Question

# Given a basis of angular momentum eigenstates | l, m >, where m is the component of angular momentum along the

z-axis:

(a) Show that the matrix elements < l', m' | L± | l, m> are non-zero only if l' = l and m' = m ± 1. (L± is the raising and lowering angular momentum operators)

(b) Using the relation L±L∓ = L^{2} − L_{z}^{2} ± (h_{bar})L_{z}, show that < l, m ± 1| L± | l, m> = h_{bar}[l(l + 1) − m(m ± 1)]^{1/2}

(c) Using the relation L± = Lx±iLy, derive the matrix representations for L_{x} = S_{x}, L_{y} = S_{y}, and L_{z} = S_{z} for a spin 1/2 system. Combine these operators together to get a matrix representation for L^{2} = S^{2 }.

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