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Blurred orbits and blurred particles: Heisenberg's 1926 helium atom

G. Gimeno, M. Xipell, Enric Pérez

TL;DR

This paper analyzes Heisenberg's 1926 helium work, focusing on the blurred orbit imagery and the early use of particle indistinguishability. It shows how Heisenberg combined matrix mechanics with Schrödinger wave functions to address two-electron systems, using symmetry considerations to distinguish para- and ortho-helium and to derive Rydberg corrections. The authors argue that Heisenberg's attempts to relate Bose–Einstein counting to quantum mechanics were incomplete and that Dirac soon clarified the role of symmetry in quantum statistics. The piece contextualizes these developments within the broader shift from classical orbital pictures to a statistical, superposition-based view of quantum systems, highlighting the historical significance for the foundations of quantum statistics and atomism.

Abstract

This work analyses the extent to which the "blurred orbits" of the current model for the atom, drafted by Heisenberg in 1926, fits the image of a bunch of wandering electrons around a nucleus. We will deal with early appearances of the concept of indistinguishable particles within the frame of quantum mechanics. There are few studies on the use of this concept in Heisenberg's 1926 papers on helium, in contrast with the large number of them on its use in Bose--Einstein's 1924 papers. We will discuss to what extent Heisenberg's approach leads to a purely statistical interpretation of this concept. We will also study the viewpoint of Dirac, who dealt with the same topic few months later. Although the indistinguishability of the electrons and indeterminacy are common explanations for the blurring of electron orbits, we argue that such an image is an oversimplification which masks interesting aspects of: a) the dynamics of bound electrons and b) the deeper implications of indistinguishability in quantum mechanics.

Blurred orbits and blurred particles: Heisenberg's 1926 helium atom

TL;DR

This paper analyzes Heisenberg's 1926 helium work, focusing on the blurred orbit imagery and the early use of particle indistinguishability. It shows how Heisenberg combined matrix mechanics with Schrödinger wave functions to address two-electron systems, using symmetry considerations to distinguish para- and ortho-helium and to derive Rydberg corrections. The authors argue that Heisenberg's attempts to relate Bose–Einstein counting to quantum mechanics were incomplete and that Dirac soon clarified the role of symmetry in quantum statistics. The piece contextualizes these developments within the broader shift from classical orbital pictures to a statistical, superposition-based view of quantum systems, highlighting the historical significance for the foundations of quantum statistics and atomism.

Abstract

This work analyses the extent to which the "blurred orbits" of the current model for the atom, drafted by Heisenberg in 1926, fits the image of a bunch of wandering electrons around a nucleus. We will deal with early appearances of the concept of indistinguishable particles within the frame of quantum mechanics. There are few studies on the use of this concept in Heisenberg's 1926 papers on helium, in contrast with the large number of them on its use in Bose--Einstein's 1924 papers. We will discuss to what extent Heisenberg's approach leads to a purely statistical interpretation of this concept. We will also study the viewpoint of Dirac, who dealt with the same topic few months later. Although the indistinguishability of the electrons and indeterminacy are common explanations for the blurring of electron orbits, we argue that such an image is an oversimplification which masks interesting aspects of: a) the dynamics of bound electrons and b) the deeper implications of indistinguishability in quantum mechanics.

Paper Structure

This paper contains 21 sections, 37 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The preceding years 1923-1925: The banishing of electron orbits
  3. Heisenberg's new approach to the helium problem
  4. Matrices and wave functions
  5. The quantitative viewpoint
  6. Synthesis of Heisenberg's papers
  7. Quantum statistics in Spring 1926
  8. Indistinguishability "à la Heisenberg"
  9. Exchanging places
  10. A new atomism
  11. Connection with the new statistics
  12. Subsequent developments: Dirac
  13. Discussion
  14. Indistinguishable/identical particles
  15. Superposition vs. indistinguishability and indeterminacy in the forsaking of orbital motions
  16. ...and 6 more sections

Figures (4)

  • Figure 1: Examples of periodic configurations of the electron pair in helium that served as classical models for the ground state: (a) Bohr, 1913; (b) and (d) Langmuir, 1921; (c) Landé, 1919; (e) Kemble, 1921 and Kramers, 1923 (See RefWorks:1433)
  • Figure 2: Heisenberg's first (classical) approach
  • Figure 3: Heisenberg's Fig. 4. An interpretation of Heisenberg's approach under quantum (matrix) considerations
  • Figure 4: Heisenberg's Fig. 5. Para- and ortho-helium scheme considering spin. Now the • (resp. +) represent antisymmetric (resp. symmetric) global state functions