The equality of Schrödinger’s Theory and Heisenberg’s S-matrix Theory

Ana Marı́a Cetto, Luis de la Peña

Chapter 4 shows that, just as Heisenberg’s matrix mechanics represented a break with classical physics and the beginning of a new physics, Schrödinger’s wave mechanics represented another, no less radical, break with classical physics and at the same time opened up another path to the quantum world. Schrödinger’s theory is shown to make substantial use of de Broglie’s waves and related phenomena. The Schrödinger equation, being wave-like, constitutes mathematically an eigenvalue problem for bounded systems, and thus naturally leads to quantization, as is illustrated with a couple of simple examples. The chapter ends with Dirac’s abstract formulation, which has the advantage of being suitable for any description, be it Heisenberg’s, Schrödinger’s or any other.

Peter Woit

Joe McEvoy, Oscar Zarate

de Gosson, Anthony Maurice

The aim of this short Note is to show that the Schrodinger and Heisenberg pictures of quantum mechanics cannot be equivalent unless one uses a quantization rule clearly stated by Born and Jordan in their famous 1925 paper. This rule is sufficient and necessary to ensure energy conservation in Heisenberg's matrix mechanics. It follows, in particular, that Schrodinger and Heisenberg mechanics yield different theories if one quantizes the Hamiltonian using the Weyl prescription.

Joe McEvoy, Oscar Zarate

John Day Dollard

The time-dependent Schrödinger theory of scattering is studied rigorously with the potential Vreplaced by e−ε|l|V. Sufficient conditions are given that the Møller wave-matrices Ω± be obtainable from this theory as ε → 0. The conjecture that this theory can be used to define a reasonable S-matrix when the Ω± do not exist is shown to be false for the Coulomb potential. It is remarked that the Goldberger-Gell-Mann switching procedure also breaks down in this case.