= Lamb shift line-centre model
{c}
{title2=$\nu_0=\nu_d-aB_r$}
Consider an original illustrative resonance model, with all numerical settings chosen here. On one isolated branch approximate the transition <frequency> by $\nu(B)=\nu_0+aB$, with known slope $a=20\ \mathrm{MHz/mT}$ and drive <frequency> $\nu_d=1300\ \mathrm{MHz}$. Choose $\nu_0=1100\ \mathrm{MHz}$, hence resonance field $B_r=10\ \mathrm{mT}$. This local linear model omits <hyperfine structure> and nonlinear level mixing; it is not the historical hydrogen <Zeeman splitting> calculation.
Use a <Lorentzian line shape> for a weak-drive current dip,
$$
I(B)=C-\frac{A}{1+4(B-B_r)^2/w_B^2},\qquad
w_B=\frac{w_\nu}{|a|}.
$$
Choose baseline $C=1$, depth $A=0.25$ in arbitrary detector signal units, and <full width at half maximum> $w_\nu=100\ \mathrm{MHz}$, so $w_B=5\ \mathrm{mT}$. The half-depth points are $B_r\pm w_B/2$; this checks the factor of four. If an exponential decay with lifetime $\tau$ alone set this width, <natural linewidth> would give $w_\nu=1/(2\pi\tau)$ and $\tau\simeq1.59\ \mathrm{ns}$. The chosen width is an explanatory input, not a measurement of a historical lifetime.
Generate 65 equally spaced fields from $2$ to $18\ \mathrm{mT}$ and add independent <Gaussian noise> with known <standard deviation> $\sigma_I=0.004$ signal units. A simultaneous <nonlinear regression> fit of $C,A,B_r,w_B$ gives $B_r\simeq10.0007\ \mathrm{mT}$, $w_B\simeq4.9305\ \mathrm{mT}$ and $\nu_0\simeq1099.986\ \mathrm{MHz}$. With residual derivatives $J_{ik}=\partial I(B_i)/\partial\theta_k$ and $\theta=(C,A,B_r,w_B)$, the local <covariance matrix> estimate is
$$
\operatorname{Cov}(\hat\theta)\simeq\sigma_I^2(J^{\mathsf T}J)^{-1}.
$$
It gives centre <standard deviation> $0.01427\ \mathrm{mT}$, or $0.285\ \mathrm{MHz}$ in the inferred intercept. This uses the stipulated noise variance and a local linear approximation, rather than treating the width as the centre error. Repeated noisy samples can locate a broad line's centre much more closely than its width.
\Image[/lamb-shift-chosen-resonance-line-fit.png]
{title=Chosen resonance line fit and zero-field intercept}
{description=Original synthetic data and fitted Lorentzian current dip, with a separate linear field-to-frequency construction. All frequencies, fields, widths, signal units and noise are chosen for this example. The zero-field intercept is not a measured Lamb splitting. The displayed uncertainties are standard deviations under the stated model.}
Include a common magnetic calibration offset with <standard deviation> $u_b=0.020\ \mathrm{mT}$, a fractional gain error $u_g=0.001$, and independent drive-frequency error $u_d=0.100\ \mathrm{MHz}$. To first order the inferred <variance> is
$$
u^2(\nu_0)\simeq a^2u^2(B_r)+a^2u_b^2+(aB_r)^2u_g^2+u_d^2.
$$
Here $u$ denotes a <standard uncertainty>, not a <frequency>. The four contributions in $\mathrm{MHz}$ are $0.285$, $0.400$, $0.200$, and $0.100$; adding their squares gives $u(\nu_0)\simeq0.540\ \mathrm{MHz}$. The calibration offset shifts all sampled fields together, so increasing the sample count cannot average it away. Uncertainty in $a$, unresolved components, asymmetric line distortion, and errors in the assumed field dependence would require additional terms or a richer model. The historical inference requires the appropriate <Zeeman splitting> branches, rather than simply drawing a straight line through one resonance.
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