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exam 1 still

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solutions/exam1/prob4
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Two operators' matrix representations are known in the |1〉, |2〉, |3〉 basis, where and b are real numbers: Two operators' matrix representations are known in the |1〉, |2〉, |3〉 basis, where a and b are real numbers, and ι is the imaginary unit:
 ≐ B̂ ≐  ≐ B̂ ≐
⎛ a 0 0 ⎞ ⎛ b 0 0 ⎞ ⎛ a 0 0 ⎞ ⎛ b 0 0 ⎞
@ -38,14 +38,24 @@ Since the operators commute, they share a set of common eigenstates. The eigenst
|a₁〉 ≐ ⎛1⎞ |a₂〉 ≐ ⎛0⎞ and |a₃〉 ≐ ⎛0⎞ |a₁〉 ≐ ⎛1⎞ |a₂〉 ≐ ⎛0⎞ and |a₃〉 ≐ ⎛0⎞
⎜0⎟ ⎜1⎟ ⎜0⎟ ⎜0⎟ ⎜1⎟ ⎜0⎟
⎝0⎠, ⎝0⎠, ⎝1⎠. ⎝0⎠; ⎝0⎠; ⎝1⎠.
The eigenvalues are already known (λ = b,b,-b.), and using the eigenvalue equations, the eigenstates can be determined. The eigenvalue equation Eigenstates of B̂ must be obtained, and if they are a linear combination of the eigenstates of Â, they will serve as the complete set of eigenstates shared by  and B̂. If not, some subset of those will be the shared basis, and this will need to be determined.
⎛ b 0 0 ⎞ ⎛ α₁ ⎞ ⎛ b α₁ ⎞ ⎛ α₁ ⎞ For B̂, the eigenvalues are already known (λ = b,b,-b.), and using the eigenvalue equations, the eigenstates can be determined. The eigenvalue equation
⎜ 0 0 -ιb ⎟ ⎜ β₁ ⎟ = ⎜ -ι b γ₁ ⎟ = b ⎜ β₁ ⎟
⎝ 0 ιb 0 ⎠ ⎝ γ₁ ⎠ ⎝ ι b β₁ ⎠ ⎝ γ₁ ⎠
reveals -ι γ₁ = β₁, which, combined with the normalization condition, will allow the determination of two eigenstates of the B̂. ⎛ b 0 0 ⎞ ⎛ α ⎞ ⎛ b α ⎞ ⎛ α ⎞
⎜ 0 0 -ιb ⎟ ⎜ β ⎟ = ⎜ -ι b γ ⎟ = b ⎜ β ⎟
⎝ 0 ιb 0 ⎠ ⎝ γ ⎠ ⎝ ι b β ⎠ ⎝ γ ⎠
reveals -ι γ = β, which, combined with the normalization condition, will allow the determination of two eigenstates of the B̂. One eigenstate is obvious from inspection,
|b₁〉 ≐ ⎛1⎞
⎜0⎟
⎝0⎠.
If β = 1 is chosen, then γ = -ι, and if β = ι, then γ = -1. The two additional eigenstates of B̂ are therefore, after normalizing,
|b₂〉 ≐ 1 ⎛ 0 ⎞ |b₃〉 ≐ 1 ⎛ 0 ⎞
√2 ⎜ 1 ⎟ √2 ⎜ ι ⎟
⎝-ι ⎠ and ⎝ -1 ⎠.