For a spin- particle with -basis

PHYSICS · Multi-part · 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 with zz-basis ∣↑⟩,∣↓⟩|{\uparrow}\rangle,|{\downarrow}\rangle, the spin operator S^x\hat S_x has matrix elements ⟨↑∣S^x∣↑⟩=0\langle{\uparrow}|\hat S_x|{\uparrow}\rangle=0, ⟨↑∣S^x∣↓⟩=ℏ/2\langle{\uparrow}|\hat S_x|{\downarrow}\rangle=\hbar/2, ⟨↓∣S^x∣↑⟩=ℏ/2\langle{\downarrow}|\hat S_x|{\uparrow}\rangle=\hbar/2, ⟨↓∣S^x∣↓⟩=0\langle{\downarrow}|\hat S_x|{\downarrow}\rangle=0. Let ∣ϕ⟩=∣↑⟩|\phi\rangle=|{\uparrow}\rangle and ∣ψ⟩=(∣↑⟩+i∣↓⟩)/2|\psi\rangle = (|{\uparrow}\rangle+i|{\downarrow}\rangle)/\sqrt2. ((a)) Insert the resolution of the identity 1=∣↑⟩⟨↑∣+∣↓⟩⟨↓∣\mathbb{1}=|{\uparrow}\rangle\langle{\uparrow}|+|{\downarrow}\rangle\langle{\downarrow}| twice to rewrite the abstract matrix element ⟨ϕ∣S^x∣ψ⟩\langle\phi|\hat S_x|\psi\rangle as an explicit double sum ∑m,n∈{↑,↓}⟨ϕ∣m⟩⟨m∣S^x∣n⟩⟨n∣ψ⟩\sum_{m,n\in\{\uparrow,\downarrow\}}\langle\phi|m\rangle\langle m|\hat S_x|n\rangle\langle n|\psi\rangle. ((b)) Evaluate this sum numerically, using ℏ=1.0546×10−34 J s\hbar = 1.0546\times10^{-34}\ \mathrm{J\,s}, giving the result to three significant figures.

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