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| classes:2009:fall:phys4101.001:lec_notes_1120 [2009/11/22 18:20] – x500_vinc0053 | classes:2009:fall:phys4101.001:lec_notes_1120 [2009/11/23 21:52] (current) – yk |
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| ===== Nov 20 (Fri) ===== | ===== Nov 20 (Fri) Spin operators, expectation values ===== |
| ** Responsible party: vinc0053, Green Suit ** | ** Responsible party: vinc0053, Green Suit ** |
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| <math>|"m"| =< l</math>\\ | <math>|"m"| =< l</math>\\ |
| S<sub>e,p,n,q,nu</sub> = 1/2 is an intrinsic property that doesn't change. You can just say S<sub>z</sub>,m<sub>s</sub>,m = +/- 1/2 <math>\hbar</math> <-(the <math>\hbar</math> is really supposed to be there, but it is usually left out of the notation).\\ | S<sub>e,p,n,q,nu</sub> = 1/2 is an intrinsic property that doesn't change. You can just say S<sub>z</sub>,m<sub>s</sub>,m = +/- 1/2 <math>\hbar</math> <-(the <math>\hbar</math> is really supposed to be there, but it is usually left out of the notation).\\ |
| S<sup>2</sup> = <math>\frac{1}{2}(\frac{1}{2}+1)\hbar</math><sup>2</sup>. Compare to the shallow well with only 2 states.\\ | S<sup>2</sup> = <math>\frac{1}{2}(\frac{1}{2}+1)\hbar^2</math>. Compare to the shallow well with only 2 states.\\ |
| <math>\psi = c_1\psi_1 + c_2\psi_2</math> => <math>\begin{pmatrix} c_1\\ c_2\end{pmatrix}</math>, then <math>\begin{pmatrix} 1\\ 0\end{pmatrix}</math> is <math>\psi_1</math> the ground state and <math>\begin{pmatrix} 0\\ 1\end{pmatrix}</math> is the first excited state <math>\psi_2</math>.\\ | <math>\psi = c_1\psi_1 + c_2\psi_2</math> => <math>\begin{pmatrix} c_1\\ c_2\end{pmatrix}</math>, then <math>\begin{pmatrix} 1\\ 0\end{pmatrix}</math> is <math>\psi_1</math> the ground state and <math>\begin{pmatrix} 0\\ 1\end{pmatrix}</math> is the first excited state <math>\psi_2</math>.\\ |
| When you have spin momentum it creates a magnetic dipole, <math>\psi_1</math> and <math>\psi_2</math> are degenerate states when without a magnetic field. Apply magnetic field and <math>\psi_1</math>, <math>\psi_2</math> take different energies.\\ | When you have spin momentum it creates a magnetic dipole, <math>\psi_1</math> and <math>\psi_2</math> are degenerate states when without a magnetic field. Apply magnetic field and <math>\psi_1</math>, <math>\psi_2</math> take different energies.\\ |
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| Q: How do we know they are degenerate without a magnetic field?\\ | Q: How do we know they are degenerate without a magnetic field?\\ |
| A: <math>\psi = \psi_nlm</math> s<sub>z</sub> +/- <math>\frac{1}{2}</math>. When it decays there's emission of photon with energy equal to the difference. The spin decay +/- <math>\frac{1}{2}</math> -> +/- <math>\frac{1}{2}</math> makes no difference of photon energy without a magnetic field, but with it, experiments started showing a difference. Two peaks on a plot of measurements (which collapse to one peak without a magnetic field).\\ | A: <math>\psi = \psi_{nlm} s_{z(=\pm\frac{1}{2}}</math>. When it decays there's emission of photon with energy equal to the difference. The spin decay <math>\pm\frac{1}{2}\rightarrow\pm\frac{1}{2}</math> makes no difference of photon energy without a magnetic field, but with it, experiments started showing a difference. Two peaks on a plot of measurements (which collapse to one peak without a magnetic field).\\ |
| We could use convention <math>\chi_+</math><sup>x</sup> = <math>\begin{pmatrix} 1\\ 0\end{pmatrix}</math> <math>\chi_-</math><sup>x</sup> = <math>\begin{pmatrix} 0\\ 1\end{pmatrix}</math> for spin up and spin down, but its confusing. If the z-component is discrete, not continuous, then it snaps into allowed spots (up or down). There can only be an even number of allowed spots, why is that?\\ | We could use convention <math>\chi_+^x</math> = <math>\begin{pmatrix} 1\\ 0\end{pmatrix}</math> <math>\chi_-^x</math> = <math>\begin{pmatrix} 0\\ 1\end{pmatrix}</math> for spin up and spin down along the //x// direction, but it's confusing. |
| S<sup>2</sup>,s<sub>z</sub>,s<sub>x</sub>,s<sub>y</sub> (sometimes the x and y components) like a microwave has a magnetic field that lines up hydrogen in H2O, then microwaves flip the spin over.\\ | |
| S<sup>2</sup> = <math>\frac{3}{4}\hbar</math><sup>2</sup><math>\begin{pmatrix} 1 0\\ 0 -1\end{pmatrix}</math> => <math>\frac{3}{4}\hbar</math><sup>2</sup><math>\begin{pmatrix} 1 0\\ 0 -1\end{pmatrix}\begin{pmatrix} 1\\ 0\end{pmatrix} = \frac{3}{4}\hbar</math><sup>2</sup><math>\begin{pmatrix} 1\\ 0\end{pmatrix}</math>\\ | |
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| | Also as experimental evidence for existence of spins, (in the S-G experiment,) if the z-component is discrete, not continuous, then it snaps into allowed spots (up or down). There can only be an odd <del>even</del> number of allowed spots, why is that?\\ |
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| | Now that we know how to represent quantum states using vectors, what else would we wisk to be able to reprent so that various calculations can be done? -> operator in terms of matrix such as S<sup>2</sup>, S<sub>z</sub>, S<sub>x</sub>, S<sub>y</sub> (sometimes the x and y components play an important role) like a microwave in MRI has a magnetic field that lines up perpendicular to the large DC B field (which is usually defined to be the //z// direction) <del>hydrogen in H2O</del>, then microwaves flip the spin over.\\ |
| | S<sup>2</sup> has eigenvalues, <math>\frac{3}{4}\hbar^2</math> for both spin up and down states. If it is represented by a matrix, <math>\frac{3}{4}\hbar^2\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}</math> then <math>\frac{3}{4}\hbar^2\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}\begin{pmatrix} 1\\ 0\end{pmatrix} = \frac{3}{4}\hbar^2\begin{pmatrix} 1\\ 0\end{pmatrix}</math>, which satisfy our desire (eigenvalue for at least <math>\begin{pmatrix} 1\\ 0\end{pmatrix}</math> vector is what we want. The same can be shown for <math>\begin{pmatrix} 0\\ 1\end{pmatrix}</math> vector, too.\\ |
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| | Similarly, <math>S_z=\frac{1}{2}\hbar\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}</math> will satisfy our desire that it is eigenvalues <math>s_z=\pm\frac{1}{2}\hbar</math> for vectors representing up/down spin states. |
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| **To go back to the lecture note list, click [[lec_notes]]**\\ | **To go back to the lecture note list, click [[lec_notes]]**\\ |