After time t, the probability of monochromatic absorption of the ground state |1> to the energy state |n> is given by:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]|<n|1>|^2=4|U_{n1}|^2\frac{\sin^2((E_n-E_1-\hbar\omega)t/2\hbar)}{(E_n-E_1-\hbar\omega)^2} [/tex]

where U is the transition matrix. The claim is that as t goes to infinity, the fraction becomes (up to factors of pi and stuff):

[tex]t*\delta(E_n-E_1-\hbar\omega) [/tex]

So if you take the probability per time by dividing by the time, it is independent of t!

But what's really going on here? It looks to me like if you divide the first equation I have by t, then as t goes to ∞, then the numerator just oscillates, and t in the denominator blows up, so the transition rate is zero.

If the claim is that [itex]E_n-E_1-\hbar\omega [/itex] is really small so that the sine term can keep growing as t increases, can't you expand the sine term in power series, and get a [itex]t^2[/itex] dependence, so that when you divide by time to get the transition probability, it depends linearly on time? So you would get that transition rate is proportional to t rather than [itex]\delta(E_n-E_1-\hbar\omega) [/itex]?

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# Why transition rate independent of time in perturbation theory?

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