Oscillations Between Quantum States of an Isolated System

1 Oscillations Between Quantum States of an Isolated System 

The time-dependent perturbation theory that we have used to treat resonance energy transfer and absorption of light assumes that we know that a system is in a given state (state 1), so that the coefficient associated with this state (Ci) is 1, while the coefficient for finding the system in a different state (C2) is zero. The resulting expression for the rate of transitions to state 2 (Eq. 2.61 or 7.8) neglects the possibility of a return to state 1. It can continue to hold at later times only if the transition to state 2 is followed by a relaxation that takes the two states out of resonance. Without such relaxations, the system would oscillate between the two states as described by the coupled equations 

Equations (10.1a, 10.1b) just restate Eq. (2.59) for a two-state system with spatial wavefunctions and 2 that are mixed by perturbation operator H. 1 and 2 are the energies of the states in the absence of the perturbation. The factor 

Equations (10.3a and 10.3b) can be solved by differentiating each of them again with respect to time straightforward substitutions then yield separate differential equations for Ci and C2 (see, e.g., [1]). Assuming again that the system is in state 1 at time zero, so that Ci(0) = 1 and 2(0) = 0, the solutions are 

From Eq. (10.4b), the probability of finding the system in state 2 at time t is 

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