The Golden Rule of Quantum Mechanics

Quantum mechanics establishes a simple expression for the rate constant of a transition between an initial and a final state when these states are only subject to a weak interaction. The formalism of weak interactions was originally developed for spectroscopic transitions, but later found application in energy and electron hansfers. 

The time-dependent Schrodinger equation, for a time-independent Hamiltonian, has the form 

In the most relevant cases for these transitions, the perturbing interaction is limited in time and space. For example, the system is unperturbed before it interacts with an electromagnetic wave, and it is free again from the perturbation after sufficient time has elapsed. This suggests that the total Hamiltonian can be considered as the sum of two terms  

No approximations have yet been made and eq. (15.24) is exact. The values of Cj determined by this equation are related to the probability of finding the system in any particular state at any later time. Unfortunately, it is not generally possible to find exact solutions to this equation. The solution of this complicated linear system invokes the perturbation approximation, and the method is called time-dependent perturbation theory. 

The solution of eq. (15.23) depends critically on the initial conditions. Assuming, for simplicity, that the system at the initial time to = is in one of the stationary states of the unperturbed Hamiltonian, and that Hq possesses only discrete energy levels, the following initial conditions are introduced  

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Eo ftbafPvi ) out of the integral in Eq. (4.8a) on the assumption that the field is essentially independent of v over the small frequency interval where hv is close to Eb — Ea- The factor Pv Vo) in the final expression therefore pertains to this interval. Additional details of the derivation are in Box 4.4. Equation (4.8c) is a special case of a general expression that is often called the golden rule of quantum mechanics, which we met in Chap. 2 and will encounter again in a variety of contexts. 

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... 

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. 

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 ... 

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 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... 

An important achievement of the early theories was the derivation of the exact quantum mechanical expression for the ET rate in the Fermi Golden Rule limit in the linear response regime by Kubo and Toyozawa [4b], Levich and co-workers [20a] and by Ovchinnikov and Ovchinnikova [21], in terms of the dielectric spectral density of the solvent and intramolecular vibrational modes of donor and acceptor complexes. The solvent model was improved to take into account time and space correlation of the polarization fluctuations [20,21]. The importance of high-frequency intramolecular vibrations was fully recognized by Dogonadze and Kuznetsov [22], Efrima and Bixon [23], and by Jortner and co-workers [24,25] and Ulstrup [26]. It was shown that the main role of quantum modes is to effectively reduce the activation energy and thus to increase the reaction rate in the inverted... 

Ao and Rammer [166] obtained the same result (and more) on the basis of a fully quantum mechanical treatment. Frauenfelder and Wolynes [78] derived it from simple physical arguments. Equation (9.98) predicts a quasiadiabatic result, = h k/ v 1 and the Golden Rule result, Pk = k/ v, in the opposite limit, which is qualitatively similar to the Landau-Zener behavior of the transition probability but the implications are different. Equation (9.98) is the result of multiple nonadiabatic crossings of the delta sink although it does not depend on details of the stochastic process Xj- t). This can be understood from the following consideration. For each moment of time, the fast coordinate has a Gaussian distribution, p Xf, t) = (xy — Xj, transition region, the fast coordinate crosses it very frequently and thus forms an effective sink for the slow coordinate. 

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. 

It is possible to evaluate the rate of energy transfer using the Fermi golden rule [3] of quantum mechanics. At the lowest level of calculation, the matrix element for the process is just the electrostatic interaction between the transition densities of the donor and of the acceptor. At large distances, this may be approximated by the interaction between the corresponding transition dipoles, which is proportional to where R is the distance between the donor and the acceptor [3]. The rate of energy transfer is proportional to the square of it, and it is usual to write the rate as... 

Equation (10.36) has the form of a classical first-order kinetic expression. In the limit of weak coupling where the golden rule applies, any quantum mechanical oscillations are damped so strongly that we can describe the kinetics simply in terms of stochastic transitions of the system between the reactant and product states. The net rate in the forward or backward direction then is proportional to the population difference between the two states ( n — 22)- Persistent oscillations of the populations evidently occur only in the opposite limit, when H21 lh is comparable to, or larger than I/T2. 

Hamiltonian matrix elements, write an expression for the time dependence of each element of p (e.g., dp ldt) in the absence of stochastic relaxations, (c) What is the relationship between / ab(0 and / ba(0 (d) Suppose that interconversions of the two basis states are driven only by the quantum mechanical coupling element but that stochastic fluctuations of the energies cause pure dephasing with a time constant 7. What are the longitudinal (Ji) and transverse (T2) relaxation times in this situation (e) Write out the Stochastic Liouville expression for the time-dependence of each element of p. (/) How would Ti and 7 2 be modified if the system also changes stochastically from state atob with rate constant and from btoa with rate constant ba ( ) I what limit does the stochastic Liouville equation reduce to the golden-rule expression ... 

Linear response theory [152] is perfectly suited to the study of fluid structures when weak fields are involved, which turns out to be the case of the elastic scattering experiments alluded to earlier. A mechanism for the relaxation of the field effect on the fluid is just the spontaneous fluctuations in the fluid, which are characterized by the equilibrium (zero field) correlation functions. Apart from the standard technique used to derive the instantaneous response, based on Fermi s golden rule (or on the first Bom approximation) [148], the functional differentiation of the partition function [153, 154] with respect to a continuous (or thermalized) external field is also utilized within this quantum context. In this regard, note that a proper ensemble to carry out functional derivatives is the grand ensemble. All of this allows one to gain deep insight into the equilibrium structures of quantum fluids, as shown in the works by Chandler and Wolynes [25], by Ceperley [28], and by the present author [35, 36]. In doing so, one can bypass the dynamics of the quantum fluid to obtain the static responses in k-space and also make unexpected and powerful connections with classical statistical mechanics [36]. 

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 ... 

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