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# HW8 - Quantum Mechanics I Physics 3143 Assignment 8 Fall...

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Quantum Mechanics I, Physics 3143: Assignment 8, Fall 2010 1 . (a) The spin-1/2 operators act on a vector space C 2 of state vectors. Find the normalized eigenvectors and eigenvalues of the “Pauli matrices,” σ x = 0 1 1 0 , σ y = 0 i i 0 and σ z = 1 0 0 1 . Clearly, these eigenvectors are also eigenvectors of the spin-1/2 angular momentum operators S x = planckover2pi1 σ x / 2, S y = planckover2pi1 σ y / 2 and S z = planckover2pi1 σ z / 2 , corresponding to the eigenvalues ± planckover2pi1 / 2 in each case. Let the eigenvectors be labelled, χ ( x ) ± , χ ( y ) ± and χ ( z ) ± , respectively. (b) The system is prepared in the state with normalized vector ψ = a b , where | a | 2 + | b | 2 = 1. Suppose a measurement of the observable S y is made. Calculate the probabilities, |( χ ( y ) ± | ψ )| 2 , to find the eigenvalues corresponding to spin up or down along the y-axis, respectively. 2. Continued from end of last problem...

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HW8 - Quantum Mechanics I Physics 3143 Assignment 8 Fall...

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