A spin-one-half particle is prepared in the pure state whose spin points at angle…

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

The Structure of Quantum Mechanics, Ch Compatible observables and uncertainty as a theorem

Problem

A spin-one-half particle is prepared in the pure state whose spin points at angle θ=60∘\theta=60^\circ from the zz-axis, in the xzxz-plane, for which (as established for spin measurement along an arbitrary axis) ⟨S^z⟩=ℏ2cos⁡θ\langle\hat S_z\rangle=\tfrac{\hbar}{2}\cos\theta, ⟨S^x⟩=ℏ2sin⁡θ\langle\hat S_x\rangle=\tfrac{\hbar}{2}\sin\theta, ⟨S^y⟩=0\langle\hat S_y\rangle=0, while S^i2=(ℏ/2)21\hat S_i^2=(\hbar/2)^2\mathbb 1 for every direction ii, so ⟨S^i2⟩=ℏ2/4\langle\hat S_i^2\rangle=\hbar^2/4 always. (a) Compute ΔSx\Delta S_x and ΔSz\Delta S_z for this state. (b) Given [S^x,S^z]=−iℏS^y[\hat S_x,\hat S_z]=-i\hbar\hat S_y, evaluate the commutator term 14∣⟨[S^x,S^z]⟩∣2\tfrac14|\langle[\hat S_x,\hat S_z]\rangle|^2 of the full uncertainty relation. (c) The Pauli matrices satisfy {σ^x,σ^z}=0\{\hat\sigma_x,\hat\sigma_z\}=0, so {S^x,S^z}=0\{\hat S_x,\hat S_z\}=0 as an operator identity. Using ⟨{ΔA^,ΔB^}⟩=⟨{A^,B^}⟩−2⟨A^⟩⟨B^⟩\langle\{\Delta\hat A,\Delta\hat B\}\rangle=\langle\{\hat A,\hat B\}\rangle-2\langle\hat A\rangle\langle\hat B\rangle, evaluate the anticommutator term. (d) Verify that (ΔSx)2(ΔSz)2(\Delta S_x)^2(\Delta S_z)^2 equals the sum of the terms in (b) and (c), and state, in one sentence, why every state of a spin-half particle saturates the full relation for any pair of spin components.

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