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


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>x2 + p2</math><math>{L_x}2 + {L_y}2(+L2_z)</math>
We tried to factorize H by <math>(x+ip)(x-ip)</math> <math>(L_x+iL_y)(L_x-iL_y) (+L2_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 (+L2_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.


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