Abstract
In Section XXI.1 the decaying state is introduced as a resonance for which the production process is ignored. Section XXI.2 gives a heuristic discussion of decay probability (decay rate) and its measurement. Section XXI.3 describes a decaying state by a generalized eigenvector of a Hermitian energy operator with complex eigenvalue. Using this novel concept, the calculation of the decay rate is very simple and is given in Section XXI.5. Section XXI.6 contains a discussion of the partial decay rates and the use of various basis systems for their calculation.
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References
A Beskow, J. S. Nilsson, Arkiv för Fysik 34, 561 (1967).
For a review of these subjects the reader is referred to L. Fonda, G. C. Ghirardi, and A. Rimini. Decay theory of unstable quantum systems. Reports on Progress in Physics 41, 587 (1978). Further methods not discussed in this review can be found in A. P. Grecos: Decaying states in quantum systems. Singularities and Dynamical Systems, edited by F. N. Pnevmatikos, North Holland, Amsterdam (1985) and
A. George, F. Henin, F. Mayne, I. Prigogine, New quantum rules for dissipative systems. Hadronic Journal 1, 520 (1978).
The first line of (3.1) has been proven in Appendix XV.A.
See, e.g., Equation (II.8.12).
As is well known to specialists, in the usual precise Hilbert space formulation one obtains deviations from the exponential decay law [L. Fonda et al. (1978)]. Deviations for large values of t follow from the condition that the spectrum of H be bounded from below which corresponds to a finite lower limit in the integrals (3.5), (3.8), (3.24), (3.26) and which is not the case if one integrates from — ∞ to + ∞ as we did to prove (3.34). Deviations from the exponential law for small times t follow from the condition that the energy in the decaying state be finite, which corresponds to the condition that the decaying state vector be in the domain of the Hilbert space operator H (or even in Φ, the domain of H) which is also not the case for the vector I fD> as shown in Problem l.
P. L. Duren, Theory of Hp-Spaces, Academic Press, New York, 1970.
C. van Winter, J. Math. Anal. 47, 633 (1974) ;
C. van Winter, Trans. Am. Math. Soc. 162, 103 (1971).
P. L. Duren, ibid., Theorem 11.8.
P. L. Duren, ibid., Theorem 11.9.
For details see A. Bohm, M. Gadella (1986).
For example, A. Bohm (1978), Chapter III.
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© 1986 Springer Science+Business Media New York
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Bohm, A. (1986). The Decay of Unstable Physical Systems. In: Quantum Mechanics: Foundations and Applications. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-01168-3_21
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DOI: https://doi.org/10.1007/978-3-662-01168-3_21
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