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Quantum_Operators_Spin - Quantum operator methods p 1 SETUP...

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Quantum operator methods p. 1 SETUP: Consider a quantum particle with some new properties that we can measure. We won’t talk much about the physics of these measurements yet, but the formalism of quantum mechanics will teach us a great deal, just from operator methods! Consider a hermitian operator S x which yields only 2 possible measurement outcomes, +1 or -1. (It will represent the measurement of “the component of spin in the x direction”. For now think of it as a measurement of some property of a particle which can be “rightwards” or “leftwards”…) With only two possible measurement outcomes, S x has only two (orthonormal) eigenvectors. People get tired of writing out complicated “ket names”, so a simple, common notation is S x + = + , S x - = - 1 - I suppose you might refer to the |+> state as a “spin right” state, and |-> as “spin left”, does that seem reasonable to you? Stare at these two eigen-equations and make sure you, and your group, understand the notation: which symbols are the eigenvectors here, what are the eigenvalues, in those equations? What can you say about, e.g. - + ? (Explain) Now consider a second observable, with corresponding Hermitian operator, S z. This operator is NOT the same as S x, although it does have the same spectrum. Since the eigenvectors of S z are different from those of S x, we need to give them a different name. Here, people conventionally name the ket with a symbol : S z = , S z = - 1 (Since S z measures the vertical, or z, component of spin, I suppose you might refer to the | > state as a “spin up”, and | > as “spin down” particle, does that seem reasonable to you?)
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