= Solution
For standard deviations of <position> and <momentum>,
$$
\boxed{\Delta x\,\Delta p_x\ge\hbar/2}.
$$
Confinement to a region of size $R$ requires a momentum spread of order $\hbar/R$, hence kinetic energy of order $\hbar^2/(2m^*R^2)$. A sufficiently small semiconducting <nanocrystal> therefore has discrete <energy levels> and an optical edge shifted above the bulk <band gap>.
An illustrative strong-confinement model is
$$
E_{\rm opt}(R)\simeq E_g+
\frac{\pi^2\hbar^2}{2R^2}\left(\frac1{m_e^*}+\frac1{m_h^*}\right)
-\frac{\alpha e^2}{4\pi\epsilon_0\epsilon_rR},
$$
where $\alpha$ is a geometry-dependent number of order unity. The positive confinement contribution grows as $R^{-2}$; <electron-hole Coulomb interaction> contributes an attractive term of order $-R^{-1}$. The uncertainty principle motivates the scale, not the exact coefficient or oscillator strength. Strong size dependence occurs near the <exciton Bohr radius>. Finite barriers, dielectric mismatch and surface states modify this model, while size distributions broaden ensemble spectra.
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