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# 5.5: Transitions to a group of states


We are often interested in the situation where transitions take place not to a single final state but to a group, $$G$$, of final states with energy in some range about the initial state energy

$E_k − \Delta E \leq E_m \leq E_k + \Delta E \nonumber$

Then the total transition probability is obtained by summing the contributions of all the final states. The number of final states in the interval between $$E_m$$ and $$E_m + dE_m$$ is $$g(E_m) dE_m$$, where the function $$g(E_m)$$ is known as the density of final states. The total transition probability for transitions to $$G$$ is then given by

$p_G(t) = \frac{1}{\hbar^2} \int^{E_k+\Delta E}_{E_k−\Delta E} |V_{mk}|^2 f(t, \omega_{mk}) g(E_m) dE_m. \nonumber$

For sufficiently large $$t$$, and $$\Delta E \gg 2\pi \hbar/t$$, we observe that essentially the only contributions to the integral come from the energy range corresponding to the narrow central peak of the function $$f(t, \omega_{mk})$$. Within this range we can neglect the variation of $$g(E_m)$$ and $$V_{mk}$$, which can therefore be taken out of the integral to give

$p_G(t) = \left[ \frac{|V_{mk}|^2}{\hbar^2} g(E_m) \right]_{E_m=E_k} \int^{E_k+\Delta E}_{E_k−\Delta E} f(t, \omega_{mk}) dE_m. \nonumber$

Furthermore, we can extend the limits on the integration to $$\pm \infty$$. Noting that $$dE_m = \hbar d\omega_{mk}$$ and using the result that

$\int^{\infty}_{−\infty} \frac{\ sin^2 x}{x^2} dx = \pi \nonumber$

we obtain for the first–order transition probability

$p_G(t) = \frac{2\pi t}{\hbar} \left[ |V_{mk}|^2 g(E_m) \right]_{E_m=E_k} \nonumber$

The transition rate, $$R$$, is just the derivative of this with respect to $$t$$ and is thus given by the so–called Fermi Golden Rule:

$R = \frac{2\pi}{\hbar} \left[ |V_{mk}|^2 g(E_m) \right]_{E_m=E_k} \nonumber$

The Fermi Golden Rule is probably the single most widely used result in quantum mechanics. The factor of $$\frac{2\pi}{\hbar}$$ depends on the choice of perturbing potential, but the $$|V_{n1}|^2 g(E_m)$$ term appears for any applied perturbation. Be careful about the density of energy states - one sometimes encounters density of frequency states (which differs by a factor of $$\hbar$$) or of wavevector states.

It may appear that need to know the density of final states, $$g(E_m)$$, but this is not always true. In cases where $$|V_{mk}| = 0$$ transitions are forbidden, and in some cases we can deduce $$g(E_m)$$ from the relative rates of related transitions.

5.5: Transitions to a group of states is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.