4604_ProblemSet6_fa07 - PHY4604 Fall 2007 Problem Set 6 PHY...

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PHY4604 Fall 2007 Problem Set 6 Department of Physics Page 1 of 2 PHY 4604 Problem Set #6 Due Wednesday November 14, 2007 (in class) (Total Points = 130, Late homework = 50%) Reading: Griffiths Chapter 4 Sections 4.4. Problem 1 (20 points): The Pauli spin matrices are given by = 0 1 1 0 x σ = 0 0 i i y σ = 1 0 0 1 z σ (a) (10 points) Show that σ i = σ i , det( σ i ) = -1, Tr( σ i ) = 0, [ σ i , σ j ] = 2i ε ijk σ k , and { σ i , σ j } = 2 δ ij . Note that [A, B] = AB – BA and {A, B} = AB +BA. (b) (10 points) Show that + = l l ijl ij j i i σ ε δ σ σ . ( Hint : see Griffiths section 4.3 and problem 4.26) Problem 2 (20 points): An electron is in the spin state >= 4 3 | i A χ . (a) (4 points) Determine the normalization constant A. (b) (8 points) Find the expectation value of S x , S y , and S z for the state | χ >, where σ r r h 2 = S . (c) (8 points) Find the “uncertainties” Δ S x , Δ S y , and Δ S z for this state.
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