Word Count : 386

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We are tasked with verifying the spin operators \(\hat{S}_x\) and \(\hat{S}_y\) for an electron. The general spin operators for a spin-\(\frac{1}{2}\) particle (like the electron) are given as: \[ \hat{S}_x = \frac{\hbar}{2} \left( |\uparrow\rangle \langle \downarrow| + |\downarrow\rangle \langle \uparrow| \right) \] \[ \hat{S}_y = \frac{\hbar}{2i} \left( |\uparrow\rangle \langle \downarrow| - |\downarrow\rangle \langle \uparrow| \right) \] Let's express these in terms of matrix representations and verify their properties. ### Step 1: Basis Vectors For a spin-\(\frac{1}{2}\) system, ______ ______ ____ _________ _____ _____ ____.
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