1999/10/04 by Toshiki Shimbori, Tsunehiro Kobayashi
Physics and Astronomy · Mathematics · #math-ph #math.MP
published as Nuovo Cim. B115 (2000) 325-342 · 18 pages
arxiv created 1999/10/04 · arxiv updated 2009/11/30
The paper deals with the one-dimensional parabolic potential barrier V(x)=V0-mγ2 x2/2, as a model of an unstable system in quantum mechanics. The time-independent Schrödinger equation for this model is set up as the eigenvalue problem in Gel'fand triplet and its exact solutions are expressed by generalized eigenfunctions belonging to complex energy eigenvalues V0∓ i\Gammavn whose imaginary parts are quantized as \Gammavn=(n+1/2)\hslashγ. Under the assumption that time factors of an unstable system are square integrable, we provide a probabilistic interpretation of them. This assumption leads to the separation of the domain of the time evolution, namely all the time factors belonging to the complex energy eigenvalues V0-i\Gammavn exist on the future part and all those belonging to the complex energy eigenvalues V0+i\Gammavn exist on the past part. In this model the physical energy distributions worked out from these time factors are found to be the Breit-Wigner resonance formulas. The half-widths of these physical energy distributions are determined by the imaginary parts of complex energy eigenvalues, and hence they are also quantized.