classes:2009:fall:phys4101.001:lec_notes_1214
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| classes:2009:fall:phys4101.001:lec_notes_1214 [2009/12/15 14:57] – ludeman | classes:2009:fall:phys4101.001:lec_notes_1214 [2009/12/17 23:11] (current) – fitch |
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| ====Fine Structure==== | ===Perturbation Theory=== |
| | What do we need in perturbation theory? We want to find the energy: |
| | <math>E=E^{(0)} + E_{r} + (E_{FS}+E_Z)</math> where <math>E^{(0)} = \frac{-13.6 \mathrm{eV}}{n^2}</math> |
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| | ===Fine Structure=== |
| Fine structure is due to two mechanisms: **relativistic correction** and **spin-orbit coupling**. In other words, a very small perturbation (correction) to the Bohr energies. | Fine structure is due to two mechanisms: **relativistic correction** and **spin-orbit coupling**. In other words, a very small perturbation (correction) to the Bohr energies. |
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| The equation of which is: <math>H_{fs}=\alpha\vec{S}\vec{L}</math> | The equation of which is: <math>H_{fs}=\alpha\vec{S}\vec{L}</math>, where <math>\alpha = \frac{e^2}{4\pi \epsilon_0}\cdot\frac{1}{m^2 c^2 r^3}</math>, like a [[http://en.wikipedia.org/wiki/Magnetic_dipole–dipole_interaction|dipole-dipole interaction in E&M]]. |
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| By considering the relativistic version of momentum: <math>p=\frac{mv} {\sqrt{1-(\frac v {c})^2}}</math>. We can derive the relativistic equation for kinetic energy: <math>T=\sqrt{m^2c^4+p^2c^2}-mc^2</math>. | By considering the relativistic version of momentum: <math>p=\frac{mv} {\sqrt{1-(\frac v {c})^2}}</math>. We can derive the relativistic equation for kinetic energy: <math>T=\sqrt{m^2c^4+p^2c^2}-mc^2</math>. |
| <math><\psi|V^2|\psi> \appr <\frac{1} {r^2}> \appr \frac{1} {(l+\frac{1} 2)n^3a^2}</math> | <math><\psi|V^2|\psi> \appr <\frac{1} {r^2}> \appr \frac{1} {(l+\frac{1} 2)n^3a^2}</math> |
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| Putting it all together and we get: <math>E^1_r=-\frac{E_n^2} {2mc^2}[\frac{4n} {l+\frac{1} 2} - 3]</math> Dividing both sides by <math>E_n</math> we get a relativistic correction of about <math>2x10^{-5}</math> | Putting it all together and we get: <math>E^1_r=-\frac{E_n^2} {2mc^2}[\frac{4n} {l+\frac{1} 2} - 3]</math> Dividing both sides by <math>E_n</math> and we get a relativistic correction of about <math>2</math>x<math>10^{-5}</math> |
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| | ===Higher-Order Degeneracy=== |
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| | Enter Notes Here |
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| | ===Spin-Orbit Coupling=== |
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| | Enter Notes Here |
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classes/2009/fall/phys4101.001/lec_notes_1214.1260910670.txt.gz · Last modified: 2009/12/15 14:57 by ludeman