Variation in the state of a system with time

If at time zero, the probability of the system being in the nth state is unity, and at time t the probability of the system being in the mth state is then the task is to discover how 

The oscillating electrical field associated with the incident wave displaces the charges in the molecule slightly. This displacement in the x direction corresponds to an energy 

Equation (25.44) tells us how the coefficient depends on the time by giving an expression for the derivative dajdt). From this equation we can obtain an expression for 

According to Eq. (25.46), thefirst term in H is proportional to fi. Since the integration in Eq. (25.44) is over coordinates only, the time-dependent factors in H and in the wave functions can be removed from the integral. Thus we see that the integrals in Eq. (25.44) are proportional to integrals of the type 

These are called the transition-moment integrals. We define the quantity by 

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