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# QM9 - 9 SPIN-1/2 PARTICLES 1 Spinors Eigenvalues and...

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9. SPIN-1/2 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. Stern-Gerlach Experiments (not yet written) Problems The simplest system with nonzero angular momentum is a spin-1/2 particle. Quarks, the building blocks of protons, neutrons, and the other baryons, as well as of the mesons, are spin-1/2 particles. Protons and neutrons, the building blocks of nuclei, are spin-1/2 particles. Electrons, which with nuclei are the building blocks of atoms, are spin-1/2 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 charge—the 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 spin-1/2 particle. Not all particles have spin 1/2. The “carriers” of the fundamental forces— the photon, gluons, W and Z bosons, and graviton—have spin 1 or (the graviton) spin 2. Particles made of an even number of spin-1/2 particles (the mesons are such) have integer spins. And particles—nuclei and atoms, for example—can 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 two-component vectors and 2-by-2 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 time-dependent field, the spin can be flipped up and down; in an inhomogeneous field, the spin-up and -down states can be separated into two beams. Such arrangements are the basis for many scientific experiments and for magnetic-resonance imaging. c 2011, Charles G. Wohl, work in progress. 1

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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 angular-momentum operators ˆ s 2 and ˆ s z : | s, m = | 1 / 2 , +1 / 2 and | 1 / 2 , 1 / 2 . These states are represented in the literature in a number of short-hand ways, among them | + z = | + = | α = | 1 = and |− z = |−⟩ = | β = | 2 = . Here we shall usually use | + z and |− z in order later to distinguish them from |− x , | + y , and other such states. The | + z and |− z states are orthogonal because they are eigenstates of ˆ s z with different eigenvalues, and we shall of course take them to be normalized. We
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QM9 - 9 SPIN-1/2 PARTICLES 1 Spinors Eigenvalues and...

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