FermiGoldenRule - Maple Help

Fermi's Golden Rule

 Overview The transition rate of a molecular process between a discrete state and a continuum of states can be estimated from Fermi's Golden Rule.  The rule was popularized by Enrico Fermi in a book entitled Nuclear Physics published in 1950, but it was first derived by Paul Dirac in 1927.  Fermi's Golden Rule predicts the transition rate between a discrete state l and a continuum of states m as follows: where $\mathrm{\rho }\left(\mathrm{E__m}\right)$is the energy density and $\left|{V}_{\mathrm{ml}}\right|$ is the absolute value of the transition element of the perturbation matrix.

Derivation

To derive Fermi's Golden Rule, we begin with the transition rate from first-order perturbation theory where the perturbation is sinusoidal with a frequency $\mathrm{\omega }$

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Switching from angular frequency to energy, multiplying by the density of final states and integrating yields

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Assuming that the energy density is a constant, we have

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But integral, whose function in the integrand is known as the sinc function, can be evaluated to a constant.  Consider the sinc function

 > $\frac{\mathrm{sin}\left(x\cdot t\right)}{x};$
 $\frac{{\mathrm{sin}}{}\left({x}{}{t}\right)}{{x}}$ (2.1)

Use the Explore function to plot the sinc function as a function of time t:

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$\mathbf{t}$

Observe that the area under the curve appears independent of t.  We can confirm this hunch by taking the integral:

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 ${{\int }}_{{-}{\mathrm{\infty }}}^{{\mathrm{\infty }}}\frac{{\mathrm{sin}}{}\left({x}{}{t}\right)}{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{\mathrm{\pi }}$ (2.2)

Therefore,

$\mathrm{\pi }$

whose substitution into the transition rate equation yields Fermi's Golden Rule

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Note that the predicted transition rate is independent of time t.

Fluorescence

To illustrate the Golden Rule, we consider the fluorescence decay of a molecule in front of a mirror, following the work of K. H. Drexhage, H. Kuhn, and F. P. Schäfer in Ref. [3].  The molecule's density of states changes significantly as its distance h from the mirror changes.  By Fermi's Golden Rule, we would expect the emission rate to vary in proportion to the changes in the density of states.

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 ${G}{≔}\left({x}{,}{y}\right){↦}\frac{{I}}{{4}}{\cdot }{\mathrm{HankelH1}}{}\left({0}{,}{2}{\cdot }{\mathrm{\pi }}{\cdot }\sqrt{{{y}}^{{2}}{+}{{x}}^{{2}}}\right)$
 ${\mathrm{emission}}{≔}\frac{{\left|{\mathrm{HankelH1}}{}\left({0}{,}{2}{}{\mathrm{\pi }}{}\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}\right){-}{\mathrm{HankelH1}}{}\left({0}{,}{2}{}{\mathrm{\pi }}{}\sqrt{{{y}}^{{2}}{+}{\left({x}{+}{2}{}{h}\right)}^{{2}}}\right)\right|}^{{2}}}{{16}}$ (3.1)

We can make an animation of the emission pattern as a function of the distance h from the mirror.

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In agreement with Fermi's Golden Rule, as the density of states increases, the molecule glows more brightly.

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References

 1 P. A. M. Dirac, "The Quantum Theory of Emission and Absorption of Radiation," Proceedings of the Royal Society A 114, 243–265 (1927).  Refer to equations (24) and (32).
 2 E. Fermi, Nuclear Physics (University of Chicago Press, Chicago, 1950). Refer to formula VIII.2.
 3 K. H. Drexhage, H. Kuhn, F. P. Schäfer, "Variation of the Fluorescence Decay Time of a Molecule in Front of a Mirror," Berichte der Bunsengesellschaft für physikalische Chemie 72, 329 (1968).