Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

We rederive the Schrödinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.

Figure 1

(a) Extremal values of the degree of Gaussianity $g$ for a fixed uncertainty $\alpha$ shown as a dashed (blue) line. It is achieved by states ${\widehat{\rho }}_{\min }$ for $g\le 1$, and ${\widehat{\rho }}_{\max }$ for $g>1$. The line connecting subsequent number states $\left|n\right.〉$ and $\left|n+1\right.〉$ is realized by mixtures of them. The Gaussianity-bounded SR relation corresponds to the part of this extremal line shown as a solid (black) line. The uncertainty $\alpha$ must have a value larger or equal to the solid line for a given $g$. (b) Magnified view of (a), where the discontinuity of the uncertainty relation becomes evident.