The Lamb Shift Completed at Order alpha (Z alpha)^4

The order-α(Zα)⁴m Lamb shift is the part of a hydrogen-like level's energy carrying one power of α from a single loop of the quantised fields and four powers of Zα from the binding; it is the leading term of the 2S₁/₂ − 2P₁/₂ splitting. Use the supplied topic model Y_out = 1085.1503*z^4/n^3, where Y_out is UNIT Cₙ (MHz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. z is Nuclear charge Z, dimensionless. Keep n (Principal quantum number n) = 2 fixed. For this exercise use only 1 ≤ z ≤ 81, at the specified test settings separated by 1. 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 = 847773670 MHz, with a bounded reading uncertainty ±467120000 MHz. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of z consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The slope of F₁ is the coefficient of q² in the electron's renormalised charge form factor near q² = 0; at one loop with a photon mass μ_γ it is (α/(3πm²))[ln(m/μ_γ) − 3/8], which depends logarithmically on that infrared regulator. Use the supplied topic model Y_out = sqrt(692.75*(-p*ln(10)-0.375)), where Y_out is RMS RADIUS √⟨r²⟩ (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; ln(x) means the natural logarithm. p is log₁₀ of photon mass μ_γ/m, dimensionless. Keep znuc (Nuclear charge Z) = 1 fixed. For this exercise use only -7.3625 ≤ p ≤ -2.6375 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 = 82.986363 fm, with a bounded reading uncertainty ±3.0238 fm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0004725, 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 high-energy contact potential is the δ³(r) term that the slope of F₁ adds to the Coulomb potential of a static nucleus; it shifts only states whose wavefunction is non-zero at the nucleus, which are the s states. Use the supplied topic model Y_out = 10^lmu, where Y_out is PHOTON MASS μ_γ (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. lmu is Photon mass, log₁₀(μ_γ / 1 eV), dimensionless. Keep z (Nuclear charge Z) = 1 fixed. For this exercise use only -2.25625 ≤ lmu ≤ 3.25625 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 = 12.775415 eV, with a bounded reading uncertainty ±1.9155 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lmu by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00055125, 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.

Over the momentum transfers an atom supplies, the one-loop magnetic form factor F₂(q²) equals its zero-momentum value, so a bound electron feels it as Pauli's term with a = F₂(0) = α/2π, adding 2a times Dirac's Darwin and spin-orbit energies. Use the supplied topic model Y_out = 4*ln((sqrt(100^lq+4)+10^lq)/2)/(10^lq*sqrt(100^lq+4)), where Y_out is F₂/F₂(0) AT PROBE (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; ln(x) means the natural logarithm. lq is Probe momentum log₁₀(|q|/m), dimensionless. Keep zn (Nuclear charge Z) = 1 fixed. For this exercise use only -2.575 ≤ lq ≤ 0.575 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 = 0.99186066, with a bounded reading uncertainty ±0.0012207. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lq by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315, 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.