Abstract
An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t → ∞ more slowly than any exponential function of t. This can be described in terms of the time-dependent decay rate γ(t) which, when considered with the Khalfin result, means that this γ(t) is not a constant for large t but that it tends to zero as t → ∞. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the “instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t → ∞) and relatively short (lifetime of the state) time regions, are different. This is a purely quantum mechanical effect. It is hypothesized that there is a possibility to detect this effect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
[1] L. P. Horwitz, J. P. Marchand, Rocky Mt. J. Math. 1, 225 (1971) http://dx.doi.org/10.1216/RMJ-1971-1-1-22510.1216/RMJ-1971-1-1-225Search in Google Scholar
[2] W. Królikowski, J. Rzewuski, Nuovo Cimento B 25, 739 (1975) http://dx.doi.org/10.1007/BF0272474910.1007/BF02724749Search in Google Scholar
[3] R. Zwanzig, Physica 30, 1109 (1964) http://dx.doi.org/10.1016/0031-8914(64)90102-810.1016/0031-8914(64)90102-8Search in Google Scholar
[4] A. Agodi, M. Baldo, E. Recami, Ann. Phys. 77, 157 (1973) http://dx.doi.org/10.1016/0003-4916(73)90413-210.1016/0003-4916(73)90413-2Search in Google Scholar
[5] F. Haake, Staistical Treatment of Open Systems by Generalized master Equations, Springer Tracts in Modern Physics Vo. 66 (Springer, Berlin, 1973) 10.1007/978-3-662-40468-3_2Search in Google Scholar
[6] K. Urbanowski, Phys. Rev. A 50, 2847 (1994) http://dx.doi.org/10.1103/PhysRevA.50.284710.1103/PhysRevA.50.2847Search in Google Scholar
[7] Y. Aharonov and D. Rohrlich, Quantum Paradoxes (Wiley- VCH, Weinheim, 2005) http://dx.doi.org/10.1002/978352761911510.1002/9783527619115Search in Google Scholar
[8] Y. Aharonov, L. Vaidman, J. Phys. A-Math. Gen. 24, 2315 (1991) http://dx.doi.org/10.1088/0305-4470/24/10/01810.1088/0305-4470/24/10/018Search in Google Scholar
[9] Y. Aharonov, L. Vaidman, Phys. Rev. A 41, 11 (1990) http://dx.doi.org/10.1103/PhysRevA.41.1110.1103/PhysRevA.41.11Search in Google Scholar PubMed
[10] Y. Aharonav, D. Albert and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988) http://dx.doi.org/10.1103/PhysRevLett.60.135110.1103/PhysRevLett.60.1351Search in Google Scholar PubMed
[11] P. C. W. Davies, arXiv:0807.1357 Search in Google Scholar
[12] V. F. Weisskopf, E. T. Wigner, Z. Phys. 63, 54 (1930) http://dx.doi.org/10.1007/BF0133676810.1007/BF01336768Search in Google Scholar
[13] V. F. Weisskopf, E. T. Wigner, Z. Phys. 65, 18 (1930) http://dx.doi.org/10.1007/BF0139740610.1007/BF01397406Search in Google Scholar
[14] M. L. Goldberger, K. M. Watson, Collision Theory (Willey, New York, 1964) 10.1063/1.3051231Search in Google Scholar
[15] S. Krylov, V. A. Fock, Zh. Eksp. Teor. Fiz.+ 17, 93 (1947) Search in Google Scholar
[16] L. A. Khalfin, Zh. Eksp. Teor. Fiz.+ 33, 1371 (1957) Search in Google Scholar
[17] L. A. Khalfin, Sov. Phys. JETP-USSR 6, 1053 (1958) Search in Google Scholar
[18] R. E. A. C. Paley, N. Wiener, Fourier transforms in the comlex domain (American Mathematical Society, New York, 1934) Search in Google Scholar
[19] R. G. Newton, Scattering Theory of Waves and Particles, 2nd edtition (Springer, New York, 1982) 10.1007/978-3-642-88128-2Search in Google Scholar
[20] L. Fonda, G. C. Ghirardii, A. Rimini, Rep. Prog. Phys. 41, 587 (1978) http://dx.doi.org/10.1088/0034-4885/41/4/00310.1088/0034-4885/41/4/003Search in Google Scholar
[21] A. Peres, Ann. Phys. 129, 33 (1980) http://dx.doi.org/10.1016/0003-4916(80)90288-210.1016/0003-4916(80)90288-2Search in Google Scholar
[22] P. T. Greenland, Nature 335, 298 (1988) http://dx.doi.org/10.1038/335298a010.1038/335298a0Search in Google Scholar
[23] D. G. Arbo, M. A. Castagnino, F. H. Gaioli, S. Iguri, Physica A 227, 469 (2000) http://dx.doi.org/10.1016/S0378-4371(99)00480-X10.1016/S0378-4371(99)00480-XSearch in Google Scholar
[24] J. M. Wessner, D. K. Andreson, R. T. Robiscoe, Phys. Rev. Lett. 29, 1126 (1972) http://dx.doi.org/10.1103/PhysRevLett.29.112610.1103/PhysRevLett.29.1126Search in Google Scholar
[25] E. B. Norman, S. B. Gazes, S. C. Crane, D. A. Bennet, Phys. Rev. Lett. 60, 2246 (1988) http://dx.doi.org/10.1103/PhysRevLett.60.224610.1103/PhysRevLett.60.2246Search in Google Scholar PubMed
[26] E. B. Norman, B. Sur, K. T. Lesko, R.-M. Larimer, Phys. Lett. B 357, 521 (1995) http://dx.doi.org/10.1016/0370-2693(95)00818-610.1016/0370-2693(95)00818-6Search in Google Scholar
[27] J. Seke, W. N. Herfort, Phys. Rev. A 38, 833 (1988) http://dx.doi.org/10.1103/PhysRevA.38.83310.1103/PhysRevA.38.833Search in Google Scholar
[28] R. E. Parrot, J. Lawrence, Europhys. Lett. 57, 632 (2002) http://dx.doi.org/10.1209/epl/i2002-00509-010.1209/epl/i2002-00509-0Search in Google Scholar
[29] J. Lawrence, Journ. Opt. B: Quant. Semiclass. Opt. 4, S446 (2002) http://dx.doi.org/10.1088/1464-4266/4/4/33710.1088/1464-4266/4/4/337Search in Google Scholar
[30] I. Joichi, Sh. Matsumoto, M. Yoshimura, Phys. Rev. D 58, 045004 (1998) http://dx.doi.org/10.1103/PhysRevD.58.04500410.1103/PhysRevD.58.045004Search in Google Scholar
[31] N. G. Kelkar, M. Nowakowski, K. P. Khemchandani, Phys. Rev. C 70, 024601 (2004) http://dx.doi.org/10.1103/PhysRevC.70.02460110.1103/PhysRevC.70.024601Search in Google Scholar
[32] M. Nowakowski, N. G. Kelkar, AIP Conf. Proc. 1030, 250 (2008) http://dx.doi.org/10.1063/1.297350810.1063/1.2973508Search in Google Scholar
[33] T. Jiitoh, S. Matsumoto, J. Sato, Y. Sato, K. Takeda, Phys Rev. A 71, 012109 (2005) http://dx.doi.org/10.1103/PhysRevA.71.01210910.1103/PhysRevA.71.012109Search in Google Scholar
[34] C. Rothe, S. I. Hintschich, A. P. Monkman, Phys. Rev. Lett. 96, 163601 (2006) http://dx.doi.org/10.1103/PhysRevLett.96.16360110.1103/PhysRevLett.96.163601Search in Google Scholar PubMed
[35] K. M. Sluis, E. A. Gislason, Phys. Rev. A 43, 4581 (1991) http://dx.doi.org/10.1103/PhysRevA.43.458110.1103/PhysRevA.43.4581Search in Google Scholar PubMed
[36] M. Abramowitz, I. A. Stegun (Eds.), Handbook of Mathematical Functions, Natl. Bur. Stand. Appl. Math. Ser. No 55 (U.S. GPO, Washington, D.C., 1964) Search in Google Scholar
[37] R. M. Corless, G. H. Gonet, D. E. G. Hare, D. J. Jeffrey, D. E. Khnut, Adv. Comput. Math. 5, 329 (1996) http://dx.doi.org/10.1007/BF0212475010.1007/BF02124750Search in Google Scholar
[38] F. W. J. Olver, Asymptotics and special functions (Academic Press, New York, 1974) Search in Google Scholar
[39] F. M. Dittes, H. L. Harney, A. Müller, Phys. Rev. A 45, 701 (1992) http://dx.doi.org/10.1103/PhysRevA.45.70110.1103/PhysRevA.45.701Search in Google Scholar
[40] W. Heitler, The Quantum Theory of Radiation (Oxford University Press, London, 1954; Dover Publications, New York, 1984) Search in Google Scholar
[41] E. W. Kolb, M. S. Turner, The Early Universe (Addison-Wesley Publ. Co., 1993) Search in Google Scholar
© 2009 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
