5.5: Transitions to a group of states

Last updated
Save as PDF

Page ID: 28771

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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.