Spin-half outcome probabilities along an arbitrary direction: the Bloch parametrization and worked cases

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin-$\frac{1}{2}$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$, and the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations). Fix a unit vector $f\in L^2({\mathbb{R}}^3;\mathbb{C})$ and a unit internal spinor $v=(v_1,v_2)\in {\mathbb{C}}^2$, and put $⟨\mathit{\sigma }{⟩}_v:=(2\mathrm{Re}(v_1\overline{v_2}),-2\mathrm{Im}(v_1\overline{v_2}),\mathrm{\mid }{v}_1{\mathrm{\mid }}^2-\mathrm{\mid }{v}_2{\mathrm{\mid }}^2)\in {\mathbb{R}}^3\mathrm{.}$ Let $\psi :=f\otimes v\in H_{1\mathrm{/}2}$ and let $S_n={\sum }_i{n}_i{S}_i$ be the directional spin of Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities. Then:

(i) $\psi$ is a unit vector, $\mathrm{\mid }⟨\mathit{\sigma }{⟩}_v\mathrm{\mid }=1$, and for every unit $n\in {\mathbb{R}}^3$ ${\mu }_{\psi }^{S_n}\left(\right\{\frac{\mathrm{\hslash }}{2}\left\}\right)=\frac{1+n\cdot ⟨\mathit{\sigma }{⟩}_v}{2},{\mu }_{\psi }^{S_n}\left(\right\{-\frac{\mathrm{\hslash }}{2}\left\}\right)=\frac{1-n\cdot ⟨\mathit{\sigma }{⟩}_v}{2};$

(ii) for the parametrization $v=(e^{i\phi }\cos (\theta \mathrm{/}2),\sin (\theta \mathrm{/}2))$ with $\theta \in [0,\pi ]$ and $\phi \in \mathbb{R}$ one has $⟨\mathit{\sigma }{⟩}_v=(\sin \theta \cos \phi ,-\sin \theta \sin \phi ,\cos \theta )$, and for the coordinate directions ${\mu }_{\psi }^{S_3}\left(\right\{\pm \frac{\mathrm{\hslash }}{2}\left\}\right)={\cos }^2\frac{\theta }{2},{\sin }^2\frac{\theta }{2},{\mu }_{\psi }^{S_1}\left(\right\{\pm \frac{\mathrm{\hslash }}{2}\left\}\right)=\frac{1\pm \sin \theta \cos \phi }{2};$

(iii) for the basis state $v=e_{1\mathrm{/}2}$ (the case $\theta =0$) the component $S_3$ has the laws ${\mu }_{\psi }^{S_3}(\{\mathrm{\hslash }\mathrm{/}2\})=1$ and ${\mu }_{\psi }^{S_3}(\{-\mathrm{\hslash }\mathrm{/}2\})=0$, while ${\mu }_{\psi }^{S_1}(\{\pm \mathrm{\hslash }\mathrm{/}2\})=\frac{1}{2}$ for both signs; and for $v=(e_{1\mathrm{/}2}+e_{-1\mathrm{/}2})\mathrm{/}\sqrt{2}$ the component $S_1$ has ${\mu }_{\psi }^{S_1}(\{\mathrm{\hslash }\mathrm{/}2\})=1$ and ${\mu }_{\psi }^{S_1}(\{-\mathrm{\hslash }\mathrm{/}2\})=0$.

All quantities are model probabilities for the pure preparation $[\psi ]$; no measurement procedure, apparatus, calibration or empirical frequency is asserted, and the example makes no claim that a laboratory device realizes the observable $S_n$.

• One preparation, all directions. The Bloch vector determines the whole family of directional laws of this preparation. These marginal laws alone do not specify joint probabilities for measurements in two directions. Such probabilities require a declared joint or sequential measurement model, including any state change between measurements; none is supplied here. No impossibility of all joint probability laws is asserted.
• No apparatus. The probabilities are Born weights of the model; whether a given Stern-Gerlach or optical apparatus realizes $S_n$ is a tested identification under the preparation and apparatus postulate P7 and is not established here.