Singular Perturbation Series in Quantum Mechanics

✍ Scribed by John Stalker

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 189 KB
• Volume: 141
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1789

✦ Synopsis

This is the first in a series of articles on singular perturbation series in quantum mechanics. In these papers we will compute transition amplitudes non-perturbatively for several simple quantum mechanical systems. This paper treats the behavior of a harmonic oscillator as a function of the spring constant. All the systems to be considered exhibit the same phenomenon: there is, in general, no power series expansion, even of an asymptotic nature, in powers of the perturbation parameter. For reasonable initial and final states there are, however, expansions of a more complicated type. In the case of the harmonic oscillator there is a log-power series expansion. This example is sufficiently simple and explicit that one can see how the logarithmic terms arise mathematically. How they arise physically is rather more mysterious. One point however is clear; the standard techniques of perturbation theory, whether expressed in terms of Dyson series, Feynman diagrams, successive approximation, or path integrals, are all structured so as to produce only power series and thus fail to detect logarithmic terms.