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| classes:2009:fall:phys4101.001:lec_notes_1116 [2009/11/18 13:20] – yk | classes:2009:fall:phys4101.001:lec_notes_1116 [2009/11/19 11:59] (current) – yk |
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| ===== Nov 16 (Mon) ===== | ===== Nov 16 (Mon) Angular momentum with raising/lowing operators ===== |
| ** Responsible party: Pluto 4ever, malmx026 ** | ** Responsible party: Pluto 4ever, malmx026 ** |
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| ^ ^ SHO ^ Angluar momentum ^ | ^ ^ SHO ^ Angluar momentum ^ |
| |The hamiltonian, H, is proportional to | <math>x^2 + p^2</math>|<math>{L_x}^2 + {L_y}^2</math>| | |The hamiltonian, H, is proportional to | <math>x^2 + p^2</math>|<math>{L_x}^2 + {L_y}^2(+L^2_z)</math>| |
| |We tried to factorize H by |<math>(x+ip)(x-ip)</math> | <math>(L_x+iL_y)(L_x-iL_y)</math> | | |We tried to factorize H by |<math>(x+ip)(x-ip)</math> | <math>(L_x+iL_y)(L_x-iL_y) (+L^2_z)</math> | |
| |Call these terms| <math>a_\pm\approx \mp ip + x</math> |<math>L_\pm\approx \pm iL_y + L_x</math> | | |Call these terms| <math>a_\pm\approx \mp ip + x</math> |<math>L_\pm\approx \pm iL_y + L_x</math> | |
| |factorization is not perfect | |factorization is not perfect so H is|<math>a_+a_-+1/2</math>|<math>L_+L_-+\hbar L_z (+L^2_z)</math>| |
| | |the extra factor in H is related to the commutator |<math>[a_+,a_-]=-1</math>|<math>[L_+,L_-] = 2{\hbar}{L_z}</math> | |
| | |while they in turn come from |<math>[x,p]=i{\hbar}</math>|<math>[L_x,L_y]=i{\hbar}{L_z}</math> | |
| | |meanwhile, these equation for the bottom rung state will be useful for other things |<math>{a_-}{\psi}=0</math>|<math>{L_-}{\psi}=0</math> | |
| | | |no top rung|<math>{L_+}{\psi}=0</math>| |
| | | from above, we can figure out, for example, |<math>\psi_0 = </math>, <math>E_0=\hbar\omega(n+1/2)</math> ...|<math>\lambda=m_{max}(m_{max}+1)</math> and <math>\lambda=m_{min}(m_{min}-1)</math> and more| |
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| | |<math>[x,p]=i{\hbar}</math>|<math>[L_x,L_y]=i{\hbar}{L_z}</math> | | For the top rung, by definition, <math>{L_z}{f_t}={\hbar}{l}{f_t}</math>; <math>{L^2}{f_t}={\lambda}{f_t}</math>. |
| no top rung | For the bottom rung, <math>{L_z}{f_b}=-{\hbar}{l}{f_b}</math>; <math>{L^2}{f_b}={\lambda}{f_b}</math>. These are also important to draw various additional conclusions such as <math>2l</math> being an integer. |
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| <math>a_\pm ={ \frac{1}{sqrt{2m{\hbar}{\omega}}}(\mp{ip}+m{\omega}x)</math> | |
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| bottom rung, <math>{a_-}{\psi}=0</math> | |
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| <math>[a_+,a_-]=-1</math> | |
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| <math>H = {\hbar}{\omega}({a_+}{a_-}-1)</math> | |
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| ===Angular Momentum=== | |
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| has a top rung, <math>{L_z}{f_t}={\hbar}{l}{f_t}</math>; <math>{L^2}{f_t}={\lambda}{f_t}</math> | |
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| <math>L_\pm =\pm{i}{\hbar}L_y + L_x</math> | |
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| bottom rung, <math>{L_z}{f_b}=-{\hbar}{l}{f_b}</math>; <math>{L^2}{f_b}={\lambda}{f_b}</math> | |
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| The hamiltonian, H, is proportional to | |
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| <math>[L_+,L_-] = 2{\hbar}{L_z}</math> | |
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| <math>L^2 = {L_\pm}{L_\mp}+{L_z}^2\mp{\hbar}{L_z}</math> | |
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