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- https://phys.libretexts.org/Learning_Objects/A_Physics_Formulary/Physics/10%3A_Quantum_PhysicsQuantum mechanics, atomic physics, Schrödinger and Dirac equations
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Introductory_Physics_II_(1112)/zz%3A_Back_Matter/10%3A_13.1%3A_Appendix_J-_Physics_Formulas_(Wevers)/1.10%3A_Quantum_PhysicsQuantum mechanics, atomic physics, Schrödinger and Dirac equations
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/13%3A_Atomic_Structure/13.05%3A_Atomic_Spectra_and_X-raysAn electron in the L shell “sees” an effective charge Z=13−1=12, because one electron in the K shell shields the nuclear charge. (Recall, two electrons are not in the K shell because the oth...An electron in the L shell “sees” an effective charge Z=13−1=12, because one electron in the K shell shields the nuclear charge. (Recall, two electrons are not in the K shell because the other electron state is vacant.) The frequency of the emitted photon can be estimated from the energy difference between the L and K shells.
- https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/08%3A_Atomic_Structure/8.06%3A_Atomic_Spectra_and_X-raysRadiation is absorbed and emitted by atomic energy-level transitions. Quantum numbers can be used to estimate the energy, frequency, and wavelength of photons produced by atomic transitions. X-ray ph...Radiation is absorbed and emitted by atomic energy-level transitions. Quantum numbers can be used to estimate the energy, frequency, and wavelength of photons produced by atomic transitions. X-ray photons are produced when a vacancy in an inner shell of an atom is filled by an electron from the outer shell of the atom. The frequency of X-ray radiation is related to the atomic number Z of an atom.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/12%3A_Time-Dependent_Perturbation_Theory/12.10%3A_Selection_Rules_(Hydrogen_Atoms)Consider x±=x±iy. It is easily demonstrated that [Lz,x±]=±ℏx±. Hence, \[\langle n,l,m|[L_z,x_+]-\hbar\,x_+|n',l',m'\rangle = \hbar\,(m-m'-1)\, \langle n,l...Consider x±=x±iy. It is easily demonstrated that [Lz,x±]=±ℏx±. Hence, ⟨n,l,m|[Lz,x+]−ℏx+|n′,l′,m′⟩=ℏ(m−m′−1)⟨n,l,m|x+|n′,l′,m′⟩=0, and ⟨n,l,m|[Lz,x−]+ℏx−|n′,l′,m′⟩=ℏ(m−m′+1)⟨n,l,m|x−|n′,l′,m′⟩=0. Clearly, ⟨n,l,m|x+|n′,l′,m′⟩ is zero unless m′=m−1, and ⟨n,l,m|x−|n′,l′,m′⟩ is zero unless m′=m+1.
