For a spin- particle

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Hilbert space and Dirac notation

Problem

For a spin-1/21/2 particle, let ∣↑⟩,∣↓⟩|{\uparrow}\rangle,|{\downarrow}\rangle be the standard zz-basis, and define ∣+x⟩=(∣↑⟩+∣↓⟩)/2|{+x}\rangle = (|{\uparrow}\rangle+|{\downarrow}\rangle)/\sqrt2, ∣−x⟩=(∣↑⟩−∣↓⟩)/2|{-x}\rangle = (|{\uparrow}\rangle-|{\downarrow}\rangle)/\sqrt2. (a) Verify that {∣+x⟩,∣−x⟩}\{|{+x}\rangle,|{-x}\rangle\} is an orthonormal basis. (b) The state ∣ψ⟩=(∣↑⟩+i∣↓⟩)/2|\psi\rangle = (|{\uparrow}\rangle+i|{\downarrow}\rangle)/\sqrt2 is given (normalised). Find c+x=⟨+x∣ψ⟩c_{+x}=\langle{+x}|\psi\rangle and c−x=⟨−x∣ψ⟩c_{-x}=\langle{-x}|\psi\rangle. (c) Check that ∣c+x∣2+∣c−x∣2=1|c_{+x}|^2+|c_{-x}|^2=1.

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