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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.
  

The original plan for today's lecture was to discuss the following things:
1) Clebsch-Gordon coefficients
2) Taking another look at the spherical harmonics (aka <math>Y_\ell^m</math>)
3) z-particle systems (I don't quite know what that means,sorry)

However, we didn't get to any of these items. Instead the lecture covered two main areas:
First, we went over some details about Discussion Problem #13,
Second, we talked about addition of angular momenta (aka <math>\vec{J}=\vec{L_1}+\vec{L_2}</math>)

Yuichi asked for student input on the discussion problem, since the TA's said a lot of students found it challenging. Class comments ranged from 'it was pretty straightforward' to 'I have no idea how to build the H matrix'.

Well, this problem involves looking at finding a matrix corresponding to the operator <math>\vec{S}\cdot\vec{L}</math>. First off, we know that <math>\vec{S}\cdot\vec{L} = {S_x}{L_x}+{S_y}{L_y}+{S_z}{L_z}</math>

To work with this more easily, let's recall that (as on p.174), we have
<math>S_x =\frac{1}{2}(S_+ + S_-) \qquad\qquad S_y = \frac{1}{2i}(S_+ - S_-)</math> , and similarly,
<math>L_x =\frac{1}{2}(L_+ + L_-) \qquad\qquad L_y = \frac{1}{2i}(L_+ - L_-) </math>

Using these expressions, we can rewrite <math>\vec{S}\cdot\vec{L}</math> in terms of just the + and - operators and the z direction operators. This is useful because we know exactly how these operators affect the vector they act on. They pull out certain constants, and the + and - operators also change the vector. (The constants for the + and - operators are given by eq 4.136 on p.172).

Now, you can use this information to find the Hamiltonian by operating on each of our 6 orthogonal basis vectors. Each one we operate on will give us one row of the Hamiltonian (and one column too because the hermitian H matrix is symmetric about its diagonal).

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