper with Heisenberg in early 1925. This paper played a key role in laying the foundations of matrix mechanics.
Abstract This chapter explores Schrödinger’s introduction of wave mechanics. After briefly covering Schrödinger’s earlier work, it turns to the investigations that led him to write a four-part paper in 1926 introducing and applying wave mechanics. The chapter provides a detailed analysis of these four papers, in which the analogy of the new theory to the Hamilton-Jacobi formulation of classical mechanics plays a central role, as well as the paper written in between the second and the third installment on the relation between wave and matrix mechanics. The chapter also discusses Schr "odinger’s original interpretation of his wave functions and the reception of his theory by the physics community.
In July 1925, exactly one century ago, Werner Heisenberg created Quantum Mechanics (QM) as an invariant-operational formalism by taking as a standpoint the intensive patterns that had been already observed by experimentalists in the lab. One of the few drawbacks of this proposal was the impossibility to conceptualize it in classical terms. Attempting to restore a somewhat classical spatiotemporal representation, six months later, Erwin Schrödinger would present a formalism grounded on a wave equation in configuration space. Taking as a standpoint Schrödinger’s wave mechanics together with the methodological guide of Bohr and logical positivists, Dirac would present in 1930 an axiomatic re-formulation of the theory of quanta that would become known as the “standard” version of QM (SQM). This new version of the theory, that is still today taught in physics classrooms all around the world, would become accepted regardless of its inconsistent narrative as a “recipe” intended (but unable) to predict (binary) measurement outcomes. After pointing out some of the many problems of SQM, in this work we propose to go back to Heisenberg’s matrix mechanics (and also to his understanding of physical theories). We will argue that his matrix mechanics, as an operationally-invariant standpoint, opens the door to a tensorial enlargement of the mathematical formalism capable to account for new phenomena.