  We calculate the effect of a spatially dependent effective mass (SPDEM)
[adapted from R. N. Costa Filho et al. Phys. Rev. A., \textbf{84} 050102
(2011)] on an electron and hole confined in a quantum well (QW). In the work of
Costa Filho et al., the translation operator is modified to include an inverse
character length scale, $\gamma$, which defines the SPDEM. The introduction of
$\gamma$ means translations are no longer additive. In nonadditive space, we
choose a `skewed' Gaussian confinement potential defined by the replacement
$x\rightarrow\gamma^{-1}\ln(1+\gamma x)$ in the usual Gaussian potential.
Within the parabolic approximation $\gamma$ is inversely related to the QW
thickness and we obtain analytic solutions to our confinement Hamiltonian. Our
calculation yields a reduced dispersion relation for the gap energy ($E_G$) as
a function of QW thickness, $D$: $E_G\sim D^{-1}$, compared to the effective
mass approximation: $E_G\sim D^{-2}$. Additionally, nonadditive space contracts
the position space metric thus increasing the occupied momentum space and
reducing the effective mass, in agreement the relation: $m_o^{*-1}\propto
\frac{\partial^2 E}{\partial \v{k}^2}$. The change in the effective mass is
shown to be a function of the confinement potential via a point canonical
transformation. Our calculation agrees with experimental measurements of $E_G$
for Si and Ge QWs.
