===== Nov 16 (Mon) Angular momentum with raising/lowing operators =====
** Responsible party: Pluto 4ever, malmx026  ** 

**To go back to the lecture note list, click [[lec_notes]]**\\
**previous lecture note: [[lec_notes_1111]]**\\
**Main quiz 3 concepts: [[Quiz_3_1113]]**\\
**next lecture note: [[lec_notes_1118]]**\\

**Main class wiki page: [[home]]**

Please try to include the following

  * main points understood, and expand them - what is your understanding of what the points were.
    * expand these points by including many of the details the class discussed.
  * main points which are not clear.  - describe what you have understood and what the remain questions surrounding the point(s).
    * Other classmates can step in and clarify the points, and expand them. 
  * How the main points fit with the big picture of QM.  Or what is not clear about how today's points fit in in a big picture.
  * wonderful tricks which were used in the lecture.\\
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====Comparison Between SHO and Angular Momentum====

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 ^ 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|

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.










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**To go back to the lecture note list, click [[lec_notes]]**\\
**previous lecture note: [[lec_notes_1111]]**\\
**Main quiz 3 concepts: [[Quiz_3_1113]]**\\
**next lecture note: [[lec_notes_1118]]**\\