Lifetimes and Spontaneous Transition Probabilities

The probability that an excited molecule in the level E. makes a transition to a lower level Ej by spontaneous emission of a fluorescene quantum hvik = E. - Ej is, according to (2.17), related to the Einstein coefficient 

When several transition paths from E to different lower levels E are possible (see Fig.2.19), the total transition probability is given by 

The decrease dN. of the population density during the time interval dt is then 

After a time = 1/A. the population density has decreased to 1/e of its initial value at t = 0. t.j is the mean spontaneous lifetime of the level E. . This can be seen immediately from the definition of the mean time 

The vadiant power emitted from molecules on the transition E - E is 

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