Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin-$s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$ and spin label $s$, together with the effective magnetic-moment coupling adopted in Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator and the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings, and assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix a real constant $B_z\in \mathbb{R}$ and use the uniform static field profile $\mathbf{B}(x):=(0,0,B_z)$, so that the Zeeman energy operator of the postulate is $H_Z=H_0-{\gamma }_s{B}_z{S}_3,H_0=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2,D(H_Z)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),$ with flow $U_Z(t)=e^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}$, and put $\omega :={\gamma }_s{B}_z$ with unit ${\mathrm{s}}^{-1}$. For a unit vector $\psi \in H_s$ put $\psi (t):=U_Z(t)\psi$ and $⟨S_i⟩(t):=⟨S_i\psi (t),\psi (t)⟩$. Then:

(a) $H_Z$ is self-adjoint on $D(H_Z)$ and $U_Z(t)=e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}{e}^{it{\gamma }_s{B}_z{S}_3\mathrm{/}\mathrm{\hslash }}(t\in \mathbb{R}),$ so in particular the transverse means satisfy, for every $t$, $⟨S_1⟩(t)=\cos (\omega t)⟨S_1⟩(0)+\sin (\omega t)⟨S_2⟩(0),⟨S_2⟩(t)=\cos (\omega t)⟨S_2⟩(0)-\sin (\omega t)⟨S_1⟩(0),$ and the transverse vector $(⟨S_1⟩(t),⟨S_2⟩(t))$ is the rotation of $(⟨S_1⟩(0),⟨S_2⟩(0))$ by the angle $-\omega t$ in the $(1,2)$-plane;

(b) the component along the field is conserved: $S_3{U}_Z(t)=U_Z(t)S_3$ and $⟨S_3⟩(t)=⟨S_3⟩(0)$ for every unit $\psi$ and every $t$; more generally the outcome law of $S_3$ is time-independent in every preparation;

(c) the conjugation identities $e^{it{H}_Z\mathrm{/}\mathrm{\hslash }}{S}_1{e}^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}=\cos (\omega t)S_1+\sin (\omega t)S_2,e^{it{H}_Z\mathrm{/}\mathrm{\hslash }}{S}_2{e}^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}=\cos (\omega t)S_2-\sin (\omega t)S_1$ hold as bounded operator identities, and with $S^2={\mathrm{\hslash }}^{2}s(s+1)I$ (the Casimir operator of the model) also $S^2{U}_Z(t)=U_Z(t)S^2$;

(d) Scope. The statement is conditional on P1, P2, P4, P5, P6 and the adopted coupling; it concerns the uniform static field $\mathbf{B}=(0,0,B_z)$ with $B_z\in \mathbb{R}$ and the declared isolated model. It excludes inhomogeneous and time-dependent fields, spin-dependent spatial potentials, radiation, relaxation and environment coupling, asserts no measured precession frequency and no measured gyromagnetic parameter, and identifies no apparatus: $\omega ={\gamma }_s{B}_z$ is a model quantity computed from assigned model parameters.

• Sign and naming of the angular frequency. The model's precession angular frequency is $\omega ={\gamma }_s{B}_z$, a signed quantity; the transverse vector rotates by $-\omega t$. Physical conventions differing by the sign of ${\gamma }_s$ or by the orientation of the field axis differ by the sign of $\omega$ and by the sense of the rotation, not by the content of the theorem.
• Energy eigenvalues of the internal factor. Combining this theorem with the spin spectrum page, the internal factor of the uniform-field model has the energy offsets $-{\gamma }_s{B}_z\mathrm{\hslash }m$, $m=-s,\dots ,s$, added to the free kinetic energy; the theorem does not assert a spectroscopic measurement or the absence of additional structure.