12.E: Time-Dependent Perturbation Theory (Exercises)
- Page ID
- 15962
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Consider the two-state system examined in Section 1.3.[ex8.1] Suppose that \[\begin{aligned} \langle 1 |\,H_1\,|1\rangle &= e_{11},\nonumber\\[0.5ex] \langle 2 |\,H_1\,|2\rangle &= e_{22},\nonumber\\[0.5ex] \langle 1|\,H_1\,|2\rangle = \langle 2|\,H_1\,|1\rangle^\ast &= \frac{1}{2}\,\gamma\,\hbar\,\exp(\,{\rm i}\,\omega\,t),\nonumber\end{aligned}\] where \(e_{11}\), \(e_{22}\), \(\gamma\), and \(\omega\) are real. Show that \[\begin{aligned} {\rm i}\,\frac{d\hat{c}_1}{dt} &= \frac{\gamma}{2}\,\exp\left[+{\rm i}\,(\omega-\hat{\omega}_{21})\,t\right]\hat{c}_2,\nonumber\\[0.5ex] {\rm i}\,\frac{d\hat{c}_2}{dt} &= \frac{\gamma}{2}\,\exp\left[-{\rm i}\,(\omega-\hat{\omega}_{21})\,t\right]\hat{c}_1,\nonumber\end{aligned}\] where \(\hat{c}_1 = c_1\,\exp(\,{\rm i}\,e_{11}\,t/\hbar)\), \(\hat{c}_2 = c_2\,\exp(\,{\rm i}\,e_{22}\,t/\hbar)\), and \[\hat{\omega}_{21} = \frac{E_2+e_{22}-E_1-e_{11}}{\hbar}.\] Hence, deduce that if the system is definitely in state 1 at time \(t=0\) then the probability of finding it in state 2 at some subsequent time, \(t\), is \[P_2(t) = \frac{\gamma^{\,2}}{ \gamma^{\,2} + (\omega-\hat{\omega}_{21})^{\,2}}\, \sin^2\left(\left[\gamma^{\,2}+ (\omega-\hat{\omega}_{21})^{\,2}\right]^{1/2} \frac{t}{2}\right).\]
- Consider an atomic nucleus of spin-\(s\) and gyromagnetic ratio \(g\) placed in the magnetic field \[{\bf B} = B_0\,{\bf e}_z + B_1\left[\cos(\omega\,t)\,{\bf e}_x-\sin(\omega\,t)\,{\bf e}_y\right],\] where \(B_1\ll B_0\). Let \(\chi_{s,m}\) be a properly normalized simultaneous eigenstate of \(S^{\,2}\) and \(S_z\), where \({\bf S}\) is the nuclear spin. Thus, \(S^{\,2}\,\chi_{s,m} = s\,(s+1)\,\hbar^{\,2}\,\chi_{s,m}\) and \(S_z \,\chi_{s,m} = m\,\hbar\,\chi_{s,m}\), where \(-s\leq m\leq s\). Furthermore, the instantaneous nuclear spin state is written \[\chi= \sum_{m=-s,s} c_m(r)\,\chi_{s,m},\] where \(\sum_{m=-s,s}|c_m|^{\,2}=1\).
- Demonstrate that \[\begin{gathered} \frac{dc_m}{dt} = \frac{{\rm i}\,\gamma}{2}\left([s\,(s+1)-m\,(m-1)]^{1/2}\,{\rm e}^{\,{\rm i}\,(\omega-\omega_0)\,t}\,c_{m-1}\right.\\[0.5ex]\left. + [s\,(s+1)-m\,(m+1)]^{1/2}\,{\rm e}^{-{\rm i}\,(\omega-\omega_0)\,t}\,c_{m+1}\right)\nonumber\end{gathered}\] for \(-s\leq m\leq s\), where \(\omega_0=g\,\mu_N\,B_0/\hbar\), \(\gamma= g\,\mu_N\,B_1/\hbar\), and \(\mu_N=e\,\hbar/(2\,m_p)\).
- Consider the case \(s=1/2\). Demonstrate that if \(\omega=\omega_0\) and \(c_{1/2}(0)= 1\) then \[\begin{aligned} c_{1/2}(t)= \cos(\gamma\,t/2), && c_{-1/2}(t)= {\rm i}\,\sin(\gamma\,t/2).\nonumber\end{aligned}\]
- Consider the case \(s=1\). Demonstrate that if \(\omega=\omega_0\) and \(c_{1}(0)= 1\) then \[\begin{aligned} c_{1}(t)&= \cos^2(\gamma\,t/2),\nonumber\\[0.5ex] c_0(t)&= {\rm i}\sqrt{2}\,\cos(\gamma\,t/2)\,\sin(\gamma\,t/2),\nonumber\\[0.5ex] c_{-1}(t)&= -\sin^2(\gamma\,t/2).\nonumber\end{aligned}\]
- Consider the case \(s=3/2\). Demonstrate that if \(\omega=\omega_0\) and \(c_{3/2}(0)= 1\) then \[\begin{aligned} c_{3/2}(t)&= \cos^3(\gamma\,t/2),\nonumber\\[0.5ex] c_{1/2}(t)&= {\rm i}\sqrt{3}\,\cos(\gamma\,t/2)\,\sin^2(\gamma\,t/2),\nonumber\\[0.5ex] c_{-1/2}(t)&= -\sqrt{3}\,\cos^2(\gamma\,t/2)\,\sin(\gamma\,t/2),\nonumber\\[0.5ex] c_{-3/2}(t)&=-{\rm i}\,\sin^3(\gamma\,t/2).\nonumber\end{aligned}\]
- Demonstrate that a spontaneous transition between two atomic states of zero orbital angular momentum is absolutely forbidden. (Actually, a spontaneous transition between two zero orbital angular momentum states is possible via the simultaneous emission of two photons, but takes place at a very slow rate .)
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)