= Solution
For a vertical interband transition, the <crystal momentum> is effectively unchanged because the <photon momentum> is small. A stimulated downward transition needs an occupied upper state and an empty lower state, with factor $f_c(E_2)[1-f_v(E_1)]$; upward <photon absorption> has factor $f_v(E_1)[1-f_c(E_2)]$. The two matrix elements agree by microscopic reversibility. Their difference is therefore
$$
f_c(1-f_v)-f_v(1-f_c)=f_c-f_v,
\qquad \boxed{\alpha(\omega)\propto g_j(\omega)[f_c(E_2)-f_v(E_1)].}
$$
A common slowly varying transition-strength factor is implicit. Broadening replaces the sharp transition by a convolution; multiple valence bands contribute separate terms.
For one isotropic pair of parabolic <energy bands>, write
$$
E_2=E_c+\frac{\hbar^2k^2}{2m_e^*},\qquad
E_1=E_v-\frac{\hbar^2k^2}{2m_h^*},\qquad
\frac1{m_r}=\frac1{m_e^*}+\frac1{m_h^*}.
$$
Thus $E_2-E_1=E_g+\hbar^2k^2/(2m_r)$. Including the usual twofold spin degeneracy, the number of paired states per volume inside a sphere of radius $k$ is $k^3/(3\pi^2)$. Differentiating with respect to transition <energy> $E$ gives the <three-dimensional parabolic interband joint density of states>
$$
\boxed{g_j^{(E)}(E)=\frac{1}{2\pi^2}\left(\frac{2m_r}{\hbar^2}\right)^{3/2}\sqrt{E-E_g}\,\Theta(E-E_g).}
$$
For density per <angular frequency>, $g_j^{(\omega)}(\omega)=\hbar g_j^{(E)}(\hbar\omega)$. Stating the density convention avoids a missing factor of $\hbar$; other degeneracies multiply the formula.
At a common carrier <temperature>, the <Fermi-Dirac distribution> is a strictly decreasing function of $E-F$. The <direct-gap semiconductor gain condition> follows:
$$
f_c(E_2)>f_v(E_1)
\iff E_2-F_c<E_1-F_v
\iff \boxed{E_g<\hbar\omega<F_c-F_v.}
$$
The lower inequality denotes a transition above the ideal band edge, where the displayed <joint density of states> is nonzero. Thus <quasi-Fermi-level splitting> must exceed the transition <photon energy>. Equality is transparency, rather than net amplification; at zero <temperature> an allowed transition must also lie within the occupied-electron and empty-valence windows.
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