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Journal of Physics A Mathematical and General

Exact long-term total transition probability rates and Fermi's 'golden rules'

Sidney Golden

Published Mar 1, 1977• 3 citations
Publisher

Abstract

Fermi's 'golden rule' expressions refer to first- and second-order time-dependent perturbation theory results for transition probability rates involving designated final states which are members of a continuum. By use of a Laplace-average formalism, it is shown that the long-term time-averaged total transition probability rate involving designated initial and final states which are all members of a continuum is exactly the sum of the pertinent 'golden rule' expressions. Generalization to higher-order perturbation theory results is suggested.

DOI
10.1088/0305-4470/10/3/008

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Fermi's 'golden rule' expressions refer to first- and second-order time-dependent perturbation theory results for transition probability rates involving designated final states which are members of a continuum. By use of a Laplace-average formalism, it is shown that the long-term time-averaged total transition probability rate involving designated initial and final states which are all members of a continuum is exactly the sum of the pertinent 'golden rule' expressions. Generalization to higher-order perturbation theory results is suggested.
We show that the usual expression found in the literature, which tries to generalize the `Fermi golden rule' beyond second order in perturbation, is (surprisingly) incorrect. After identifying the weak steps of the two usual derivations, we derive a new expression of this generalized golden rule, intrinsically very different from the previous one, even though its form may look similar. We show that this new result already affects the next nonzero term of the transition probability expansion. From a direct comparison of this new result with its exact value - as given by the exact evolution operator - we also show that the usual approaches are inherently quite questionable beyond the second order in perturbation.
Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.
The rates of charge transfer and energy transfer are essential for understanding nanoscale phenomena and processes. Here we introduce the general conditions for the emergence of rate processes in quantum mechanics. We refer to the generic scenario in which a transition is induced between eigenstates of a given Hamiltonian by a weak perturbation. Analysis reveals that when the initial and final states are pure states, the transition rate is time-dependent and fails to reach a finite constant value. If, however, the final state is a mixed ensemble that is sufficiently wide and dense in energy, a rate constant emerges, given by Fermi’s Golden Rule (FGR). When the initial state is also mixed, for example, a thermal equilibrium state, a thermal rate constant is formulated in terms of transition rates between specific eigenstates, summed over the final eigenstates, and averaging over the thermal distribution of the initial eigenstates.
Using second-order, time-dependent perturbation theory, we calculate the intermediate and final state amplitudes for transitions from an initial to a final state via an intermediate resonance. When our complex energies are replaced by real energies our results are identical to the real energy final state amplitudes obtained by other authors, thus justifying the usual phenomenological replacement of real energies with complex ones when considering resonance behavior. We then derive the Fermi Golden Rule No. 1 with complex energies.
In this paper we prove the following: (1) The basic error of time-dependent perturbation theory is using the sum of first finite order of perturbed solutions to substitute the exact solution in the divergent interval of the series for calculating the transition probability. In addition quantum mechanics neglects the influence of the normality condition in the continuous case. In both cases Fermi golden rule is not a mathematically reasonable deductive inference from the Schrodinger equation. (2) The transition probability per unit time deduced from the exact solution of the Schrodinger equation is zero, which cannot be used to describe the transition processes.
Abstract Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from − ∞ to + ∞ , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time , even when the first order perturbation approximation itself is still valid.