We believe, that alpha particle goes out of nuclei with probability ~exp(-gamma*t), but this is only asymptotic formula. Can we choose some potential, where the formula will be "exact"?
I checked the references. My task is of other type: at time t=0 all wave function is inside barrier. Then it partly goes outside of barrier. I want to know probability to be inside barrier on time t. It must be sorta exponent.
It is "quasi-stationary" state. The total probability is not conserved because of probability flow at infinity.
Total probability is integral from 0 to infinity. If it's = 1 in initial time t=0, then it will sonserved in the future t>0 as followed from Shrodinger equation.
Gert Van der Zwan,
As I understand, scattering and decay are different processes with different parameters. In classical textbook on quantum mechanics I find out decay formula, derived from pass metohd of complex integration, but this is approximation.
This time dependent problem was solved explicitly for the delta barrier: R.G. Winter, Phys. Rev. 123 (4) 1503 (1961). The probability to remain in the trap was found to decay as t^{-3/2}, not exponentially, as one might have expected.
You can read our papers on the problem of quantum decay: There we found an analytical solution of the time evolution of quantum decay of an arbitrary initial state located within a finite range arbitrary potential. We showed that the decay is exponential for a long period of time, but there is a crossover to an asymptotically nonexponential power-law decay.
1) G. García-Calderon, J. L. Mateos, and M. Moshinsky. "Resonant Spetra and the Time Evolution of the Survival and Nonescape Probabilities".
Phys. Rev. Lett. 74 (3) (1995) 337.
2) G. García-Calderon, J. L. Mateos, and M. Moshinsky. Survival and Nonescape Probabilities for Resonant and Nonresonant Decay".
Annals of Physics 249 (1996) 430.
Also, G. García-Calderon, J. L. Mateos, and M. Moshinsky. Phys Rev. Lett. 80 (1998) 4354; Phys. Rev. Lett. 90 (2003) 028902.
Best regards.
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