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- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/09%3A_Spin_Angular_Momentum/9.05%3A_Spin_PrecessionThe expectation value of the spin angular momentum vector subtends a constant angle α with the z -axis, and precesses about this axis. This behavior is actually equivalent to that predicted by clas...The expectation value of the spin angular momentum vector subtends a constant angle α with the z -axis, and precesses about this axis. This behavior is actually equivalent to that predicted by classical physics.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/06%3A_Perturbative_Approaches/6.03%3A_Fine_Structure_of_atomic_LevelsIt is straightforward (and hence left for the reader :-) to prove that all off-diagonal elements of the set (49) are equal to 0 . Thus we may use Eq. (27) for each set of the quantum numbers \(\{n, l,...It is straightforward (and hence left for the reader :-) to prove that all off-diagonal elements of the set (49) are equal to 0 . Thus we may use Eq. (27) for each set of the quantum numbers {n,l,m} : \[\begin{aligned} E_{n, l, m}^{(1)} & \equiv E_{n, l, m}-E_{n}^{(0)}=\left\langle n l m\left|\hat{H}^{(1)}\right| n l m\right\rangle=-\frac{1}{2 m c^{2}}\left\langle\left(\hat{H}^{(0)}-\hat{U}(r)\right)^{2}\right\rangle_{n, l, m} \\ &=-\frac{1}{2 m c^{2}}\left(E_{n}^{2}-2 E_{n}\langle\hat{…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/05%3A_Quantum_Electrodynamics/5.04%3A_The_Electron-Photon_InteractionNow, we replace it by the vector potential operator derived in Section 5.3: \[\hat{\mathbf{A}}(\hat{\mathbf{r}},t) = \begin{cases} \displaystyle \sum_{\mathbf{k}\lambda} \sqrt{\frac{\hbar}{2\epsilon_0...Now, we replace it by the vector potential operator derived in Section 5.3: \[\hat{\mathbf{A}}(\hat{\mathbf{r}},t) = \begin{cases} \displaystyle \sum_{\mathbf{k}\lambda} \sqrt{\frac{\hbar}{2\epsilon_0\omega_{\mathbf{k}}V}}\, \Big(\hat{a}_{\mathbf{k}\lambda} \, e^{i(\mathbf{k}\cdot\mathbf{r} - \omega_{\mathbf{k}} t)} + \mathrm{h.c.}\Big)\, \mathbf{e}_{\mathbf{k}\lambda}, & (\mathrm{finite}\;\mathrm{volume}) \\ \displaystyle \int d^3k \sum_{\lambda} \sqrt{\frac{\hbar}{16\pi^3\epsilon_0\omega_{\ma…
