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Unformatted text preview: previous index next The Energy-Time Uncertainty Principle: Decaying States and Resonances Michael Fowler 10/1/08 Model of a Decaying State The momentum-position uncertainty principle p x = has an energy-time analog, . Evidently, though, this must be a different kind of relationship to the momentum- position one, because t is not a dynamical variable, so this cant have anything to do with non- commutation. E t = To illustrate the meaning of the equation E t = , let us reconsider -decay, but with a slightly simplified potential to clarify whats going on. Specifically, we replace the combined nuclear force/electrostatic repulsion barrier with a square barrier, high enough and thick enough that there is a small probability per unit time of the particle tunneling out of the well. V V ( r ) Radial potential for toy model of alpha decay r 0 well r exit r If the barrier thickness were increased to infinity (keeping r fixed) there would be a true bound state having energy E , and for E well below V , having approximately an integral number of half wavelengths in the well. For a barrier of finite thickness, there is of course some nonzero probability of the particle escapingso no longer a true bound state, but for a thick barrier the difference may be hard to detect. As with the -decay analysis, well look at this semiclassically, picturing the particle as bouncing off the walls backwards and forwards inside, time between hits, and at each hit probability of penetration some small number . Therefore, the probability that the particle is still in the well after a time t n = is ( ) ( ) 1 n P n = . Since really is very small for - 2 decay (less than 10-12 ), we can conveniently write P as a function of time by using the formula ( ) / lim 1 x x e = ....
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