The Non-Relativistic Limit: g = 2, Spin-Orbit Coupling and the Darwin Term

Eliminating the small component means solving the lower two rows of the Dirac equation for χ in terms of the upper pair φ, χ ≈ (σ⋅π/2mc)φ for a slow electron, so that φ obeys a closed two-component equation of its own. Use the supplied topic model Y_out = sqrt(t*(t+1021.998))/(t+1021.998), where Y_out is FREE EXACT |χ|/|φ| (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. t is Free electron's kinetic energy T, measured in keV. Keep z (Nuclear charge Z of the 1s ion) = 92 fixed. For this exercise use only 25.7188 keV ≤ t ≤ 179.281 keV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.32342261, with a bounded reading uncertainty ±0.0094344. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0153563 keV, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The Pauli equation, iħ∂φ/∂t = [π²/2m − (qħ/2m)σ⋅B + V]φ, is the two-component equation the upper Dirac components obey at leading order in v/c; its σ⋅B term is the energy of a spin moment (q/m)S, which is g = 2. Use the supplied topic model Y_out = (1/3+2*gs/3)*13.996*b, where Y_out is D2 ½ → −½ SHIFT (GHz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Magnetic field B, measured in T. Keep gs (Spin g-factor gₛ) = 2 fixed. For this exercise use only 0.3275 T ≤ b ≤ 1.2725 T on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 21.08614 GHz, with a bounded reading uncertainty ±1.2896 GHz. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000945 T, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The Dirac Landau levels are the exact energies E = √(m²c⁴ + p_z²c² + 2nħ|q|Bc²) of a spin-½ Dirac particle in a uniform magnetic field, where each level n ≥ 1 holds orbit n with its spin moment along B and orbit n − 1 with its spin moment against B. Use the supplied topic model Y_out = 510.999*(sqrt(1+2*b)-1), where Y_out is n = 1 PAIR AT g = 2 (keV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. b is Field B, measured in B_c. Keep gs (Spin g-factor g (range exaggerated)) = 2 fixed. For this exercise use only 0.257188 B_c ≤ b ≤ 1.79281 B_c on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 429.5498 keV, with a bounded reading uncertainty ±23.056 keV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000153562 B_c, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

An anomalous magnetic moment is the part of a spin-½ particle's moment beyond the g = 2 that minimal coupling gives; writing g = 2(1 + a), it enters the Dirac equation through Pauli's term, proportional to σ^(μν)F_(μν), with coefficient μ′ = aqħ/2m. Use the supplied topic model Y_out = 2*(1+a), where Y_out is g = 2(1 + a) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. a is Anomaly a = (g − 2)/2, dimensionless. Keep n (Principal quantum number n) = 2 fixed. For this exercise use only 0.0002125 ≤ a ≤ 0.0017875 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 2.0023465, with a bounded reading uncertainty ±0.00018428. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 1.575e-7, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.