classes:2009:fall:phys4101.001:lec_notes

Nov 16 (Mon) Angular momentum with raising/lowing operators

Responsible party: Pluto 4ever, malmx026

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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.

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.

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