Background

Abstract

Quantum mechanics was born in two different guises, both inspired by the problem of explaining the existence of stable energy levels in an atom. Heisenberg had started from the idea that, in some sense, the position coordinates x⁽ⁱ⁾ and momentum coordinates p⁽ⁱ⁾of electrons should be replaced by doubly indexed arrays x ⁽ⁱ⁾ _nm ,p ⁽ⁱ⁾ _nm where the indices n, m label the energy levels and the numbers x ⁽ⁱ⁾ _nm ,p ⁽ⁱ⁾ _nm are related to the intensity of spectral lines for a transition between levels n and m. It was quickly realized by Born and Jordan that, mathematically, the double indexed schemes should be regarded as infinite matrices and the essential relations could be simply expressed in terms of matrix multiplication. Schrödinger’s starting idea is seen in the title of his first paper on the subject: “Quantization as an eigenvalue problem”. Following up the suggestion by de Broglie he assumed that classical mechanics results from a more fundamental wave theory by the same approximation by which geometric optics results from wave optics. He replaced the Hamilton function of classical mechanics by a differential operator acting on a wave function. Its eigenfunctions should correspond to the stationary states, its eigenvalues to the allowed energy levels.