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classes:2009:fall:phys4101.001:lec_notes_1116 [2009/11/19 08:52] – ykclasses: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 so H is|<math>a_+a_-+1/2</math>|<math>L_+L_-+\hbar L_z (+L^2_z)</math>| |factorization is not perfect so H is|<math>a_+a_-+1/2</math>|<math>L_+L_-+\hbar L_z (+L^2_z)</math>|
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 |while they in turn come from |<math>[x,p]=i{\hbar}</math>|<math>[L_x,L_y]=i{\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> | |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> | +| |no top rung|<math>{L_+}{\psi}=0</math>| 
-| from above, we can figure out, for example, |<math>\psi_0 = </math>...|<math>\lambda=m_{max}(m_{max}+1)</math> and <math>\lambda=m_{min}(m_{min}-1)</math> and more|+| 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|
  
 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>.   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>.  
classes/2009/fall/phys4101.001/lec_notes_1116.1258642320.txt.gz · Last modified: 2009/11/19 08:52 (external edit)

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