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--- classes:2009:fall:phys4101.001:lec_notes_1125 [2009/11/29 10:59] – x500_razi0001
+++ classes:2009:fall:phys4101.001:lec_notes_1125 [2009/11/29 22:20] (current) – yk
@@ Line 67: / Line 67: @@
 Wrong! When you separate the particles based on their x-direction spin, you lose all your knowledge of the z-direction spin. In fact, you cause the z-direction spin to become indeterminate. This is because x-direction spin and z-direction spin are incompatible observables.
 ====Section 2: Addition of angular momenta====
 ===Part 1: Why do it? Why do we care about adding angular momenta?===
+Example: when looking at fine structure, <math>\vec{L}</math> and <math>\vec{S}</math> will be important and useful.
+In a magnetic field, the hamiltonian will have an extra term, so that
+<math>\hat{H} = \alpha \vec{L} \cdot \vec{S} + (\frac{p^2}{2m} + V)</math>
+, and <math>\alpha</math> can be determined from E&M, which can be seen in chapter 6.
+Notice that we are going to ignore the <math>(\frac{p^2}{2m} + V)</math> term of the hamiltonian since that part isn't changing from the magnetic field; we're not interested in that part of it right now.
+Anyway, if the magnetic field we apply is in the <math>\hat{z}</math> direction, then we have
+<math>\vec{B}=B_z</math> and <math>\hat{H}=-\alpha B_z S_z</math>
+In that case, which is associated with the Zeeman Effect, <math>S_z</math> is an eigenfunction of <math>\hat{H}</math>
+If the magnetic field is not on, then the <math>\vec{L} \cdot \vec{S}</math> becomes important, and
+<math>[\vec{L} \cdot \vec{S},S_z] \neq 0</math>
+So, we want to find eigenfunctions of <math>\vec{L} \cdot \vec{S}</math>, and in some situations, we may also want to find <math>\vec{S_1} \cdot \vec{S_2}</math> eigenfunctions.
+Well, if we let <math>\vec{J} = \vec{L}+\vec{S}</math>, we'll have the following...
+<math>[J^2,\vec{L} \cdot \vec{S}] = 0</math> and
+<math>[J_z,\vec{L} \cdot \vec{S}] = 0</math>,
+and then we can use <math>\vec{J}</math> to play with <math>\vec{L} \cdot \vec{S}</math> in a simple useful way that will come up in Chapter 6.
+In short, let's just trust that this addition of angular momentum technique we develop in this chapter will be very useful later, and move on to the technique itself.
+===Part 2: How to add angular momenta===
+The basic idea of the following section is to define <math>J^2</math> and then show how to find the eigenvalues of <math>J^2</math> for the basic spin-up and spin-down eigenfunctions. In the lecture, and in what follows, we just found the eigenvalues for the combined state <math>\uparrow\,_1\uparrow\,_2</math>. At the end, Yuichi just wrote out a matrix that included the other eigenvalues for the other combined spin states. They can be found in a similar manner to what is shown below. I hope my writeup makes some sense.
+First of all, we know that
+<math>\left|S_1\right|^2=\frac{1}{2}(\frac{1}{2}+1)\hbar^2 = \frac{3}{4}\hbar^2</math> and by the same method
+<math>\left|S_2\right|^2 = \frac{3}{4}\hbar^2</math>.
+For each of the two Hilbert spaces (one for S1 and one for S2) we have a pair of spin-up and spin-downs represented as follows:
+<math>\chi_1^+ = \begin{pmatrix} 1\\ 0\end{pmatrix}_1 \:\:</math>
+<math>\chi_1^- = \begin{pmatrix} 0\\ 1\end{pmatrix}_1</math> \\
+<math>\chi_2^+ = \begin{pmatrix} 1\\ 0\end{pmatrix}_2 \:\:</math>
+<math>\chi_2^- = \begin{pmatrix} 0\\ 1\end{pmatrix}_2</math> \\
+Remember these can also be represented with up and down arrows, which will be done below.\\
+We have to be careful with these since each pair is associated with a different Hilbert space. If we want to be able to represent combined states of S1 and S2, one way to do it is to use a column 4-vector and define it like this...\\
+<math>\uparrow\uparrow = \begin{pmatrix} 1\\ 0 \\0\\0\end{pmatrix} \:\:
+\uparrow\downarrow = \begin{pmatrix} 0\\ 1 \\0\\0\end{pmatrix} \:\:
+\downarrow\uparrow = \begin{pmatrix} 0\\ 0 \\1\\0\end{pmatrix} \:\:
+\downarrow\downarrow = \begin{pmatrix} 0\\ 0 \\0\\1\end{pmatrix}</math>
+Now, let's try adding <math>\vec{S_1}</math> and <math>\vec{S_2}</math> when they are both in the spin up state....\\
+<math>\vec{J}=\vec{S_1} + \vec{S_2}</math> so \\
+<math>J^2\uparrow\uparrow=(\vec{S_1} + \vec{S_2})^2\uparrow\uparrow = (S_1^2+S_2^2+2S_1 \cdot S_2)\uparrow\uparrow</math> (This last bit works because S1 and S2 commute.)
+Now, how do we find this sum? \\
+Well, we already know that \\
+<math>\vec{S_1}^2 \uparrow\;_1\uparrow\:_2 = (\frac{3}{4}\hbar^2\uparrow\;_1)\uparrow\;_2</math>  and, similarly,\\
+<math>\vec{S_2}^2 \uparrow\;_1\uparrow\:_2 = (\frac{3}{4}\hbar^2\uparrow\;_2)\uparrow\;_1</math>  \\
+but what about <math>2\vec{S_1}\cdot\vec{S_2}\uparrow\,_1\uparrow\,_2</math> ?? \\
+Well,\\
+<math>2\vec{S_1}\cdot\vec{S_2}\uparrow\,_1\uparrow\,_2 = 2(\vec{S_{1x}}\,\vec{S_{2x}}+\vec{S_{1y}}\,\vec{S_{2y}}+\vec{S_{1z}}\,\vec{S_{2z}})\uparrow\,_1\uparrow\,_2</math> \\
+Now, we know all of the spin matrices in the above expression, so we can just plug them all in and solve, and we will get \\
+<math>J^2\uparrow\uparrow = \frac{1}{2}\hbar^2\uparrow\uparrow</math>.\\
+//Yuichi// I mis-stated here.  It should have been:<math>2\vec{S_1}\cdot\vec{S_2}\uparrow\uparrow = \frac{1}{2}\hbar^2\uparrow\uparrow</math>.\\
+Using a similar technique, we can also find (with corrections)...
+<math>2\vec{S_1}\cdot\vec{S_2}\uparrow\downarrow = -\frac{1}{2}\hbar^2\uparrow\downarrow + \hbar^2\uparrow\downarrow \\
+\vec{S_1}\cdot\vec{S_2}\downarrow\uparrow = -\frac{1}{2}\hbar^2\downarrow\uparrow + \hbar^2\downarrow\uparrow \\
+\vec{S_1}\cdot\vec{S_2} \downarrow\downarrow = \frac{1}{2}\hbar^2\downarrow\downarrow \\</math>
+We can use our 4 element column vector notation described above to represent <math>J^2</math> using a single matrix, which is \\
+<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>
+//Yuichi// and this should have been <math>2\vec{S_1}\cdot\vec{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>
-Me or the other guy will finish the notes later; happy Thanksgiving!
+\\
+\\
+Happy Thanksgiving!
 ------------------------------------------
 **To go back to the lecture note list, click [[lec_notes]]**\\

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