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Unformatted text preview: rmula to make the calcula2on easy. 20 Spherical Harmonics. 21 Summary We have used separa2on of variables in Cartesian coordinates and in spherical coordinates to solve the Schrodinger equa2on for bound states in 3 dimensions. u༇  In Cartesian, we have one quantum number per dimension. u༇  For spherical symmetry, we have the quantum numbers l and m and the spherical harmonics Ylm for the angular coordinates and one quantum number for the radial coordinate. u༇  The spherical harmonics are orthonormal. u༇  We have the solu2ons to the Hydrogen problem. u༇  We will use these states to understand atoms and the periodic table. u༇  This simple solu2on to Hydrogen in accurate to a few parts in 10000 for energies. u༇  In this solu2on the energy only depends on the principle quantum number n. u༇  But the states will be split apart by a few parts in 10000. u༇  The study of Hydrogen was extremely important for the understanding of QM and QED. u༇  22 Chapter 9: Atoms u༇  Hydrogen in a Magne2c eld u༇  Electron Spin u༇  Hydrogen Fine Structure u༇  Total Angular Momentum J u༇  Exchange Symmetry u༇  Mul2 electron Atoms u༇  Some of these effects are easy to confuse with others, so we need to think clearly. 23 Correc2ons to Simple Hydrogen Spectrum u༇  Splikng degenerate states in a magne2c field o  Zeeman Effect u༇  Electron Spin u༇  Spin- Orbit Interac2on o  Total Angular Momentum u༇  Hydrogen Fine Srtucture o  rela2vis2c correc2on o  + Spin- Orbit o  +… o  States of Total J2 are the energy eigenstates u༇  Anomalous Zeeman effect (includes spin) o  splits the eigenstates above 24 Magne2c Moment due to e Current 25 Magne2c Moment Propor2onal to L u༇  Magne2c Moment ~ L u༇ Bohr Magneton u༇ Torque µB=5.788×10−5 eV/T u༇ Energy Quan2zed 26 Zeeman Effect u༇  Atom in a B field The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. u༇  Choose z direc2on along field u༇  Energy shiGed by o ΔE=µBBml o  µ =5.788×10−5 eV/T B u༇ Usual Spectra limited to Δl=±1 and Δm=0,±1 and Δs=0 due to photon emission rules The image cannot be displayed. Your computer may not have enough memory to open th...
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