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International Journal of Scientific Research in Physics and Applied Sciences

The Theory of Spontaneous Emission and Manifestation of Fermis Golden Rule

Jyotisman Saikia, B. Goswami, Gauranga Dhar Baruah

Published Dec 31, 2018• 0 citations
Publisher

Abstract

In the present work we have worked out a vector model for spontaneous emission based on the principle of moving vectors.This simplified approach reflects the work of Weisskopf and Wigner.Normalized probability of a transition to an upper level versus detuning has been plotted.Further we have shown that the areas which have been generated at phasor diagrams in different values of decay constants for moving vectors are directly proportional to time.This is a manifestation of Fermi's Golden rule.Fermi's Golden Rule in quantum mechanics is well known.But it is for the first time that we have worked out the concept of moving vectors using the transition probabilities and the concept of phasor diagrams to represent the Fermi's Golden rule.

DOI
10.26438/ijsrpas/v6i6.189192

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In the present work we have worked out a vector model for spontaneous emission based on the principle of moving vectors.This simplified approach reflects the work of Weisskopf and Wigner.Normalized probability of a transition to an upper level versus detuning has been plotted.Further we have shown that the areas which have been generated at phasor diagrams in different values of decay constants for moving vectors are directly proportional to time.This is a manifestation of Fermi's Golden rule.Fermi's Golden Rule in quantum mechanics is well known.But it is for the first time that we have worked out the concept of moving vectors using the transition probabilities and the concept of phasor diagrams to represent the Fermi's Golden rule.
An elementary account is given of the absorption and emission of radiation by molecules employing a particular formulation of Schrödinger theory which facilitates the treatment of radiative line widths and level shifts. The familiar Fermi golden rule for transition rates, the Rabi Bloch-Siegert absorption profile, and the Weisskopf-Wigner approximation for spontaneous decay are obtained under appropriate conditions in the context of a common perspective. Comparisons are made with other recently devised perturbative and nonperturbative formulations of electromagnetic transition phenomena which correctly include radiative line widths and level shifts, and the importance of proper treatment of the normalization and phase of the Schrödinger wave function in this connection is emphasized.
We reconsider the problem of the spontaneous emission of light by an excited atomic state. We scrutinize the survival probability of this excited state for very short times, in the so-called Zeno regime, for which we show that the dynamics is dictated by a coherent, in-phase, response of the on-shell and off-shell vacuum modes. We also develop a perturbative approach in order to interpolate between different temporal regimes: the Zeno, golden rule (linear), and Wigner-Weisskopf (exponential) regimes. We compare results obtained with the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{x}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ interaction Hamiltonians, using successively the dipole approximation and the exact coupling.
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.
The recent first-principles derivation of Langevin-Bloch equations of motion for near-resonant atoms embedded in a linear dielectric host leads to a clear, direct physical reinterpretation of fundamental physical principles as applied to one of the most basic problems of quantum mechanics: the renormalization of the spontaneous emission rate of an atom in a dielectric host. Here we show the role of the Lorentz local field, the fluctuation-dissipation theorem, the wave number and wave vector, the density of states, Fermi's golden rule, the electromagnetic energy density, canonical quantization, and the classical constitutive relations in this context.