= Signed group velocity of a magnetic-gradient doublet
{title2=$v_y=\hbar q\alpha/[2m\sqrt{2\hbar q\alpha k}]$}
Differentiating $E_k\simeq\sqrt{\hbar^3q\alpha k/2}/m$ gives the displayed <group velocity> when $q\alpha k>0$. Its sign is the sign of $q\alpha$, equivalently of k on that branch. Writing only a positive square root of $q\alpha/k$ would lose this sign. Exactly, the <Hellmann–Feynman theorem> gives $v_y=\langle\hbar k-q\alpha x^2/2\rangle/m$. Evaluating this small drift from a localized wavefunction requires enough anharmonic accuracy: a leading centred Gaussian alone need not reproduce it.
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