Mean transition time , probability moments

Alternatively, the definition of the mean transition time (5.4) may be obtained on the basis of consideration of optimal estimates [54]. Let us define the transition time i) as the interval between moments of initial state of the system and abrupt change of the function, approximating the evolution of its probability Q(t.X(t) with minimal error. As an approximation consider the following function v /(f,xo, ) = flo(xo) + a (xo)[l(f) — l(f — i (xo))]. In the following we will drop an argument of ao, a, and the relaxation time d, assuming their dependence on coordinates of the considered interval c and d and on initial coordinate x0. Optimal values of parameters of such approximating function satisfy the condition of minimum of functional ... 

If the experimental data are sufficiently accurate, then from the susceptibility function 3 /(m- t) it is possible to evaluate the transit time probability density, (Puu id-, t), by means of numerical inverse Fourier transformation. If the experimental data are not very accurate, then it is still possible to evaluate the first two or three moments and cumulants of the transit time. Since the mean transit time is nonnegative, it follows that (puu d < 0 r) = 0, and thus the characteristic function of the probability density (Puu id-, t) is identical with the susceptibility function in the frequency domain, Sr/MdsT t), experimentally accessible from eqs. (12.116)-(12.117). It follows that the moments < 0" (r) > , m = 1, 2,..., and the cumulants m = 1,2,...,... 

The transition probabilities depend on the mean squared interaction energy relative to the mechanism which causes the transition, times the value of the spectral density at the required frequencies (Eq. (3.14)). The square of the dipolar interaction energy is, as usual (see Eq. (1.4) and Appendix V), proportional to (p, 1 1x2/r3)2, where p and p2 are the magnetic moments of the two spins. The actual equations are... 

Fig. 35). The potential energy curves and the transition dipole moment are taken from [117]. The time evolution of the populations on the ground and excited states is shown in Fig. 36 More than 86% of the initial state is excited to the B state within the period shorter than a few femtoseconds. The integrated total transition probability V given by Eq. (173) is P = 0.879, which is in good agreement with the value 0.864 obtained by numerical solution of the original coupled Schroedinger equations. This means that the population deviation from 100% is not due to the approximation, but comes from the intrinsic reason, that is, from the spread of the wavepacket. Note that the LiH molecule is one of the...

The transition probability density becomes Dirac delta function for coinciding moments of time (physically this means small variation of the state during small period of time) ... 

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