Using the notation \(\left|j,m_j\right\rangle\) we have \[\begin{aligned} &\left|2p_{3/2}, 3/2\right\rangle=\psi_{211}(r)\left|\uparrow\right\rangle \\ &\left|2p_{3/2}, 1/2\right\rangle=\sqrt{\frac{2}{3}}\psi_{210}(r)\left|\uparrow\right\rangle+\sqrt{\frac{1}{3}}\psi_{211}(r)\left|\downarrow\right\rangle \\ &\left|2p_{3/2}, -1/2\right\rangle=\sqrt{\frac{2}{3}}\psi_{210}(r)\left|\downarrow\right\rangle+\sqrt{\frac{1}{3}}\psi_{21-1}(r)\left|\uparrow\right\rangle \\ &\left|2p_{3/2}, -3/2\right\rangle=\psi_{21-1}(r)\left|\downarrow\right\rangle \\ \\ &\left|2p_{1/2}, 1/2\right\rangle=\sqrt{\frac{1}{3}}\psi_{210}(r)\left|\uparrow\right\rangle-\sqrt{\frac{2}{3}}\psi_{211}(r)\left|\downarrow\right\rangle \\ &\left|2p_{1/2}, -1/2\right\rangle=\sqrt{\frac{1}{3}}\psi_{210}(r)\left|\downarrow\right\rangle-\sqrt{\frac{2}{3}}\psi_{21-1}(r)\left|\uparrow\right\rangle \\ \\ &\left|2s_{1/2}, 1/2\right\rangle=\psi_{200}(r)\left|\uparrow\right\rangle \\ &\left|2s_{1/2}, -1/2\right\rangle=\psi_{200}(r)\left|\downarrow\right\rangle \end{aligned}\] These are all of the degenerate \(n=2\) levels. Recall the transition rate \(\Gamma_{f\leftarrow i}\) satisfies \[\Gamma_{f\leftarrow i}=\frac{4\omega^3e^2}{3\hbar c^3}|\left\langle\psi_f\middle|\mathbf{r}\middle|\psi_i\right\rangle|^2\] in the dipole approximation. Also, recall that the lifetime of states satisfies \[\tau_{f\leftarrow i}=\frac{1}{\Gamma_{f\leftarrow i}}\] Calculate \(\tau_{1s_{1/2}\leftarrow2p_{3/2}},\tau_{1s_{1/2}\leftarrow2p_{1/2}},\) and \(\tau_{1s_{1/2}\leftarrow2s_{1/2}}\) in the dipole approximation for each \(m_j\) value. Express your final answer in seconds.
Consider a five state system with the following \(\hat{H}\): \[\hat{H}_0=\begin{pmatrix}E_0^0 & 0 & 0 & 0 & 0 \\ 0 & E_0^0 & 0 & 0 & 0 \\ 0 & 0 & E_0^0 & 0 & 0 \\ 0 & 0 & 0 & E_0^0 & 0 \\ 0 & 0 & 0 & 0 & E_0^1\end{pmatrix}, \ \hat{V}=\begin{pmatrix}0 & 0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 & c \\ 0 & 0 & 0 & a & d \\ 0 & 0 & a & 0 & e \\ b & c & d & e & f\end{pmatrix}\] with \(a,b,c,d,e,f\) all real. This has four degenerate and one nondegenerate level for \(\hat{H}_0\). Compute all energies to order \(V^2\). Note the degeneracy is only partially lifted to first order.