Rigged Hilbert spaces and time asymmetry: The case of the upside-down simple harmonic oscillator
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The upside-down simple harmonic oscillator system is studied in the contexts of quantum mechanics and classical statistical mechanics. It is shown that in order to study in a simple manner the creation and decay of a physical system by way of Gamow vectors we must formulate the theory in a time-asymmetric fashion, namely using two different rigged Hilbert spaces to describe states evolving toward the past and the future. The spaces defined in the contexts of quantum and classical statistical mechanics are shown to be directly related by the Wigner function.
KeywordsHilbert Space Wigner Function Liouville Equation Liouvillian Operator Generalize Eigenvector
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- Aquilano, R., and Castagnino, M. (1996).Modern Physics Letters A, in press.Google Scholar
- Bohm, A., and Gadella, M. (1989).Dirac Kets, Gamow Vectors and Gel'fand Triplets: The Rigged Hilbert Space Formulation of Quantum Mechanics, Springer-Verlag, Berlin.Google Scholar
- Cohen-Tannoudji, C., Diu, B., and Laloe, F. (1977).Quantum Mechanics, Vol. 1, Wiley, New York.Google Scholar
- Davies, P. C. W. (1994). Stirring up trouble, inPhysical Origin of Time Asymmetry J. J. Haliwellet al., eds., Cambridge University Press, Cambridge.Google Scholar
- Gel'fand, I. M., and Shilov, N. Y. (1964).Generalized Functions, Vol. 4,Applications of Harmonic Analysis, Academic Press, New York.Google Scholar
- Merzbacher, E. (1970).Quantum Mechanics, 2nd ed., Wiley, New York.Google Scholar
- Reichenbach, H. (1956).The Direction of Time, University of California Press, Berkeley.Google Scholar