# Transition rates when interaction with a measuring device is taken into account

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This is essentially a reiteration of this question posed on SE, which has not received a complete answer yet.

In some circumstances (cfr. Qmechanic's answer in the link above), in time dependent perturbation theory, one can define a “transition rate” beetween eigenstates of the unperturbed energy of the system. By definition, this rate refers to the free unitary evolution of the (perturbed) system, i.e. when no measurement is being made upon the system.

On the other hand, if the number of such transitions is being somehow counted, this means that the system is interacting with some device which leads to decoherence beetween the “initial” and “final” states of perturbation theory. For example, if the process in question is the beta decay of a neutron, the number of decays in a sample of neutrons could be monitored by detecting the protons emitted in the decay. It is clear that a complete quantum description of a transition process requires the interaction with the measuring device to be taken into account and, in general, the resulting (non-unitary) dynamic for the state of the system can be very different from the simple one described by a transition rate.

Therefore, my question is: under what assumptions about the interaction with the measuring device (i.e. on the interaction hamiltonian, the decoherence time of initial and final states etc.)  is it still possible to meaningfully talk of a “transition rate”?

In brief, the transition rate describes this situation: $$H=H_0 + H_{\text {pert}},\qquad \lvert i \rangle \longrightarrow \sqrt {1-wt} \lvert i \rangle+\sqrt{wt}\lvert f \rangle,$$ where $i$ and $f$ are eigenstates of $H_0$. Taking into account the interaction with a measuring device, we should have something like $$H=H_0+H_{\text {pert}} +H_{\text {int}} ,\qquad \lvert i \rangle \longrightarrow (1-wt) \lvert i \rangle \langle i \rvert +wt \lvert f \rangle \langle f \rvert.$$ Under what assumptions on $H_{\text{int}}$ is the above equation true?

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