Time-dependent perturbation theory, golden rule of

From the definition of 6k it should be noted that Bk and k are Hermitian conjugates. According to the Golden Rule of time-dependent perturbation theory the probability per unit time, Wt f k, co), that the field (k, to) induces a transition in the system from the initial state f> to the final state / ) is given by... 

Much more is becoming known about the rates of the physical processes in competition with proton exchange reactions in excited states. (For an excellent review see Henry and Siebrand, 1973.) The factors which determine the rate constants (k) for internal conversion and intersystem crossing are neatly summarized in the Golden Rule of time-dependent perturbation theory ... 

Several quantum mechanical calculations have been made for electron transfer processes between metals and atoms or ions " in gaseous medium. In all the cases, the considerations concern the transition of electrons from a metal state to a bound atomic state or to a free continuum state or vice versa. The calculations of transition probabilities in the cited works have been based on Fermi s golden rule of time-dependent perturbation theory. However, it was pointed out by Gadzuk that the use of the golden rule usually presents a difficult problem if an estimate of the transition probability is desired, because it requires evaluation of a matrix element one must specify initial and final state (wave functions) and an interaction. This is not as straightforward as it seems. In a transition, e.g., between an atomic and a conduction band metal state, the initial and final states are eigenfunctions of different Hamiltonians. It seems meaningless to evaluate matrix elements, if the initial and final states are solutions of different Hamiltonians. 

Thus, to have a proper estimate of photocurrent from expression (75), it is desirable to calculate the transition probability, T E,hv)y using the time-dependent perturbation theory. The general expression of T E, hv) in terms of Fermi s golden rule of time-dependent perturbation theory... 

The Golden Rule of time-dependent perturbation theory now yields, for the probability of scattering per unit time from state i// k) to y/ k -I- q),... 

The asymmetric dependence of the activation energy on the driving force (energy-gap law) of ET reactions can be reproduced using the golden rule of time-dependent perturbation theory. The golden rule allows for the calculation of the transition rate from an initial to a final electronic state subject to a weak perturbation applied for a short period of time. The ET rate constant was first expressed in terms of the golden rule in the 1970 s ... 

Fermi called eq. (15.36) the Golden Rule of time-dependent perturbation theory because of its prevalence in radiationless transitions. Sometimes it is referred to as Fermi s Golden Rule. 

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

For a quantitative treatment of establishing connections between vibronic coupling and vibrational progressions in electronic spectra, band profiles from vibronic wavefunctions must be calculated using common procedures of time-dependent perturbation theory and Fermi s golden rule [57], For emission, e.g., the transition rate which is the transition probability per unit time summed over... 

Not only is the master equation more convenient for mathematical operations than the original Chapman-Kolmogorov equation, it also has a more direct physical interpretation. The quantities W(y y ) At or Wnn> At are the probabilities for a transition during a short time At. They can therefore be computed, for a given system, by means of any available approximation method that is valid for short times. The best known one is time-dependent perturbation theory, leading to Fermi s Golden Rule f)... 

Because of the time dependence of the vector potential A(rJ( t), the photon-atom interaction also depends on time. Hence, time-dependent perturbation theory has to be applied. The golden rule (so called by Fermi [Fer50], see also [Dir47, Sch55, LLi58]) for the transition rate w then yields for the change from an initial atomic state i> to a final atomic state f>... 

The method proposed by Fermi (1934) for calculating the / decay of a nucleus is based on the time-dependent perturbation theory. The small value of the weak-interaction constant makes it possible to restrict oneself to the first order in perturbation theory and to use the so-called Fermi Golden Rule... 

The transition probability per unit time given by the time-dependent perturbation theory, that Fermi named Golden Rule in view of its prevalence in radiationless transitions, has the form... 

This expression is the exact form of Fermi s Golden Rule, familiar in time-dependent perturbation theory where F[, 0)) is approximated by o) (Merzbacher, 1970). p( ,) is the density of final states. 

The time-dependent perturbation theory of the rates of radiative ET is based on the Born-Oppenheimer approximation [59] and the Franck Condon principle (i.e. on the separation of electronic and nuclear motions). The theory predicts that the ET rate constant, k i, is given by a golden rule -type equation, i.e., it is proportional to the product of the square of the donor-acceptor electronic coupling (V) and a Franck Condon weighted density of states FC) ... 

A word of caution is needed here. The golden-rule expression, Eq. (12.33) or (12,43), was obtained for the rate of decay of a level interacting with a continuous manifold (Section 9,1), not as a perturbation theory result but under certain conditions (in particular a dense manifold of final states) that are not usually satisfied for optical absorption, A similar expression is obtained in the weak coupling limit using time-dependent perturbation theory, in which case other conditions are not... 

We shall need this later to apply a well known result of quantum mechanics, involving time dependent perturbation theory, Fermi s Golden Rule [1]. 

Inclusion of quantum effects on the nuclear dynamics can be accomplished by using Fermi s Golden Rule (134), which is really a manifestation of first-order time-dependent perturbation theory and conservation of energy during a transition. In this level of refinement, and formally allowing for the inclusion of solvent effects, the rate is given by... 

According to Fermi s golden rule [40, 42], the integral intensity A of the absorption band of the normal mode is proportional to the probability per unit time of a transition between an initial state i and a final state j. Within the framework of the first (dipole) approximation of time-dependent perturbation quantum theory [46, 65], this probability is proportional to the square of the matrix element of the Hamiltonian H = —E p, where E is the electric field vector and p is the electric dipole moment, resulting in the absorption... 

This formula of the time-dependent perturbation theory was originally derived by Dirac, but it is often referred to as Fermi s golden rule No. 2. ... 

In summary, we have made an attempt to classify existing MQC strategies in formulations resulting from (i) a partial classical limit, (ii) a connection ansatz, and (iii) a mapping formalism. In this overview, we shall focus on essentially classical formulations that may be relatively easily applied to multidimensional surface-crossing problems. On the other hand, it should be noted that there also exists a number of essentially quantum-mechanical formulations which at some point use classical ideas. A well-known example are formulations that combine quantum-mechanical time-dependent perturbation theory with a classical evaluation of the resulting correlation functions, e.g. Golden Rule type formulations.Furthermore, several... 

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