Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Consolidation of Matrix Mechanics: Born–Jordan, Dirac and the Three-Man-Paper

Anthony Duncan A, Michel Janssen B

Published Aug 17, 2023• 0 citations
Publisher

Abstract

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

DOI
10.1093/oso/9780198883906.003.0005

Related papers

Heisenberg's Matrix Quantum Mechanics

Caio Lima Firme

This chapter introduces Heisenberg’s trajectory from old quantum mechanics towards his quantum matrix mechanics. It shows his partnership with Born and for the study of the helium atom and molecules under the old quantum theory and his work on the core model to account for the anomalous Zeeman effect in collaboration with Landé. The collaboration between Heisenberg Kramers to present the treatment between matter and radiation under the scope of the old quantum mechanics is also displayed. Then, afterwards, it is shown the birth of the matrix mechanics by its three initial papers in 1925: the Heisenberg’s paper where he replaced the motion of the electron into its transition; the Born and Jordan’s paper where they placed Heisenberg equation into matrices and found the corresponding equation of motion and the commutator between linear momentum and position of the electron; the Heisenberg, Jordan and Born’s paper where they obtained the diagonal elements of the Hamiltonian matrix (the energy of a quantum state) and its application to one-electron harmonic oscillator. Dirac’s contribution to matrix mechanics is also introduced followed the application in the quantum angular momentum (184 words).

CRC Press eBooks•Mar 21, 2022•0 citations

Matrix Aspects of Quantum Mechanics

F.J. Duarte

As mentioned in Chapter 1 , the Heisenberg matrix mechanics provides one of the three main avenues to quantum mechanics. This approach to quantum mechanics was disclosed in three papers authored by Heisenberg (1925) , Born and Jordan (1925) , and Born et al. (1926) . An iconic result from the Heisenberg–Born–Jordan contribution was the commutation rule ( 14.1 ) p q − q p = h 2 π i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315216720/70121b16-f2c6-4331-b1ba-eecb806b93da/content/eq595.tif"/>

CRC Press eBooks•Oct 29, 2013•0 citations

Born–Jordan Quantization and the Equivalence of the Schrödinger and Heisenberg Pictures

Maurice A. de Gosson

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.

Foundations of Physics•Sep 4, 2014•24 citations

The 1925 Born and Jordan paper “On quantum mechanics”

William A. Fedak, Jeffrey J. Prentis

The 1925 paper “On quantum mechanics” by M. Born and P. Jordan, and the sequel “On quantum mechanics II” by M. Born, W. Heisenberg, and P. Jordan, developed Heisenberg’s pioneering theory into the first complete formulation of quantum mechanics. The Born and Jordan paper is the subject of the present article. This paper introduced matrices to physicists. We discuss the original postulates of quantum mechanics, present the two-part discovery of the law of commutation, and clarify the origin of Heisenberg’s equation. We show how the 1925 proof of energy conservation and Bohr’s frequency condition served as the gold standard with which to measure the validity of the new quantum mechanics.

American Journal of Physics•Jan 12, 2009•47 citations

Born--Jordan Quantization and the Equivalence of Matrix and Wave Mechanics

Maurice A. de Gosson

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg's theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger's wave mechanics cannot be equivalent to Heisenberg's more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born-Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born--Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.

arXiv:1405.2519•May 11, 2014•1 citation

The invention of quantum mechanics—matrix mechanics

Robert Golub, S. K. Lamoreaux

Abstract Driven by the idea that a physical theory should only deal with observable quantities, Werner Heisenberg, a young post-doc working with Max Born in Göttingen, who had earlier done his Ph.D. work under Sommerfeld in Munich, was led to introduce a new type of object which represented the transitions between Bohr-Sommerfeld orbits. Max Born realized and recognized these objects as matrices. Together with Pascual Jordan, another post-doc, they worked out a complete theory, matrix mechanics. They completed this work in a few months. We discuss the role of matrix mechanics in the development of quantum mechanics.

•Feb 28, 2023•0 citations
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
This chapter introduces Heisenberg’s trajectory from old quantum mechanics towards his quantum matrix mechanics. It shows his partnership with Born and for the study of the helium atom and molecules under the old quantum theory and his work on the core model to account for the anomalous Zeeman effect in collaboration with Landé. The collaboration between Heisenberg Kramers to present the treatment between matter and radiation under the scope of the old quantum mechanics is also displayed. Then, afterwards, it is shown the birth of the matrix mechanics by its three initial papers in 1925: the Heisenberg’s paper where he replaced the motion of the electron into its transition; the Born and Jordan’s paper where they placed Heisenberg equation into matrices and found the corresponding equation of motion and the commutator between linear momentum and position of the electron; the Heisenberg, Jordan and Born’s paper where they obtained the diagonal elements of the Hamiltonian matrix (the energy of a quantum state) and its application to one-electron harmonic oscillator. Dirac’s contribution to matrix mechanics is also introduced followed the application in the quantum angular momentum (184 words).
As mentioned in Chapter 1 , the Heisenberg matrix mechanics provides one of the three main avenues to quantum mechanics. This approach to quantum mechanics was disclosed in three papers authored by Heisenberg (1925) , Born and Jordan (1925) , and Born et al. (1926) . An iconic result from the Heisenberg–Born–Jordan contribution was the commutation rule ( 14.1 ) p q − q p = h 2 π i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315216720/70121b16-f2c6-4331-b1ba-eecb806b93da/content/eq595.tif"/>
The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.
The 1925 paper “On quantum mechanics” by M. Born and P. Jordan, and the sequel “On quantum mechanics II” by M. Born, W. Heisenberg, and P. Jordan, developed Heisenberg’s pioneering theory into the first complete formulation of quantum mechanics. The Born and Jordan paper is the subject of the present article. This paper introduced matrices to physicists. We discuss the original postulates of quantum mechanics, present the two-part discovery of the law of commutation, and clarify the origin of Heisenberg’s equation. We show how the 1925 proof of energy conservation and Bohr’s frequency condition served as the gold standard with which to measure the validity of the new quantum mechanics.
The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg's theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger's wave mechanics cannot be equivalent to Heisenberg's more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born-Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born--Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.
Abstract Driven by the idea that a physical theory should only deal with observable quantities, Werner Heisenberg, a young post-doc working with Max Born in Göttingen, who had earlier done his Ph.D. work under Sommerfeld in Munich, was led to introduce a new type of object which represented the transitions between Bohr-Sommerfeld orbits. Max Born realized and recognized these objects as matrices. Together with Pascual Jordan, another post-doc, they worked out a complete theory, matrix mechanics. They completed this work in a few months. We discuss the role of matrix mechanics in the development of quantum mechanics.