 Solution

For standard deviations of position and momentum,Confinement to a region of size $R$ requires a momentum spread of order $\hslash /R$, hence kinetic energy of order ${\hslash }^2/(2m^{\ast }{R}^2)$. 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 iswhere $\alpha$ 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.