= Solution
Write the dispersive band as $E_p=Ak^2$ with $A>0$ and define $k_c=\sqrt{E_{\rm fl}/A}$. The uncoupled sketch is an upward-opening parabola and a horizontal <flat band> at positive energy. They cross on the sphere $k=k_c$ in three dimensions, or at $\pm k_c$ in a one-dimensional cut:
$$
\boxed{E_p(k_c)=E_{\rm fl},\qquad \alpha=0.}
$$
Without mixing the crossing is allowed. The left panel of the figure shows this case.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2022-q5-hybridisation.png]
{title=Avoided crossings and Fermi-volume redistribution in a two-band model}
{description=Original model calculations. The first three panels use an exactly flat second band and show zero, weak and strong hybridisation in units of its energy. The fourth uses a finite heavy-band curvature and fixed total electron density; two Fermi radii separate before the inner pocket vanishes. It distinguishes the general two-surface result from the eventual single-surface limit.}
For dimensional clarity $A=\hbar^2/(2m_\ell)$ in the usual <band effective mass> convention. The printed product $m_p\hbar^2k^2$ is consistent only if $m_p$ is a curvature coefficient with inverse-mass units; it cannot simultaneously denote the effective mass itself. Using $A$ avoids relying on that notation.
Back to article page
