Quantum Science pp 277-294 | Cite as

# 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.

## Keywords

Dirac Equation Adjoint Equation Schroedinger Equation Orthogonal Idempotent Symplectic Matrice## Preview

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## References

- 1.H. Weyl, “über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklungen willkürlicher Funktionen,” Mathematische Annalen 68, 220–269 (1910).MathSciNetMATHCrossRefGoogle Scholar
- 2.H.V. McIntosh, Quantization as an Eigenvalue Problem, in Group Theory and its Applications (E.M. Loebl, ed.), Academic Press, New York, 1975. pp 333–368.Google Scholar
- 3.George D. Birkhoff and Rudolph E. Langer, “The Boundary Problems and Developments Associated with a System of Ordinary Linear Differential Equations of the First Order,” Proceedings of the American Academy of Arts and Sciences 58, 49–128 (1923).Google Scholar
- George D. Reprinted in Collected Mathematical Papers of George David Birkhoff, Dover Publications Inc., New York, 1968. pp. 345–42b.Google Scholar
- 4.F.R. Gantmacher, The Theory, of Matrices, Chelsea Publishing Company, New York, 1959. Volume II, pp. 125–131.Google Scholar