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If I have a particle that oscillates between states A and B, and I calculate the probability of transition (or of rather being in state B after an IC of being in A @ t=0)

and that probability goes like:

Pb = C1 Sin^2 (C2 t)

And I want to find the average/mean lifetime of a particle that starts in A what do I do?

It seems like I could consider only C1, since on average the Sin^2 term goes to (1/2)

So <Pb> == C1 (1/2)

And therefor <Pa> = 1-C1 *(1/2)

but I feel like I'm making an error in not considering the frequency of oscillation. If this were a particle-antiparticle system oscillating between the two, wouldn't the oscillation frequency matter?

Or do you say that since in a part-antipart system, the period of oscillation must be very long (as in a neutron osc), therefor in any given realistic time interval we can say that the Sin term goes like:

Sin[C2 t]^2 -->> C2^2 t^2

the ol' sin[x]->x for small x series expansion.

So the approximate probability

Pb == C1 C2^2 t^2 for t<<periodicity

then Pa = 1-Pb

But now what would the mean lifetime of a particle in A be thanks to one of these probabilities?

Or am I doing this all wrong?

Thanks

-KJH

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# Mean lifetime of a particle (quantum)

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