5. Consider the addition of orbital angular momentum (L) and spin angular momentum (S) for the case /=1, s=1/2. The eigenfunctions of L2, S², L₂ and Sz are products of the space Ym and spin X± functions (with /=1). Derive the eigenfunctions of the operators L²,S², J², Jz in terms of the first set of eigenfunctions, These functions are product functions of the form JmJ = YmX+ or linear combinations of these products, Hint start by applying the lowering operator to the state with maximum values of J and MJ. J₁|jm) = ħ√√j(j + 1) − m(m±1)|j(m±1)) J_ = L_ + S_ Repeated application of this method will give you four of the six eigenfunctions. The remaining two can be found by determining orthogonal eigenfunctions to those already found with mJ = 1/2 and --1/2. {similar form to the linear combinations for combining two spin 1/2 particles}.

5. Consider the addition of orbital angular momentum (L) and spin angular momentum (S) for the case /=1, s=1/2. The eigenfunctions of L2, S², L₂ and Sz are products of the space Ym and spin X± functions (with /=1). Derive the eigenfunctions of the operators L²,S², J², Jz in terms of the first set of eigenfunctions, These functions are product functions of the form JmJ = YmX+ or linear combinations of these products, Hint start by applying the lowering operator to the state with maximum values of J and MJ. J₁|jm) = ħ√√j(j + 1) − m(m±1)|j(m±1)) J_ = L_ + S_ Repeated application of this method will give you four of the six eigenfunctions. The remaining two can be found by determining orthogonal eigenfunctions to those already found with mJ = 1/2 and --1/2. {similar form to the linear combinations for combining two spin 1/2 particles}.