Quantum mechanics Golden Rule

This equation can be used for solving many problems. Its only restriction is that the perturbation is weak. Because of the general applicability of this equation it is named Fermi s Golden Rule of quantum mechanics. This rule has been applied In many... 

The microscopic rate constant is derived from the quantum mechanical transition probability by considering the system to be initially present in one of the vibronic levels on the initial potential surface. The initial level is coupled by spin-orbit interaction to the manifold of vibronic levels belonging to the final potential surface. The microscopic rate constant is then obtained, following the Fermi-Golden rule, as ... 

In the formalism of quantum mechanics, the probability that an electron in the initial state is transferred into the final state, is given by Fermi s Golden Rule ... 

Energy transfer in solution occurs through a dipole-dipole interaction of the emission dipole of an excited molecule (donor) and the absorptive moment of a unexcited molecule (acceptor). Forster<40) treated the interaction quantum mechanically and derived and expression for the rate of transfer between isolated stationary, homogeneously broadened donors and acceptors. Dexter(41) formulated the transfer rate using the Fermi golden rule and extended it to include quadrupole and higher transition moments in either the donor or the acceptor. Following the scheme of Dexter, the transfer rate for a specific transition is... 

Electron transfer reactions have also been treated from the quantum mechanical point of view in formal analogy to radiationless transitions, considering the weakly interacting states of a supermolecule AB the probability (rate constant) of the electron transfer is given by a golden rule expression of the type17... 

Those who wish to extend their skills in this direction here should investigate I crmi s golden rule, a general quantum mechanical expression for transition probability. It runs... 

The spectral density can equally well be written directly in terms of quantum-mechanical transition probabilities in the golden rule form. 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

The quantum mechanical approach is based on time-dependent perturbation theory and is derived from Fermi s Golden Rule for non-radiative decay processes [1]. 

The second theoretical approach is quantum mechanical in nature and is based on the Fermi Golden Rule expression for nonradiative decay processes [45,46]... 

In other words, the ability of the solvent to absorb a quantum of energy h >0 (or its classical equivalent) is determined quite literally by the ability of the solvent to respond to the solute dynamics at a frequency o> = oj(). One can derive this relation quantum mechanically by assuming that the solvent s effect on the solute can be handled perturbatively within Fermi s golden rule (1), but it is actually more general than that. Perhaps it is worth pausing to see how the same basic result appears in a purely classical context. 

A strict quantum mechanical calculation of a tunneling system the size of a protein quickly becomes intractably complex. Fortunately, relatively simple theory has been successful at organizing and predicting electron tunneling rates in proteins. When the donor and acceptor redox centers are well separated, non-adiabatic electron transfer theory applies Fermiis Golden Rule, in which the rate of electron transfer is proportional to two terms, one electronic. Hah, and the other nuclear, FC (Devault, 1980). 

The quantum mechanical treatment of non-adiabatic electron transfers are normally considered in terms of the formalism developed for multiphonon radiationless transitions. This formalism starts from Fermi s golden rule for the probability of a transition from a vibronic state Ay of the reactant (electronic state A with vibrational level v) to a set of vibronic levels B of the product... 

Approaches beyond classical theory were clearly needed, and in the 1960s and 1970s two largely complementary quantum-mechanical approaches— the golden rule and superexchange—were developed. Modem computing power has essentially fused these two theories and greatly enhanced their reach and realism. 

The golden rule rate expression is a standard quantum-mechanical result for the relaxation rate of a prepared state z) interacting with a continuous manifold of states 1/ The result, derived in Section 9.1, for this rate is the Fermi golden rule... 

In Section 6.2,3 we have seen that a simple quantum mechanical theory based on the golden rule yields an expression for the absorption lineshape that is given essentially by the Fourier transform of the relevant dipole correlation function (/ii(0)/ii(Z)). Assuming again that ix is proportional to the displacement x of the oscillator from its equilibrium position we have... 

Fortunately, the solution to this problem was one of the early successes of quantum mechanics and is so central to modem physics that it carries its own name, Fermi s Golden Rule. The transition rate between the initial and final states of a system depends on the strength of the coupling between the initial and final states and on the number of ways the transition can happen, the density of the final states [1]. This rule only applies to scattering problems where the incident wavefunction is weakly perturbed by the presence of the sample. If the scattering is not strong we can suppose that the total waveftinction does not differ substantially from the incident wavefunction this supposition is the Bom approximation. 

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

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