= Stern-Gerlach discrete and classical profiles
{c}
{title2=$d(v)=\mu_BG\ell(D+\ell/2)/(Mv^2)$}
For an equally populated ideal pair of <angular momentum eigenstates>, the <magnetic moment> component takes two values, $\pm\mu_B$, and the displacements at fixed <velocity> are $\pm d(v)$. Compare a specified classical alternative: every <chemical atom> carries a permanent <magnetic moment> of fixed magnitude $\mu_B$, with independently isotropic initial orientations and no dissipative alignment. If $u=\cos\theta$, the sphere's area element is $2\pi\,du$, so $u$ has a <uniform distribution> on $[-1,1]$. With precession preserving this component in the ideal geometry, the displacement $d(v)u$ fills the interval $[-d(v),d(v)]$ continuously. A pair of bands therefore tests allowed components rather than merely proving that <silver> is magnetic. Other classical preparation assumptions would be different models and must be stated.
To include finite instrumental width, choose a normalized <Gaussian distribution>
$$
\phi_\sigma(z)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{z^2}{2\sigma^2}\right)
$$
with <standard deviation> $\sigma=0.035\,\mathrm{mm}$, representing a combined field-off spatial width. At fixed <velocity> the model <probability density functions> are
$$
p_{\rm two}(z\mid v)=\frac{\phi_\sigma(z-d(v))+\phi_\sigma(z+d(v))}{2},\qquad
p_{\rm classical}(z\mid v)=\frac12\int_{-1}^1\phi_\sigma(z-d(v)u)\,du.
$$
Each has unit <integral> and zero <mean>; the different shapes, rather than a different average displacement, distinguish them. The discrete model has <variance> $\sigma^2+d(v)^2$, whereas the classical model has <variance> $\sigma^2+d(v)^2/3$. Their common mean also explains why the raw average landing position is insufficient to identify the internal <quantum state>.
Now choose a <uniform distribution> of forward <velocities> from $400$ to $600\,\mathrm{m\,s^{-1}}$, explicitly as an example rather than the oven's actual distribution. Average either conditional <probability density function> over that interval. The positive-channel displacement ranges from $0.1776$ to $0.3995\,\mathrm{mm}$, since slower <chemical atoms> spend longer accelerating and drifting. The two components broaden into bands even though their internal <magnetic moment> values remain discrete. The corresponding classical profile continues to populate the center. The displayed strip darkness is proportional to predicted deposited number density, with equal total arrivals; it is not a photograph or a calibrated optical response of a developed film.
\Image[/stern-gerlach-discrete-and-continuous-profiles.png]
{title=Discrete magnetic components and continuous classical deflections}
{description=Original model predictions with the chosen flight parameters, comparing one forward speed with a finite speed interval. Both alternatives include the same Gaussian spatial width. Synthetic deposition strips use a common density scale; none of the curves or strips is measured data.}
Increasing the spatial width until it is comparable with $d$ can hide the two maxima. At fixed <velocity>, differentiating the two-state <probability density function> twice gives $p_{\rm two}''(0)=\phi_\sigma(d)(d^2-\sigma^2)/\sigma^4$: the central minimum becomes a central maximum when $\sigma>d$. Conversely, a broad distribution of <velocities> need not create a new internal component: it spreads each existing component through the $v^{-2}$ dependence. Further instrumental limits include collisions, aperture clipping of strongly deflected trajectories, asymmetric pole geometry, and nonlinear film development. Field-off widths and a mapped <magnetic field> <gradient> are needed to separate such effects. For a modern counting experiment, channel calibration and efficiency corrections belong to <Stern-Gerlach expectation estimation>; spatial separation alone does not supply those statistics.
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