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--- classes:2009:fall:phys4101.001:lec_notes_1116 [2009/11/17 17:34] – kuehler
+++ classes:2009:fall:phys4101.001:lec_notes_1116 [2009/11/19 11:59] (current) – yk
@@ Line 1: / Line 1: @@
-===== Nov 16 (Mon)  =====
+===== Nov 16 (Mon) Angular momentum with raising/lowing operators =====
 ** Responsible party: Pluto 4ever, malmx026  **
@@ Line 20: / Line 20: @@
 ====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 ocsillator (SHO).
-===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).
-<math>[x,p]=i{\hbar}</math>
+^ ^ 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|
-no top rung
+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.
-<math>a_\pm ={ \frac{1}{sqrt{2m{\hbar}{\omega}}}(\mp{ip}+m{\omega}x)</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>

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