A Further Discussion of Resonance

The probabilities a aA and a aB of finding the system in state A and state B, respectively, at time t are hence 

We see that these probabilities vary harmonically between the values 0 and 1. The period of a cycle (from a aA = 1 to 0 and back to 1 again) is seen to be h/2H AB, and the frequency 2H AB/h, this being, as stated in Section 416, just 1/6 times the separation of the levels due to the perturbation. 

Now let us switch in the coupling at the time t = 0, and then switch it out again at the time t = t We now, at times later 

This sequence of experiments can be repeated over and over, each time starting with the system in the state ni = 1, n2 = 0 and allowing the coupling to be operative for the length of time i. In this way we can find experimentally the probability of finding the system in the various states ru = 1, 2 = 0 i = 0, 2 = 1 m = 0, n2 = 0 etc. after the perturbation has been operative for the length of time t.  

The same probabilities are given directly by our application of the method of variation of constants. The probability of transition to states of considerably different energy as the result of a small perturbation acting for a short time is very small, and we have neglected these transitions. Our calculation shows that the probability of finding the system in the state B depends on the value of t in the way given by Equation 41-12, varying harmonically between the limits 0 and 1. 

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