Responsible party: Anaximenes, Dark Helmet

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This lecture turned into mostly review. I could use some help remembering specifically what we talked about. I've put in the major topics below, and I'm working on adding the details. If anyone can add something I've forgotten in the mean-time, that'd be great.

Major topics:

• Brief discussion of Stern-Gerlach device
• vector representation
• First steps in adding angular momentum
• including matrix representation
• The last quiz will be on 12/11 instead of 12/04
• continued the addition of angular momentum

As incorrectly stated in previous lecture:

<math>J^2 = \frac{1}{2}\hbar^2 \begin{pmatrix} 1 & 0 & 0 & 0
0 & -1 & 2 & 0
0 & 2 & -1 & 0
0 & 0 & 0 & 1 \end{pmatrix} </math>

What this should have said was

<math>2(S_1)(S_2)= \frac{1}{2}\hbar^2 \begin{pmatrix} 1 & 0 & 0 & 0
0 & -1 & 2 & 0
0 & 2 & -1 & 0
0 & 0 & 0 & 1 \end{pmatrix} </math>

<math>J^2=(S_1+S_2)^2=(S_1)^2+(S_2)^2+2(S_1)(S_2)</math>

So we missed the <math>(S_1)^2</math> and <math>(S_2)^2</math> terms which bring:

<math>\frac{1}{2}\hbar^2 \begin{pmatrix} 1 & 0 & 0 & 0
0 & 1 & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1 \end{pmatrix} </math>

which gives us the <math>J^2</math> matrix:

<math>J^2 = \frac{1}{2}\hbar^2 \begin{pmatrix} 4 & 0 & 0 & 0
0 & 2 & 3 & 0
0 & 3 & 2 & 0
0 & 0 & 0 & 4 \end{pmatrix} </math>

We then found some of the Eigenvalues and Eigenstates of <math>2(S_1)(S_2)</math>

<math> \begin{pmatrix} 1-{\lambda} & 0 & 0 & 0
0 & -1-{\lambda} & 2 & 0
0 & 2 & -1-{\lambda} & 0
0 & 0 & 0 & 1-{\lambda} \end{pmatrix} </math>

more will be coming when i figure out how to make arrows

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