Advanced Quantum Theory Amath 473, 673 / Phys 454 Problem Set 3 This assignment is due at the beginning of class on Friday, Oct. 29, 2010. PLEASE NOTE: Late assignments will not be accepted due to the fact that the solutions to this assignment will (hopefully) be posted online shortly after class on the due date. 1. Functions of Operators. a) Show that exp(-iθσ i ) = 1 cos( θ )-iσ i sin( θ ) holds for any Pauli operator σ i . (Hint: Match the series expansions for the relevant functions and use elementary properties of the Pauli operators to simplify the expressions.) b) Show that exp( iθσ x / 2) σ z exp(-iθσ x / 2) = cos( θ ) σ z + sin( θ ) σ y . 2. Rotation Operators. Consider a spin-1/2 particle initially in the m = +1 / 2 eigenstate of S z = ¯ hσ z / 2 , i.e., | ψ (0) i = | 1 / 2 , +1 / 2 i , and then subject to the dynamical transformation U ( t ) = exp(-iωtS x ) . a) Derive an explicit expression for the spin’s state after time
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