Probability distribution transition time moments

Ao and Rammer [166] obtained the same result (and more) on the basis of a fully quantum mechanical treatment. Frauenfelder and Wolynes [78] derived it from simple physical arguments. Equation (9.98) predicts a quasiadiabatic result, = h k/ v 1 and the Golden Rule result, Pk = k/ v, in the opposite limit, which is qualitatively similar to the Landau-Zener behavior of the transition probability but the implications are different. Equation (9.98) is the result of multiple nonadiabatic crossings of the delta sink although it does not depend on details of the stochastic process Xj- t). This can be understood from the following consideration. For each moment of time, the fast coordinate has a Gaussian distribution, p Xf, t) = (xy — Xj, transition region, the fast coordinate crosses it very frequently and thus forms an effective sink for the slow coordinate. 

Under these assumption, the operation process may be described by the vector [p (0)]ixv of probabilities of the system operation process staying in particular operations states at the initial moment = 0, the matrix Pbiit)]vxv of the probabihties of the system operation process transitions between the operation states and the matrix [//w(0]vxv of the distribution functions of the conditional sojourn times Ou of the system operation process at the operation states or equivalently by the matrix [hbi(t) vxv of the distribution functions of the conditional sojourn times of the system operation... 

In the following we will assume that the transition moments in both absorption and emission coincide with a molecular axis M of the molecule, whose direction is specified by the spherical polar angle = (a, 6) in the reference frame (fig. 2). Let us introduce the angular functions > t ), the orientation distribution of M at time t (M in fig. 2), and V(Q tl, t ), the conditional probability density o finding at position Q a time t a vector M which was at position at time t. ... 

In this equation, (t) is the probability that the oseillator will be in state n at the moment t, JV is the rate of transfer between two states of the oseillator (from n to m) and E is the designation for the eontinuum of states (eorresponding to energy greater than the dissoeiation one), in whieh the bond is eonsidered broken. Using the method for the average time of the first transition, the authors find an expression for the average time of desorption as a funetion of the initial state distribution of the oseillator P (0) ... 

T 21 times t is the probability of finding the molecule in - 2 after time t when the radiation energy is distributed uniformly over a range of frequencies. The transition rate T2i is, therefore, proportional to the intensity of the radiation field and the square of the dipole moment matrix. The lifetime of the transition, or time per transition, is the reciprocal of T2i. 

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