= Solution
Write $L=10^{-9}\,\mathrm m$ and use the <Heisenberg uncertainty relation> separately in the three Cartesian directions. Since a <position> coordinate is confined to an interval of length $L$, <Popoviciu's inequality on variances> gives $\Delta x_i\leq L/2$. Consequently $\Delta p_i\geq\hbar/L$, and the <expectation value> of the <kinetic energy> obeys
$$
\langle K\rangle=\sum_{i=1}^3\frac{(\Delta p_i)^2+\langle p_i\rangle^2}{2m_e}\geq\frac{3\hbar^2}{2m_eL^2}=1.83\times10^{-20}\,\mathrm J.
$$
Thus an uncertainty-based lower estimate is $\boxed{K\gtrsim0.11\,\mathrm{eV}}$. This <confinement energy bound from position support> is not sharp: the exact <ground-state energy> for a cubical <infinite square well> is $3\pi^2\hbar^2/(2m_eL^2)=1.13\,\mathrm{eV}$. The <Heisenberg uncertainty principle> establishes the confinement scale $\hbar^2/(m_eL^2)$; identifying a box size with an approximate <position> <standard deviation> changes its numerical coefficient.
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