Question

1. More on Spin-1/2 system: (10 points) The rising and lowering operators for a spin-1/2 system are defined as: S+ S + iSy and S S iSy, respectively. They satisfy the following properties: Š+㈩-0, Š+|-)-치+), s-I+) = 최-), s-I-》 = 0, where lt) are the usual eigenstates of the S, operator. a) Invert the definitions of S+ and ś, to express Sa and Šy in terms of St and S. b) Find the matrix representations of Š+ and Š in the usual basis |+) c) Derive the matrix representations of S and Sy in the usual basis |t). d) Check that the spin matrices obey the fundamental commutation relation for angular mo e) Write the 2x2 matrix representations of Sa, Sy, and S in the basis of eigenstates of Sy.

Answer 1