852_2010hw4 - PHYS852 Quantum Mechanics II Spring 2010...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 4 Topics covered: rotation with spin, exchange symmmetry 1. A vector pointing in the θ,φ direction, can be formed by starting with a vector pointing along vectore z , then applying an active rotation by θ about the y-axis, followed by a rotation by φ about the z-axis. (a) Verify this for an ordinary vector, by starting with the vector (0 , , 1) T and using R y ( θ ) = cos θ 0 sin θ 1 − sin θ 0 cos θ ; R z ( φ ) = cos φ − sin φ sin φ cos φ 1 (1) (b) Thus for a spin-1/2 system, the spin-up state with respect to the θ,φ direction can be found in the basis of S z eigenstates, by starting with the spin-up state along vector e z , and applying unitary rotation operators, i.e. | ↑ θφ ) = e- i planckover2pi1 φS z e- i planckover2pi1 θS y | ↑ z ) . (2) In this way, find the states | ↑ θφ ) and | ↓ θφ ) ....
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This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.

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852_2010hw4 - PHYS852 Quantum Mechanics II Spring 2010...

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