12.E: Time-Dependent Perturbation Theory (Exercises)

\[\begin{align} \langle 1 |\,H_1\,|1\rangle &= e_{11},\nonumber\\[4pt] \langle 2 |\,H_1\,|2\rangle &= e_{22},\nonumber\\[4pt] \langle 1|\,H_1\,|2\rangle = \langle 2|\,H_1\,|1\rangle^\ast &= \frac{1}{2}\,\gamma\,\hbar\,\exp(\,{\rm i}\,\omega\,t),\nonumber\end{align} \nonumber \]

where \(e_{11}\), \(e_{22}\), \(\gamma\), and \(\omega\) are real. Show that

\[\begin{align} {\rm i}\,\frac{d\hat{c}_1}{dt} &= \frac{\gamma}{2}\,\exp\left[+{\rm i}\,(\omega-\hat{\omega}_{21})\,t\right]\hat{c}_2,\nonumber\\[4pt] {\rm i}\,\frac{d\hat{c}_2}{dt} &= \frac{\gamma}{2}\,\exp\left[-{\rm i}\,(\omega-\hat{\omega}_{21})\,t\right]\hat{c}_1,\nonumber\end{align} \nonumber \]

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

\[\hat{\omega}_{21} = \frac{E_2+e_{22}-E_1-e_{11}}{\hbar}. \nonumber \]

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

\[P_2(t) = \frac{\gamma^2}{ \gamma^2 + (\omega-\hat{\omega}_{21})^2}\, \sin^2\left(\left[\gamma^2+ (\omega-\hat{\omega}_{21})^2\right]^{1/2} \frac{t}{2}\right). \nonumber \]

\[{\bf B} = B_0\,{\bf e}_z + B_1\left[\cos(\omega\,t)\,{\bf e}_x-\sin(\omega\,t)\,{\bf e}_y\right], \nonumber \]

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

\[\chi= \sum_{m=-s,s} c_m(r)\,\chi_{s,m}, \nonumber \]

where \(\sum_{m=-s,s}|c_m|^2=1\).

1. Demonstrate that

   \[\begin{gathered} \frac{dc_m}{dt} = \frac{{\rm i}\,\gamma}{2}\left([s\,(s+1)-m\,(m-1)]^{1/2}\,\rm e^{\,{\rm i}\,(\omega-\omega_0)\,t}\,c_{m-1}\right.\\[4pt]\left. + [s\,(s+1)-m\,(m+1)]^{1/2}\,\rm e^{-{\rm i}\,(\omega-\omega_0)\,t}\,c_{m+1}\right)\nonumber\end{gathered} \nonumber \]

   for \(-s\leq m\leq s\), where \(\omega_0=g\,\mu_N\,B_0/\hbar\), \(\gamma= g\,\mu_N\,B_1/\hbar\), and \(\mu_N=e\,\hbar/(2\,m_p)\).

2. Consider the case \(s=1/2\). Demonstrate that if \(\omega=\omega_0\) and \(c_{1/2}(0)= 1\) then

   \[\begin{align} c_{1/2}(t)= \cos(\gamma\,t/2), && c_{-1/2}(t)= {\rm i}\,\sin(\gamma\,t/2).\nonumber\end{align} \nonumber \]

3. Consider the case \(s=1\). Demonstrate that if \(\omega=\omega_0\) and \(c_{1}(0)= 1\) then

   \[\begin{align} c_{1}(t)&= \cos^2(\gamma\,t/2),\nonumber\\[4pt] c_0(t)&= {\rm i}\sqrt{2}\,\cos(\gamma\,t/2)\,\sin(\gamma\,t/2),\nonumber\\[4pt] c_{-1}(t)&= -\sin^2(\gamma\,t/2).\nonumber\end{align} \nonumber \]

4. Consider the case \(s=3/2\). Demonstrate that if \(\omega=\omega_0\) and \(c_{3/2}(0)= 1\) then

   \[\begin{align} c_{3/2}(t)&= \cos^3(\gamma\,t/2),\nonumber\\[4pt] c_{1/2}(t)&= {\rm i}\sqrt{3}\,\cos(\gamma\,t/2)\,\sin^2(\gamma\,t/2),\nonumber\\[4pt] c_{-1/2}(t)&= -\sqrt{3}\,\cos^2(\gamma\,t/2)\,\sin(\gamma\,t/2),\nonumber\\[4pt] c_{-3/2}(t)&=-{\rm i}\,\sin^3(\gamma\,t/2).\nonumber\end{align} \nonumber \]