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# ps4 - Physics 212 Spring 2010 Problem Set 4 1 C-TDL Ch X...

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Physics 212 Spring 2010 Problem Set 4 1. C-TDL Ch. X problem 1 2. C-TDL Ch. X problem 3 3. Show, using a plane wave as your original eigenstate, that the momentum operator “generates” a translation in space. Prove that a unitary operator U may be written in terms of a Hermitian operator G , where U = exp( iθG ), where θ is arbitrary. 4. “Adding” l = 1 with s = 1 / 2: Use raising and lowering operators as well as orthogonality to express the combined basis states in terms of the individual basis states | l = 1 , m l , s = 1 / 2 , m s i . 5. Shankar 12.5.7 6. Sakurai Ch. 3 Problem 8 (Sakurai is on reserve in the library if you want to see the problems in their original form. If that is inconvenient, they are reproduced here): Consider a sequence of Euler rotations represented by D 1 / 2 ( α, β, γ ) = exp ( - 3 α 2 ) exp ( - 2 β 2 ) exp ( - 3 γ 2 ) . [The (1 / 2) superscript on D means that we are in a spin 1/2 system, so the matrices are the Pauli matrices.] Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single rotation about some axis by an angle θ . Find θ .
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