For any vector operator V one can define raising and lowering operators as (a) Using Equation 6 33, show that (b) Show that, if is an eigenstate of L 2 and L z with eigenvalues ( 1) 2 and respectively, then either V is zero or V is also an eigenstate of L 2 and L z with eigenvalues ( 1) ( 2) 2 and ...

a First If youre comfortable working with index notation the next part could be quicker bu ...

For any vector operator V one can define raising and lowering operators as (a) Using Equation 6.33,

Question:

For any vector operator V̂ one can define raising and lowering operators as

(a) Using Equation 6.33, show that

(b) Show that, if Ψ is an eigenstate of L̂² and L̂_z with eigenvalues ℓ(ℓ+1) ћ² and ℓћ respectively, then either V̂+Ψ is zero or V̂+Ψ is also an eigenstate of L̂² and L̂_z with eigenvalues (ℓ+1) (ℓ+2) ћ² and (ℓ+1) ћ respectively. This means that, acting on a state with maximal mℓ = ℓ, the operator V̂+ either “raises” both the ℓ and m values by 1 or destroys the state.