Consider a three-dimensional vector space spanned by an orthonormal basis ∣1⟩, ∣2⟩, ∣3⟩. Kets ∣α⟩ and ∣β⟩ are given by
∣α⟩=i∣1⟩−2∣2⟩−i∣3⟩,∣β⟩=i∣1⟩+2∣3⟩.
(a) Construct ⟨α∣ and ⟨β∣ (in terms of the dual basis ⟨1∣, ⟨2∣, ⟨3∣).
(b) Find ⟨α∣β⟩ and ⟨β∣α⟩, and confirm that ⟨β∣α⟩=⟨α∣β⟩∗.
(c) Find all nine matrix elements of the operator A^≡∣α⟩⟨β∣, in this basis, and construct the matrix A. Is it hermitian?