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--- classes:2009:fall:phys4101.001:lec_notes_1118 [2009/11/16 00:03] – created yk
+++ classes:2009:fall:phys4101.001:lec_notes_1118 [2009/11/19 11:57] (current) – yk
@@ Line 1: / Line 1: @@
-===== Nov 18 (Wed)  =====
+===== Nov 18 (Wed) closure for L, Starting Spins =====
 ** Responsible party: Ekrpat, chap0326 **
@@ Line 8: / Line 8: @@
 **Main class wiki page: [[home]]**
-Please try to include the following
+==Topics covered in lecture:==
-  * main points understood, and expand them - what is your understanding of what the points were.
+* Making sense of the Del Operator
-    * 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.\\
-\\
+* Beginning spin
+===Making sense of the Del Operator===
+When shown that
+<math>\mathbf{\vec{ \nabla}} = \mathbf{\hat r} \partial r + \mathbf{\hat \theta} {\partial \theta \over r} + \mathbf{\hat \phi}{\partial \phi \over r\sin(\theta)} </math>
+we were asked "does this make sense?" (DTMS)
+We then recalled that <math>\mathbf{\vec{ \nabla}}</math> comes from
+<math>\mathbf{\vec{ \nabla}} = \mathbf{\hat x}\partial x + \mathbf{\hat y}\partial y + \mathbf{\hat z}\partial z</math>
+and that you could use a brute force type method to get the solution we are looking for from that. Also explained today was another method that involved introducing another Cartesian system: x',y',z', where
+<math>\mathbf{\vec{ \nabla}} = \mathbf{\hat x'}\partial x' + \mathbf{\hat y'}\partial y' + \mathbf{\hat z'}\partial z'</math>,
+and rotating the coordinates such that it lined up with the r-vector, <math>\theta</math> and <math>\phi</math>. Then it is possible to make a direct transformation from the [x',y',z'] coordinates to the [<math>\mathbf{\hat r}, \theta, \phi</math>] coordinates.
+We find that <math>\partial x' = \partial r</math>, <math>\partial y' = {\partial \theta \over r}</math>, and <math>\partial z' = {\partial \phi \over r\sin(\theta)}</math>
+Thus in a more elegant, though perhaps not as readily apparent way, we arrive at the desired expression:
+<math>\mathbf{\vec{ \nabla}} = \mathbf{\hat r} \partial r + \mathbf{\hat \theta} {\partial \theta \over r} + \mathbf{\hat \phi}{\partial \phi \over r\sin(\theta)} </math>
+===Starting Spin===
+We first made the note that we started with a quantum state represented by <math>\psi_n</math> which implies that it is a function of x, <math>\psi_n(x)</math>. Now we use a more abstract representation |<math>\psi_n</math>>, which doesn't necessarily imply that <math>\psi_n</math> is a function of x.
+Then from this notation, |<math>\psi_n</math>>, we can go to a matrix vector notation:
+<math>\psi_n</math> ~ |<math>\psi_n</math>> ~ <math>\Sigma c_n \psi_n</math> ~ <math>$\begin{pmatrix} c_1\\ c_2\\ c_3\\ etc. \end{pmatrix}$</math>
+ In spin we have something analogous:
+~|<math>\chi</math>> ~ <math>c_+\chi_+ + c_-\chi_-</math> which goes to <math>$\begin{pmatrix} c_+\\ c_-\end{pmatrix}$</math>
+We paused here to understand what characterizes <math>\chi_+</math>.
+When you operate with the z-component of the angular momontum, <math>L_z</math>, you get:
+<math>L_z \chi_+ = \frac{\hbar}{2} \chi_+ </math>
+This says to us that <math>\chi_+</math> is an eigenvector of  the <math>L_z</math> operator with an eigenvalue of <math>\frac{\hbar}{2}</math>
+Then we said that
+<math>\chi_+</math> ~ <math>|s, s_z></math> = <math>| \frac{1}{2}, \frac{1}{2}></math>
+We also made note that the book also calls |<math>s, s_z</math>>:
+<math>|s, s_z></math> = <math>|s, m></math> = <math>|s, m_s></math>
+and wondered why we use all these different notations. To explain we recalled from angular momentum that
+<math>{Y_l}^m </math> ~ <math>{F_l}^m</math> ~<math> |l, m></math>
+If you apply <math>L^2</math> on it
+<math>L^2 |l, m> = l(l+1){\hbar}^2 |l,m></math>
+<math>L_z |l, m> = m{\hbar} |l,m></math>
+<math>L^2 \chi_+ = s(s+1){\hbar}^2 \chi_+</math>
+we can begin to see how it makes sense. The 'm' is related to the magnetic quantum number, <math>m_s</math> is to help distinguish between the spin part of the magnetic. <math>s_z</math> indicates we are talking about the z-component.
+Yuichi says to take a good look at section 4.4 to find what does not make sense to us, so that we know what questions to ask next lecture.

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