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(a) The angular momentum operators are defined as usual by Î = ✰ × p. They satisfy the relation [Lx, Ly] =iL₂ and the cyclic permutations in x, y, z of this relation. Define Î+ = Îxtily. Check the following relations: [2+, |Ĺ|²] = 0, as well as L+L₂=(L₂¬Î) L+₂ [2+,L_] = 2L₂, L† = L_, (10) εΗ = |Î|² − ο(ο +Î), Î-Î+ = |Î|² − ο(Îz – Î) . (11) (b) Argue that we can find joint eigenvectors for Îz and β. Suppose Y₁μ is such a vector with unit norm, where is the eigenvalue of L² and u that of L₂. (c) Show that if L+V₁, does not vanish, then it must be an eigenvector for 1₂ with eigen- value μ ± 1 and an eigenvector of L2 with eigenvalue 2. (d) Using the relations from a) or otherwise, show that |||²+₁₁μ||²=2-µ(µ±1). (12) Use this to argue that λ2 ≥ μ(μ+1), and give a criterion in terms of μ, λ when L+Y₁μ vanishes. (e) What are the possible values for 2 and μ?