Spin and two-level systems

A spin-$1/2$ system has $S^2=3\hbar^2/4$ and possible outcomes $\pm\hbar/2$ for a spin component along any chosen axis. Spin is an intrinsic representation of rotation symmetry. A literal charged ball rotating around its own axis is not the model and does not explain the measured angular-momentum algebra. In a suitable magnetic-field-gradient experiment, discrete spin projections can be correlated with separated spatial detection outcomes.

In the z basis, write $S_i=\hbar\sigma_i/2$, where the Pauli matrices satisfy the same angular-momentum commutators after this scaling. A two-state approximation in another system may use the same matrices without being a fundamental electron spin. It is valid when other levels are sufficiently far away or weakly coupled for the experiment. Leakage, extra levels, and environmental coupling are limitations to test rather than features silently included by a two-component column.

After removing an overall phase, any normalized two-level pure state can be written using angles $\theta$ and $\phi$. Its Bloch coordinates are the expectation values of the three Pauli matrices. They form a unit vector for a pure state. Opposite points represent orthogonal states, not two locations of a material particle. The sphere is a visualization of the state space and measurement statistics.

A spin measurement along a unit direction $\mathbf n$ has positive-outcome probability $(1+\mathbf r\cdot\mathbf n)/2$, where $\mathbf r$ is the state's Bloch vector. For a z-up state and an axis at angle $\beta$ from z, this becomes $\cos^2(\beta/2)$. The half-angle is characteristic: a $90^\circ$ change of measurement axis gives equal outcomes rather than a deterministic sideways classical projection. Mixed states occupy the ball's interior, which our pure-state lab does not generate.

Take the effective Hamiltonian $H=\hbar\Omega\sigma_z/2$, with signed angular frequency $\Omega$ defined by that Hamiltonian. Its two energy components acquire opposite phases. After discarding a common phase, the relative phase advances by $\Omega t$. The Bloch vector therefore rotates around z, while z-outcome probabilities stay fixed. A magnetic moment in a field can produce this form, with its sign and frequency determined by the physical coupling convention.

This motion is not a little sphere spinning in space. It is a compact representation of changing expectation values. A measurement of a component still produces one of two allowed values, rather than reading off a continuous component of the drawn arrow. Decoherence can shrink an averaged Bloch vector and relaxation can change populations, but a closed pure-state precession model does neither. Those effects require an open-system extension and additional parameters.