The Fine Structure of Hydrogen and the Lamb Shift

The fine-structure hierarchy is the ordering of hydrogen's energy corrections by their powers of the dimensionless coupling α = e²/(4πε₀ħc) ≈ 1/137 and of the mass ratio mₑ/mₚ, each rung being mc² times a higher power and therefore a smaller effect. Use the supplied topic model Y_out = (n*137.036/z)^2, where Y_out is mc² ÷ BOHR BAR (dimensionless). 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 = 30.046184, with a bounded reading uncertainty ±6008.3. 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 relativistic kinetic-energy correction is the first-order shift produced by H′ᵣ = −p⁴/(8m³c²), the leading amount by which the true kinetic energy √(p²c² + m²c⁴) − mc² falls below p²/2m. Use the supplied topic model Y_out = 4*n/(min(l,n-1)+0.5)-3, where Y_out is ⟨p⁴⟩/⟨p²⟩² (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; min(a,b) selects the smaller value. l is Orbital quantum number l, dimensionless. Keep n (Principal quantum number n) = 3 fixed. For this exercise use only 0 ≤ l ≤ 2, 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 = 5, with a bounded reading uncertainty ±1.536. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of l 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.

Spin-orbit coupling is the energy of the electron's spin magnetic moment in the magnetic field the moving nucleus produces in the electron's rest frame, reduced by Thomas precession to H′_so = (e²/(8πε₀))(1/(m²c²r³)) S⋅L. Use the supplied topic model Y_out = 10.949*(g-tau), where Y_out is PREDICTED 2P DOUBLET (GHz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. g is Spin g-factor g, dimensionless. Keep tau (Thomas precession (0 off, 1 on)) = 0 fixed. For this exercise use only 0.31875 ≤ g ≤ 2.68125 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 = 19.268871 GHz, with a bounded reading uncertainty ±1.5132 GHz. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown g by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00023625, 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 first-order spin-orbit shift of a hydrogen level is the expectation value of Hₛₒ in a coupled state |n, l, ½, j, mⱼ⟩: the eigenvalue of S⋅L, fixed by j and l, times the radial average of e²/(8πε₀m²c²r³). Use the supplied topic model Y_out = -350.377*min(l,1)/(4*(nr+l+1)^3*max(l,1)*(l+0.5)), where Y_out is SHIFT OF j = l − ½ (GHz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; min(a,b) selects the smaller value; max(a,b) selects the larger value. l is Orbital quantum number l, dimensionless. Keep nr (Radial nodes nᵣ = n − l − 1) = 0 fixed. For this exercise use only 1 ≤ l ≤ 4, 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 = -0.13034859 GHz, with a bounded reading uncertainty ±0.58085 GHz. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of l 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.