7. (15 points) Consider g), an eigenstate of L2 and L with eigenvalues labeled by m...

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7. (15 points) Consider g), an eigenstate of L2 and L with eigenvalues labeled by m and I in the usual way. Show L+9") is also an eigenvector of those operators, where L+ = L+iLy is the angular momentum ladder operator. Find the new eigenvalues of L2 and L for (L+9)). Remember that commutators are lincar [A+ aB,C] = [A,C]+a[B,C] a R. s Lincar vector space, C: is closed under some addition and scalar multiplication a(|A) + |B)) = a|A) +a|B) (a+B)|A) = a|A) + B|A) Both operations are commutative and associative There is a zero element |4) + |0) = |A) V|A) EL There is a unique inverse VA) EC that is |A) +|- A) = |0) N A basis can express any vector in the space VA) EC, |A) = Akek) (A|B) C (A|B) = (B|A)* Inner Product: (AA)>0 with equality |A) = |0) (V (a|A)+B|B)) = a(V|A) + B(V|B) 7. (15 points) Consider g), an eigenstate of L2 and L with eigenvalues labeled by m and I in the usual way. Show L+9") is also an eigenvector of those operators, where L+ = L+iLy is the angular momentum ladder operator. Find the new eigenvalues of L2 and L for (L+9)). Remember that commutators are lincar [A+ aB,C] = [A,C]+a[B,C] a R. s Lincar vector space, C: is closed under some addition and scalar multiplication a(|A) + |B)) = a|A) +a|B) (a+B)|A) = a|A) + B|A) Both operations are commutative and associative There is a zero element |4) + |0) = |A) V|A) EL There is a unique inverse VA) EC that is |A) +|- A) = |0) N A basis can express any vector in the space VA) EC, |A) = Akek) (A|B) C (A|B) = (B|A)* Inner Product: (AA)>0 with equality |A) = |0) (V (a|A)+B|B)) = a(V|A) + B(V|B)