Transition time moments, probability

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...

Moments of Transition Time. Consider the probability Q(t, xo) of a Brownian particle, located at the point xo within the interval (c, d), to be at the time t > 0 outside of the considered interval. We can decompose this probability to the set of moments. On the other hand, if we know all moments, we can in some cases construct a probability as the set of moments. Thus, analogically to moments of the first passage time we can introduce moments of transition time i9 (c,xo, d) taking into account that the set of transition events may be not complete, that is, lim Q(t,xo) < 1 ... 

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 ... 

Finally, for additional support of the correctness and practical usefulness of the above-presented definition of moments of transition time, we would like to mention the duality of MTT and MFPT. If one considers the symmetric potential, such that (—oo) = <1>( I oo) = +oo, and obtains moments of transition time over the point of symmetry, one will see that they absolutely coincide with the corresponding moments of the first passage time if the absorbing boundary is located at the point of symmetry as well (this is what we call the principle of conformity [70]). Therefore, it follows that the probability density (5.2) coincides with the probability density of the first passage time wT(f,xo) w-/(t,xo), but one can easily ensure that it is so, solving the FPE numerically. The proof of the principle of conformity is given in the appendix. 

We consider the process of Brownian diffusion in a potential cp(x). The probability density of a Brownian particle is governed by the FPE (5.72) with delta-function initial condition. The moments of transition time are given by (5.1). 

Thus, we have proved the principle of conformity for both probability densities and moments of the transition time of symmetrical potential profile and FPT of the absorbing boundary located at the point of symmetry. 

The average energy flux in the evanescent wave is given by the real part of the Poynting vector S = (c/47t)ExH. However, the probability of absorption of energy per unit time from the evanescent wave by an electric dipole-allowed transition of moment pa in a fluorophore is proportional to lnfl - El2. Note that Re S and pa E 2 are not proportional to each other they have a different dependence on 0. 

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... 

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,...,... 

Prior to excitation, the molecule is in the lowest vibrational state of the lowest electronic state (ground state), Sq. The absorption of a photon is governed by optical selection rules [2]. Its probability is proportional to the square of the transition dipole moment. The most severe restriction concerns the spin conservation. Further restrictions reflect the symmetry and overlap of corresponding wave functions. Regardless of the probability (reflected by the molar absorption coefficient), the single act of transition to a higher excited state due to absorption of a photon belongs to the fastest processes that occur in nature (except nuclear processes) and proceeds on timescales shorter than 10 s [3]. In this short time, neither the position of... 

We will find the probability P(t) for the system to pass the point q = q0/2 up to the moment of time t. This probability gives the upper estimate for the transition probability since, in principle, there are trajectories for which the system goes back to the left potential well after crossing the top of the potential barrier. However, if the contribution of these trajectories is small, as is the case for not too strong an interaction with the thermal bath at large narrow barriers, P(t) is close to the exact value of the transition probability. 

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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