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
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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
Use the Explore function to plot the sinc function as a function of time t:
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Observe that the area under the curve appears independent of t. We can confirm this hunch by taking the integral:
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Therefore,
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.