Probability, of transition

If a system is initially in a state m, conservation of probability requires that the total probability of transitions out of state m be obtainable from the decrease in the probability of being in state m. Prove this to the lowest order by using the results of exercise 2, i.e. 

With the assumption that near Xc the diabatic terms are linear, and is independent of x, the probability of transition between the diabatic terms at > F+ (x ) depends on the parameter... 

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a ... 

Thus, for a transition between any two vibrational levels of the proton, the fluctuation of the molecular surrounding provides the activation. For each such transition, the motion along the proton coordinate is of quantum (sub-barrier) character. Possible intramolecular activation of the H—O chemical bond is taken into account in the theory by means of the summation of the probabilities of transitions between all the excited vibrational states of the proton with a weighting function corresponding to the thermal distribution.3,36 Incorporation in the theory of the contribution of the excited states enabled us in particular to improve the agreement between the theory and experiment with respect to the independence of the symmetry factor of the potential in a wide region of 8[Pg.135]

As was already noted in [9], the primary effect of the YM field is to induce transitions (Cm —> Q) between the nuclear states (and, perhaps, to cause finite lifetimes). As already remarked, it is not easy to calculate the probabilities of transitions due to the derivative coupling between the zero-order nuclear states (if for no other reason, then because these are not all mutually orthogonal). Efforts made in this direction are successful only under special circumstances, for example, the perturbed stationary state method [64,65] for slow atomic collisions. This difficulty is avoided when one follows Yang and Mills to derive a mediating tensorial force that provide an alternative form of the interaction between the zero-order states and, also, if one introduces the ADT matrix to eliminate the derivative couplings. 

Interestingly, the flavin molecule is significantly easier to excite than the other heterocycles investigated. The first singlet excitation energy is only 3.0 eV (413 nm), and the probability of transition is intermediate. The second band with possible singlet excitation lies at 3.8 eV. Hence, if one intends to construct systems that are more readily excited, substituted flavins seem to be a more appropriate route than the furocoumarins, whereas the latter are easier to ionize. 

Working with Markov chains, confusion is bound to arise if the indices of the Markov matrix are handled without care. As stated lucidly in an excellent elementary textbook devoted to finite mathematics,24 transition probability matrices must obey the constraints of a stochastic matrix. Namely that they have to be square, each element has to be non-negative, and the sum of each column must be unity. In this respect, and in order to conform with standard rules vector-matrix multiplication, it is preferable to interpret the probability / , as the probability of transition from state. v, to state s (this interpretation stipulates the standard Pp format instead of the pTP format, the latter convenient for the alternative 5 —> Sjinterpretation in defining p ), 5,6... 

Electronic factors describe the probability of transitions between one electronic state and another. 

Probability of transitions. The Beer-Lambert Law. Oscillator strength... 

In the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to electron coordinates, this expression can be rewritten as the product of two terms < f i M vP2> 2 Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

The Einstein coefficients characterize the probability of transition of a molecule between two energy levels Ei and E2 (Scheme B3.2). Bu is the induced absorption coefficient (see Chapter 2), B21 is the induced emission coefficient and A21 is the spontaneous emission coefficient. The emission-induced process E2 —> Ei occurs at exactly the same rate as the absorption-induced process Ei —> E2, so that B12 = B 21. 

But even if the hght is of the proper frequency, the intrinsic probabilities of transitions are highly variable. The experimentally observed intensity of a transition is proportional to the square of a matrix element ... 

The Markov model uses the clinical data to calculate the probability of transitioning from a severe disease state ( OFF time >25% of the day), to a less severe disease state ("OFF time <25% of the day) for entacapone therapy. This enables a calculation of the total amount of time a cohort of patients will... 

The probability of transitions from given energy levels of a fixed atomic population (e.g. between the lower level i and upper level j) was expressed by Einstein in the form of three coefficients. These are termed transition probabilities as follows ... 

We now multiply on the left by and integrate, where m indexes die stationary state 4> for which we are interested in measuring die probability of transition. This gives... 

The main purpose of p.m. of the second kind is to calculate the probabilities of transitions from one state to others induced by the perturbation. In what follows we restrict ourselves to the calculation of the probabiliey wvft A ) that the system falls into one of those states where the value of the energy H0 lies in the interval... 

It is seen that as a result of integration over a range of frequencies, the probability of transition to the state m in time t becomes proportional tfit... 

We assume that the absorbing gas is of a uniform composition and in thermal equilibrium. The absorption coefficient, which is defined by Lambert s law, Eq. 3.1, is expressed in terms of the probabilities of transitions between the stationary states of the supermolecular system, in response to the incident radiation. Assuming the interaction of radiation and matter may be approximated by electric dipole interaction, i.e., assuming the wavelengths of the radiation are large compared with the dimensions of molecular complexes, the transition probability between the initial and... 

It should be noted that (4.28) is only an approximation for the nuclear wave function. The perturbation terms (4.36) will mix into the nuclear wave function small contributions from harmonic-oscillator functions with quantum numbers other than v. These anharmonicity corrections to the vibrational wave function will add further to the probability of transitions with At) > 1. 

Thus, the difference of energy, A°i/ R), is proportional to the square root of the probability of transition through the potential barrier. 

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