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classes:2009:fall:phys4101.001:lec_notes_1116

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classes:2009:fall:phys4101.001:lec_notes_1116 [2009/11/17 17:51] – kuehlerclasses: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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 ====Comparison Between SHO and Angular Momentum==== ====Comparison Between SHO and Angular Momentum====
  
-In lecture, we just went over the basics of angular momentum and how it compared to the equations we previously learned for the simple harmonic oscillator (SHO).+In lecture, we just went over the basics of angular momentum and how it compared to the equations (concepts) we previously learned for the simple harmonic oscillator (SHO).
  
-===SHO:===+^ ^ SHO ^ Angluar momentum ^ 
 +|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) (+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> | 
 +|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|
  
-<math>[x,p]=i{\hbar}</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>.   
- +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.
-no top rung +
- +
-<math>a_\pm ={ \frac{1}{sqrt{2m{\hbar}{\omega}}}(\mp{ip}+m{\omega}x)</math> +
- +
-bottom rung, <math>{a_-}{\psi}=0</math> +
- +
-<math>H<\math> is proportional to  <math>x^2 + p^2<\math> +
- +
-<math>[a_+,a_-]=-1</math> +
- +
-<math>H = {\hbar}{\omega}({a_+}{a_-}-1)</math> +
- +
- +
-===Angular Momentum=== +
- +
-<math>[L_x,L_y]=i{\hbar}{L_z}</math> +
- +
-has a top rung, <math>{L_z}{f_t}={\hbar}{l}{f_t}</math>; <math>{L^2}{f_t}={\lambda}{f_t}</math> +
- +
-<math>L_\pm =\pm{i}{\hbar}L_y + L_x</math> +
- +
-bottom rung, <math>{L_z}{f_b}=-{\hbar}{l}{f_b}</math>; <math>{L^2}{f_b}={\lambda}{f_b}</math> +
- +
-<math>H<\math> is proportional to <math>{L_x}^2 + {L_y}^2<\math> +
- +
-<math>[L_+,L_-] = 2{\hbar}{L_z}</math> +
- +
-<math>L^2 = {L_\pm}{L_mp}+{L_z}^2+{\hbar}{L_z}</math>+
  
  
classes/2009/fall/phys4101.001/lec_notes_1116.1258501861.txt.gz · Last modified: 2009/11/17 17:51 by kuehler

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