# How does decay rate depend on energy difference?

From Fermi's golden rule, one knows that transition rate is

$$\Gamma_{i \to f} = \frac{2\pi}{\hbar} \left \lvert \langle f | H' | i \rangle \right \rvert^2 \rho \, .$$

However, can we make the assumption below?

Assuming that all we know is that there exists some perturbiative term in your Hamiltonian that mixes the eigenstates of your system, is it generally true that higher the energy difference between the two states, the higher the transition rate?

For example, if we have an energy diagram like in the image below, how would one go about choosing the $$\epsilon = E1-E2$$ to maximize or minimize the decay rate?

• It depends on which changes more, the matrix element or $\rho$. There's no simple answer. – DanielSank Oct 19 '18 at 16:26
• No. Various examples from atomic physics, for example see lifetimes of widely studied Rb 85/87. – Alexander Oct 19 '18 at 16:36