= Solution
In <Fermi's golden rule>,
$$
\Gamma=\frac{2\pi}{\hbar}|M|^2\rho,
$$
$\Gamma$ is a transition <decay rate>, measured in inverse time; $\hbar$ is the <reduced Planck constant>; $M=\langle f|H_{\mathrm{int}}|i\rangle$ is the transition <matrix element> of the interaction part of the <Hamiltonian>; and $\rho$ is the final <density of states> per unit <energy>, evaluated subject to <conservation of energy>. Final <spin> and other discrete labels must be summed, with initial labels averaged when the preparation requires it. If $M$ varies across the final <phase space>, $|M|^2\rho$ denotes the corresponding integral rather than a product of constants.
A convention warning: many <quantum field theory> texts use $\Gamma$ for an <energy> <decay width>. That convention is $\Gamma_{\mathrm{width}}=\hbar\Gamma_{\mathrm{rate}}$; the formula as printed uses the rate convention.
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