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35 - Physics 6210/Spring 2007/Lecture 35 Lecture 35...

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Physics 6210/Spring 2007/Lecture 35 Lecture 35 Relevant sections in text: § 5.6 Fermi’s Golden Rule First order perturbation theory gives the following expression for the transition prob- ability: P ( i n, i 6 = n ) = 4 | V ni | 2 ( E n - E i ) 2 sin 2 ( E n - E i ) t h . We have seen that the “energy conserving” transitions (if there are any available) become dominant after a sufficiently large time interval. Indeed, the probabilties for energy non- conserving transitions are bounded in time, while the probability for energy conserving transitions grow quadratically with time (for as long as the approximation is valid). Here “large time” means that the elapsed time is much larger than the period of oscillation of the transition probability for energy-non-conserving transitions T := 2 π ¯ h | E n - E i | . Note that the typical energy scale for atomic structure is on the order of electron-volts. This translates into a typical time scale T 10 - 15 s , so “large times” is often a very good approximation. In the foregoing we have been tacitly assuming the final state is an energy eigenstate coming from the discrete part of the energy spectrum. If the final state energies lie in a continuum (at least approximately) we get a qualitatively similar picture, but the details change. We shall see that the transition probability at “large times” still favors energy conserving transitions, but it will only grow linearly with time because the width of the probability distribution about such transitions is becoming narrower with time. We can see this by supposing the probability is a continuous function of Δ E = E n - E i (see discussion below) and considering the large t limit via the identities lim α →∞
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