The n-dimensional Heisenberg uncertainty inequality

Given: Countable Choice, $n\ge 1$, $f\in H^1({\mathbb{R}}^n;\mathbb{C})$ with $xf\in L^2({\mathbb{R}}^n;{\mathbb{C}}^n)$, its weak derivatives $D_{j}f$, and its Plancherel transform $\widehat{f}={\mathcal{F}}_{2}f$.

Countable Choice is the hypothesis carried by the Sobolev, multiplier, and Plancherel interfaces below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

The integer-order Fourier characterization identifies $H^1$ with $W^{1,2}$ and hence supplies every $D_{j}f\in L^2$ (Integer-order W^{k,2} and H^k agree with equivalent norms).

For a weak derivative in $L^2$, ${\mathcal{F}}_2(D_{j}f)(\xi )=2\pi i{\xi }_j{\mathcal{F}}_{2}f(\xi )$ almost everywhere (Distributional derivatives are polynomial Fourier multipliers).

Plancherel is a complex-linear isometry on $L^2$ (Plancherel theorem).

Cauchy–Schwarz for finite tuples in complex $L^2$ gives ${\sum }_{j=1}^n{a}_j{b}_j\le ({\sum }_j{a}_j^2{)}^{1\mathrm{/}2}({\sum }_j{b}_j^2{)}^{1\mathrm{/}2}$ for nonnegative real $a_j,b_j$ (Complex completeness, density, and inner product: the consumer interface).

The coordinate estimate $\mathrm{\parallel }{x}_{j}f{\mathrm{\parallel }}_2\mathrm{\parallel }{D}_{j}f{\mathrm{\parallel }}_2\ge \frac{1}{2}\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$ holds on this $H^1$-with-finite-spatial-moment domain (The coordinate inequality $\mathrm{\parallel }{x}_{j}f{\mathrm{\parallel }}_2\mathrm{\parallel }{D}_{j}f{\mathrm{\parallel }}_2\ge \frac{1}{2}\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$).