Decay Rates Fermi Golden Rule

If the perturbation V does not depend on time. Equation 7.33 can be easily integrated. If the initial time is chosen as to = 0, we obtain 

For the time-dependent picture, Equation 7.36 serves quite well. The probability for a certain continuum state, Cm(t), increases with time. This means that the probability of a state of type (Is oos) also increases with time. The latter state is a so-called Auger state. Thus, the (2s) state decays by moving one electron to Is and ejecting the other. The total energy of the states before and after is the same. 

The largest contribution to the decay is for small values of For a given time t in Equation 7.36, the function (1 - cos (Unj O/o) goes to E for ( mk 0. We thus obtain 

For a large time, there is an oscillatory behavior that leaves positive and negative contributions even for small Let us assume a density of states p(E), which is constant in a region around Eg. The rate of decrease of the total probability of the state T g, which was equal to unity at time t = 0, may be obtained by integrating over the range of continuous energy levels (E = ha)  

The decay rate W is the derivative of this function with respect to time  

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Irreversibility is an everyday phenomenon in nature but it remains one of the central issues in theoretical physics. In the early days of quantum physics, open system evolution was described using the Fermi Golden Rule, which leads inherently to an exponential decay of an excited quantum state. The underlying assumption is a continuum of final states that forces all the Poincare recurrences to infinity, and hence introduces irreversibility into the solution of an initial-value problem. While the Golden Rule yields the decay rate of the ex-... 

Let us consider an "impurity" atom moving in an atomic BEC. Atoms of another isotope or the same isotope but in a different internal (hyperfine) state can be viewed as impurities as long as their density is small enough not to modify considerably the BEC excitation spectrum. At "supercritical" velocities, namely, above the speed of sound in the BEC, the impurity atom is decelerated due to phonon creation in the BEC. The rate of such a process according to the standard Fermi golden rule (i.e., assuming exponential decay of the amplitude of the initial state) has been calculated for both a uniform BEC, [Timmermans... 

This two-state quantum beat example is identical to the doorway mediated non-radiative decay problem frequently encountered in polyatomic molecule Intramolecular Vibrational Redistribution (IVR), Inter-System Crossing (ISC), Internal Conversion (IC), and compound anticrossings. There is a single, narrow bright state. It couples to a single, broad, and dark doorway state. The width of the doorway state is determined by the rate of its Fermi Golden Rule decay into a quasi-continuum of dark states. 

A complex Heff model is constructed by associating an amplitude decay rate, Tj/2, with the zero-order energy, e3 — iTj/2, of each active-space basis state. The Tj values may be derived from a state-space Fermi Golden Rule treatment of the average squared interaction strength of the j-th active-space basis state with the approximately isoenergetic basis states in the inactive space (I)... 

We use the Fermi golden rule (Equation 7.39) to derive the decay rate due to Fbrster interaction and obtain... 

Our model assumptions lead to exponential decay of the probability that the system remains in the initial state, where the decay rate k is given by the so-called Fermi golden rule formula,... 

The conflicting serial/parallel models for IVR/VP are not readily distinguished until time resolved experiments can be performed on the systems of interest. Both models can relate the relative intensities of the emission features to the various model parameters, but the serial process seems more in line with a simple, conventional [Fermi s Golden Rule for IVR (Avouris et al. 1977 Beswick and Jortner 1981 Jortner et al. 1988 Lin 1980 Mukamel 1985 Mukamel and Jortner 1977) and RRKM theory for VP (Forst 1973 Gilbert and Smith 1990 Kelley and Bernstein 1986 Levine and Bernstein 1987 Pritchard 1984 Robinson and Holbrook 1972 Steinfeld et al. 1989)], few parameter approach. Time resolved measurements do distinguish the models because in a serial model the rises and decays of various vibronic states should be linked, whereas in a parallel one they are, in general, unrelated. Moreover, the time dependent studies allow one to determine how the rates of the IVR and VP processes vary with excitation energy, density of states, mode properties, and isotropic substitution. 

For a given microwave frequency a , the right hand side of (6.2.47) is constant. Thus, (6.2.47) can be solved immediately with the result Pn(i) = exp(—A t). Therefore, A has the physical meaning of a one-photon decay rate to the continuum. The expression (6.2.48) is a form of Fermi s golden rule. 

An alternate approach is to note that in quantum mechanics, according to Fermi s golden rule [22], the decay probability for an excited atom, that is, the photon emission rate F, is given by... 

Another way to view the situation is that a very rapid decay of state 2 broadens the homogeneous distribution of energies associated with this state. As Fig. 10.1 illustrates, the amplitude of oscillations between states 1 and 2 falls off rapidly as the two states move out of resonance. According to Fermi s golden rule (Eq. 7.10) the average rate depends on the density of... 

In the last section, we used the stochastic Liouville equation to find the steady-state rate of transitions between two weakly-coupled quantum states, on the assumption that coherence decayed rapidly relative to the rate of the transitions. The resulting expression (Eq. 10.36) reproduces Fermi s golden rule. We can use the same... 

To explain isomerism we turn to Fermi s Golden Rule which relates the transition rate to the wave functions of the initial and any final states as well as the density of final states in a given energy interval. In short, a decay can only happen if a suitable final state exists, and, if it does, the transition rate is higher the more the wave function of the final state resembles that of the initial state. We therefore expect isomers to occur in several classes ... 

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