852HW2 - PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 2 1. Consider the most general normalized spin-1/2 state: | ) = c + | + ) + c- |) , where S z |) = planckover2pi1 2 |) . a.) Compute ( S x ) , ( S y ) and ( S z ) . b.) Compute the variances S x , S y , and S z . c.) Prove that S x = planckover2pi1 2 | c 2 + c 2- | , S y = planckover2pi1 2 | c 2 + + c 2- | and S z = planckover2pi1 | c + || c- | . Hint: Use the fact that ( | c + | 2 + | c- | 2 ) 2 = 1 2 = 1. 2. The unitary rotation operator for spin Hilbert space is U R ( vector ) = e- i planckover2pi1 vector vector S . The component of spin along the direction given by , in spherical polar coordinates is found by taking S z and rotating first by about the y-axis, then by about the z-axis, giving S = U R ( vectore z ) U R ( vectore y ) S z U R ( vectore y ) U R ( vectore z )....
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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