The two-level Zeeman toy model: sector energies, level splitting and spin precession

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-$\frac{1}{2}$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$, together with the effective magnetic-moment coupling of Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator with parameter $\gamma :={\gamma }_{1\mathrm{/}2}\in \mathbb{R}$, and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix $B_z\in \mathbb{R}$ and the uniform field $\mathbf{B}=(0,0,B_z)$, let $H_Z=H_0-\gamma B_z{S}_3$ on $D(H_Z)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^2)$ be the model's Zeeman operator with flow $U_Z(t)=e^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}$ and put $\omega :=\gamma B_z$. Then:

(i) for every unit $f\in L^2({\mathbb{R}}^3;\mathbb{C})$ and each sign, the product flow satisfies $U_Z(t)(f\otimes e_{\pm })=e^{\pm i\gamma B_{z}t\mathrm{/}2}\left(e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}f\right)\otimes e_{\pm }(t\in \mathbb{R}),$ and the transverse spin means of such a product state precess as in Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component. If in addition $f\in D(H_0)=W^{2,2}({\mathbb{R}}^3;\mathbb{C})$, then $H_Z(f\otimes e_{\pm })=\left(H_{0}f\right)\otimes e_{\pm }\mp \frac{\gamma \mathrm{\hslash }{B}_z}{2}f\otimes e_{\pm },⟨H_Z{⟩}_{f\otimes e_{\pm }}:=⟨H_Z(f\otimes e_{\pm }),f\otimes e_{\pm }⟩=⟨H_{0}f,f⟩\mp \frac{\gamma \mathrm{\hslash }{B}_z}{2};$ the displayed expectation is defined only for these domain vectors.

(ii) toy truncation. Fix a real energy parameter $E_0\in \mathbb{R}$ in joules. Suppose in addition that, on a declared two-dimensional internal subspace, the free part is replaced by the scalar $E_{0}I$ (a truncation assumption of the toy model, not a theorem of the full model): then the truncated operator is $H_Z^{(2)}=E_{0}I-\gamma B_z{S}_3=(E_0-\frac{\gamma \mathrm{\hslash }{B}_z}{2})e_{+}{e}_{+}^{\ast }+(E_0+\frac{\gamma \mathrm{\hslash }{B}_z}{2})e_{-}{e}_{-}^{\ast }$ in the basis $(e_{+},e_{-})$, with energy levels $E_0\mp \frac{\gamma \mathrm{\hslash }{B}_z}{2}$ and signed level difference $\mathrm{\Delta }E:=\gamma \mathrm{\hslash }{B}_z=\mathrm{\hslash }\omega$; its evolution is $e^{-it{H}_Z^{(2)}\mathrm{/}\mathrm{\hslash }}=e^{-it{E}_0\mathrm{/}\mathrm{\hslash }}\left(\begin{array}{cc}{e}^{i\omega t\mathrm{/}2}& 0\\ 0& e^{-i\omega t\mathrm{/}2}\end{array}\right),$ so the relative phase of the two amplitudes accumulates at the angular frequency $\omega$ (equivalently at the ordinary frequency $\mathrm{\mid }\omega \mathrm{\mid }\mathrm{/}(2\pi )$);

(iii) in the truncation the conjugation identities of the precession theorem reduce to explicit matrix identities, $V(t{)}^{\ast }{\sigma }_{1}V(t)=\cos (\omega t){\sigma }_1+\sin (\omega t){\sigma }_2,V(t{)}^{\ast }{\sigma }_{2}V(t)=\cos (\omega t){\sigma }_2-\sin (\omega t){\sigma }_1,$ with $V(t)=e^{-it{H}_Z^{(2)}\mathrm{/}\mathrm{\hslash }}{e}^{it{E}_0\mathrm{/}\mathrm{\hslash }}$;

(iv) the toy model contains no driving field: its diagonal evolution preserves the two sector weights, hence there is no spin flip between $e_{+}$ and $e_{-}$ under $H_Z^{(2)}$ alone. Statements about spin-flip transitions, resonance or spectroscopy require a separately adopted drive or coupling, belong to the driven-dynamics and light-matter pages, and are not derived here. No apparatus, calibration, measured frequency or measured value of $\gamma$ is asserted.

• What the truncation discards. Replacing $H_0$ by $E_{0}I$ discards the kinetic and any potential structure of the spatial factor; the truncated model is a toy description of the internal degree of freedom, and no spatial dynamics or position-space measurement model follows from it.
• Frequency conventions. $\omega =\gamma B_z$ is an angular frequency in ${\mathrm{s}}^{-1}$; the ordinary frequency is $\mathrm{\mid }\omega \mathrm{\mid }\mathrm{/}(2\pi )$ in hertz, with the radian dimensionless, in the conventions of Dimensions, SI unit conventions and quantity types of the framework.