= Confinement energy bound from position support
{title2=$\langle K\rangle\geq d\hbar^2/(2mL^2)$}
Suppose each Cartesian coordinate of a particle of mass $m$ is supported in an interval of length $L$, in $d$ dimensions. <Popoviciu's inequality on variances> gives $\Delta x_i\leq L/2$, and the <Heisenberg uncertainty relation> gives $\Delta p_i\geq\hbar/L$. Since $\langle p_i^2\rangle=(\Delta p_i)^2+\langle p_i\rangle^2$,
$$
\boxed{\langle K\rangle\geq\frac{d\hbar^2}{2mL^2}.}
$$
This is a rigorous but non-sharp confinement estimate. For a cubical <infinite square well>, separation of variables gives the stronger exact ground-state value $d\pi^2\hbar^2/(2mL^2)$. One must not identify a support length with a uniquely prescribed <standard deviation> or claim that the uncertainty estimate is the exact ground-state energy.
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