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Adjacent nodes of a wavefunction are separated by roughly $a/n$, so a locally sinusoidal wavefunction has typical wavevector $k\sim \pi n/a$. The kinetic-energy estimate from wavefunction nodes is thereforeEquivalently, $⟨K⟩={\hslash }^2\int |{\psi }^{\prime }{|}^{2}dx/(2m\int |\psi {|}^{2}dx)$. The robust scaling is ${\hslash }^2{n}^2/(m{a}^2)$; the precise coefficient depends on how the extent is defined and how the oscillations vary near classical turning points. For few nodes, use an order-one offset to $n$: the nodeless ground state still has confinement energy bound from position support of order ${\hslash }^2/(m{a}^2)$, rather than zero.

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 $\Delta x_i\le L/2$. Consequently $\Delta p_i\ge \hslash /L$, and the expectation value of the kinetic energy obeysThus an uncertainty-based lower estimate is $\boxed{K\gtrsim 0.11\mathrm{eV}}$. This confinement energy bound from position support is not sharp: the exact ground-state energy for a cubical infinite square well is $3{\pi }^2{\hslash }^2/(2m_e{L}^2)=1.13\mathrm{eV}$. The Heisenberg uncertainty principle establishes the confinement scale ${\hslash }^2/(m_e{L}^2)$; identifying a box size with an approximate position standard deviation changes its numerical coefficient.