5.5: Transitions to a group of states
- Page ID
- 28771
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.