 Signed group velocity of a magnetic-gradient doublet

Differentiating $E_k\simeq \sqrt{{\hslash }^{3}q\alpha k/2}/m$ 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 $v_y=⟨\hslash k-q\alpha x^2/2⟩/m$. Evaluating this small drift from a localized wavefunction requires enough anharmonic accuracy: a leading centred Gaussian alone need not reproduce it.