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:

$\frac{{\mathrm{dP}}_m}{\mathrm{dt}}\= \frac{2 \mathrm{\pi }}{\mathrm{\ℏ}}\mathrm{\ρ}\left(\mathrm{E\_\_m}\right){\left|V_{\mathrm{ml}}\right|}^2$

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.

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 $\mathrm{\omega }$ 

$\frac{{\mathrm{dP}}_m}{\mathrm{dt}}\= \frac{2}{{\mathrm{\ℏ}}^2{\mathrm{\ω}}_{\mathrm{ml}}}\sin \left(\left({\mathrm{\ω}}_{\mathrm{ml}}-\mathrm{\ω}\right)t\right){\left|V_{\mathrm{ml}}\right|}^2$ .

Switching from angular frequency to energy, multiplying by the density of final states $\mathrm{\rho }\left(\mathrm{E\_\_m}\right)$and integrating yields

$\frac{{\mathrm{dP}}_m}{\mathrm{dt}}\= \frac{2}{{\mathrm{\ℏ}}^{}}{\int }_{-\infty }^{\+\infty }\mathrm{\rho }\left(\mathrm{E\_\_m}\right)\frac{\sin \left(\frac{\left(E_m-E_l-\mathrm{\ℏ\ω}\right)t}{\mathrm{\ℏ}}\right)}{\left(E_m-E_l\right)}{\left|V_{\mathrm{ml}}\right|}^2 \ⅆE_m$ .

Assuming that the energy density is a constant, we have

$\frac{{\mathrm{dP}}_m}{\mathrm{dt}}\= \frac{2}{{\mathrm{\ℏ}}^{}}\mathrm{\rho }\left(\mathrm{E\_\_m}\right){\left|V_{\mathrm{ml}}\right|}^2{\int }_{-\infty }^{\+\infty }\frac{\sin \left(\frac{\left(E_m-E_l-\mathrm{\ℏ\ω}\right)t}{\mathrm{\ℏ}}\right)}{\left(E_m-E_l\right)} \ⅆE_m$ .

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

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

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

Therefore,

${\int }_{-\infty }^{\+\infty }\frac{\sin \left(\frac{\left(E_m-E_l-\mathrm{\ℏ\ω}\right)t}{\mathrm{\hslash }}\right)}{\left(E_m-E_l\right)} \ⅆE_m \=$$\mathrm{\pi }$

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

$\frac{{\mathrm{dP}}_m}{\mathrm{dt}}\= \frac{2 \mathrm{\pi }}{\mathrm{\hslash }}\mathrm{\rho }\left(\mathrm{E\_\_m}\right){\left|V_{\mathrm{ml}}\right|}^2$ .

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

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.    

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

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