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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 Δ ⋅ Δ ≥ = 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 can’t have anything to do with non- commutation. E t Δ ⋅Δ ≥ = To illustrate the meaning of the equation E t Δ ⋅Δ ≥ = , let us reconsider α-decay, but with a slightly simplified potential to clarify what’s 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 escaping—so no longer a true bound state, but for a thick barrier the difference may be hard to detect. As with the α-decay analysis, we’ll look at this semiclassically, picturing the particle as bouncing off the walls backwards and forwards inside, time τ between hits, and at each hit probability of penetration some small number ε . Therefore, the probability that the particle is still in the well after a time t n τ = is ( ) ( ) 1 n P n τ ε = − . Since ε really is very small for α- 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 ε ε ε − → = − ....
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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