Spin-½ eigenvector along n^\hat{n} (Pauli formalism)

physics

Problem

Construct ∣S⋅n^;+⟩|\mathbf{S}\cdot\hat{n};+\rangle such that (S⋅n^)∣S⋅n^;+⟩=ℏ2∣S⋅n^;+⟩(\mathbf{S}\cdot\hat{n})|\mathbf{S}\cdot\hat{n};+\rangle = \tfrac{\hbar}{2}|\mathbf{S}\cdot\hat{n};+\rangle, where n^=sin⁡βcos⁡α x^+sin⁡βsin⁡α y^+cos⁡β z^\hat{n} = \sin\beta\cos\alpha\,\hat{x} + \sin\beta\sin\alpha\,\hat{y} + \cos\beta\,\hat{z} (β\beta is the polar angle from z^\hat{z}, α\alpha the azimuthal angle). Express your answer as a linear combination of ∣+⟩|+\rangle and ∣−⟩|-\rangle. Treat it as a straightforward eigenvalue problem; do not use rotation operators.

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