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Unformatted text preview: 9. SPIN1/2 PARTICLES 1. Spinors. Eigenvalues and Eigenstates 2. The Polarization Vector 3. Magnetic Moments and Magnetic Fields 4. Time Dependence. Precessing the Polarization 5. Magnetic Resonance. Flipping the Polarization 6. SternGerlach Experiments (not yet written) Problems The simplest system with nonzero angular momentum is a spin1/2 particle. Quarks, the building blocks of protons, neutrons, and the other baryons, as well as of the mesons, are spin1/2 particles. Protons and neutrons, the building blocks of nuclei, are spin1/2 particles. Electrons, which with nuclei are the building blocks of atoms, are spin1/2 particles. And so are muons and neutrinos. Thus a particle with spin 1/2 is an important system. The spin of a particle is as intrinsic to it, and as invariable, as is its mass and chargethe electron always has spin 1/2. A particle will also have other properties such as momentum and orbital angular momentum, but here we are only concerned with the spin of a spin1/2 particle. Not all particles have spin 1/2. The carriers of the fundamental forces the photon, gluons, W and Z bosons, and gravitonhave spin 1 or (the graviton) spin 2. Particles made of an even number of spin1/2 particles (the mesons are such) have integer spins. And particlesnuclei and atoms, for examplecan have higher angular momenta. In the ground states of carbon, nitrogen, oxygen, iron, silver, and platinum, the six, seven, eight, 26, 47, and 78 electrons conspire to have total angular momenta of 0, 3/2, 2, 4, 1/2, and 3. We start with the twocomponent vectors and 2by2 operators for s=1/2. We learn how to express any spin state in terms of the up and down states with respect to any direction, and how to associate a polarization vector with the spin. An angular momentum usually has an associated magnetic moment, and magnetic moments interact with magnetic fields. In a static magnetic field, the spin precesses; in a properly arranged timedependent field, the spin can be flipped up and down; in an inhomogeneous field, the spinup and down states can be separated into two beams. Such arrangements are the basis for many scientific experiments and for magneticresonance imaging. c 2011, Charles G. Wohl, work in progress. 1 9 1. SPINORS. EIGENVALUES AND EIGENSTATES States as vectors When s = 1 / 2 (we shall use s , for spin, instead of j here), there are just two states that are simultaneously eigenstates of the angularmomentum operators s 2 and s z :  s,m =  1 / 2 , +1 / 2 and  1 / 2 , 1 / 2 . These states are represented in the literature in a number of shorthand ways, among them  + z =  + =  =  1 = and  z =  =  =  2 = ....
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This note was uploaded on 01/28/2011 for the course PHY 137b taught by Professor Lee during the Spring '09 term at University of California, Berkeley.
 Spring '09
 Lee
 mechanics, Polarization

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