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--- classes:2009:fall:phys4101.001:lec_notes_1130 [2009/12/06 16:34] – ely
+++ classes:2009:fall:phys4101.001:lec_notes_1130 [2009/12/06 16:54] (current) – ely
@@ Line 19: / Line 19: @@
 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>
+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
  ((Please try to include the following

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