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European Journal of Physics

The road to matrix mechanics: II. Ladenburg’s quantum interpretation of optical dispersion

L. Crivellari

Published Jun 30, 2016• 0 citations
Publisher

Abstract

Abstract This paper reviews Ladenburg’s development of the phenomenological theory of radiative transitions between the stationary states of an atom put forward by Einstein in 1917. The historical background as well as the far reaching outcomes of his work are considered and discussed; among them the Kramers–Heisenberg quantum dispersion theory that paved the way to Heisenberg’s formulation of matrix mechanics and the quantum-mechanical calculation of the spectral line profiles.

DOI
10.1088/0143-0807/37/5/055405

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Abstract This paper reviews Ladenburg’s development of the phenomenological theory of radiative transitions between the stationary states of an atom put forward by Einstein in 1917. The historical background as well as the far reaching outcomes of his work are considered and discussed; among them the Kramers–Heisenberg quantum dispersion theory that paved the way to Heisenberg’s formulation of matrix mechanics and the quantum-mechanical calculation of the spectral line profiles.
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 Another main stream in the initial formulation of the quantum theory was concerned with the interpretation of atomic spectral lines. The interaction of electromagnetic radiation with atoms was discussed by Einstein f4] in 1917. His theory of the absorption and emission of light by an atom depends upon simple phenomenological considerations but it leads to predictions that are reproduced by the more formal quantum mechanics developed later. The use of quantum theory is, of course, essential for even a gross description of the nature of the states of an atom and the theory is outstandingly successful in its ability to predict the finest details of atomic energy-level structures.
The quantum theoretical description of microscopic systems discussed so far is based on differential operators and the solution of the underlying eigenvalue equations. This approach is called Schrödinger approach or wave mechanics. Heisenberg developed another equivalent approach called matrix mechanics. Because the trajectory of an electron is unobservable and moreover line spectra implied discrete allowed states, Heisenberg discarded the idea of electron orbits and formulated a theory for line spectra of excited systems based on observables. When Heisenberg arranged the allowed set of transitions in a matrix form, to his surprise, each element of the matrix was found to be related to the probability of a particular transition. This gave the birth to matrix mechanics. Then Heisenberg, Born and Jordan worked out the rules of these matrices.
The report by Born and Heisenberg on ‘quantum mechanics’ may seem surprisingly difficult to the modern reader. This is partly because Born and Heisenberg are describing various stages of development of the theory that are quite different from today's quantum mechanics. Among these, it should be noted in particular that the theory developed by Heisenberg, Born and Jordan in the years 1925–6 and known today as matrix mechanics (Heisenberg 1925b [1], Born and Jordan 1925 [2], Born, Heisenberg and Jordan 1926 [4]) a differs from standard quantum mechanics in several important respects. At the same time, the interpretation of the theory (the topic of Section II of the report) also appears to have undergone important modifications, in particular regarding the notion of the state of a system. Initially, Born and Heisenberg insist on the notion that a system is always in a stationary state (performing quantum jumps between different stationary states). Then the notion of the wave function is introduced and related to probabilities for the stationary states. At a later stage, probabilistic notions (in particular, what one now calls transition probabilities) are extended to arbitrary observables, but it remains somewhat unclear whether the wave function itself should be regarded as a fundamental entity or merely as an effective one. This may reflect the different routes followed by Born and by Heisenberg in the development of their ideas. The common position presented by Born and Heisenberg emphasises the probabilistic aspect of the theory as fundamental, and the conclusion of the report expresses strong confidence in the resulting picture.