Quantization and a Green’s Function for Systems of Linear Ordinary Differential Equations

Abstract

One of the traditional puzzles for students of quantum mechanics is the reconciliation of the quantification principle that wave functions must be square integrable with the reality that continuum wave functions do not respect this requirement. By protesting that neither are they quantized the problem can be sidestepped, although some ingenuity may still be required to find suitable boundary conditions. Further difficulties await later on when particular systems are studied in more detail. Sometimes all the solutions, and not just some of them, are square integrable. Then it is necessary to resort to another principle, such as continuity or finiteness of the wave function, to achieve quantization. Examples where such steps have to be taken can be found both in the Schroedinger equation and the Dirac equation. The ground state of the hydrogen atom, the hydrogen atom in Minkowski space, the theta component of angular momentum, all pose problems for the Schroedinger equation. Finiteness of wavefunction alone is not a reliable principle because it fails in the radial equation of the Dirac hydrogen atom. Thus the quantizing conditions which have been invoked for one potential or another seem to be quite varied.