This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: previous index next The EnergyTime Uncertainty Principle: Decaying States and Resonances Michael Fowler 10/1/08 Model of a Decaying State The momentumposition uncertainty principle p x Δ ⋅ Δ ≥ = has an energytime 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 αdecay, 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 αdecay 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 1012 ), we can conveniently write P as a function of time by using the formula ( ) / lim 1 x x e ε ε ε − → = − ....
View
Full
Document
This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.
 Fall '07
 MichaelFowler
 Energy, Momentum

Click to edit the document details