### Unusual position-momentum uncertainty relations

Consider a one-dimensional quantum particle, and let $$X$$ and $$P$$ represent its position and momentum operators, respectively. $$X$$ and $$P$$ are assumed to satisfy the canonical commutation relation, $[X,P] = i \hbar$ with $$\hbar$$ being the reduced Planck constant.

For the following discussion, it is convenient to introduce dimensionless position and momentum as $x = \frac{X}{L} \qquad \text{and} \qquad p = \frac{L P}{\hbar} \,,$ where $$L$$ is some (chosen for convenience, but otherwise arbitrary) length scale. In terms of $$x$$ and $$p$$, the canonical commutation relation reads $[x, p] = i \,.$

The operators $$x$$ and $$p$$ fulfill the Heisenberg uncertainty relation: $\Delta x \Delta p \ge \frac{1}{2} \,,$ where $$\Delta x \ge 0$$ and $$\Delta p \ge 0$$ are uncertainties in the particle's position and momentum, respectively. (The Heisenberg uncertainty relation is a straightforward consequence of a more general Robertson-Schrödinger uncertainty relation.)

In fact, there are (infinitely) many other position-momentum uncertainty relations. For instance, $$\Delta x$$ and $$\Delta p$$ satisfy the following two simple, symmetric inequalities: $\Delta x + \Delta p \ge \sqrt{2} \,,$ $(\Delta x)^2 + (\Delta p)^2 \ge 1 \,.$ These inequalities however are less precise than the Heisenberg one (which, in turn, is less precise than the Robertson-Schrödinger inequality).

The validity of the above uncertainty relations can be easily established from the following figure.

All points of the $$\Delta x$$-$$\Delta p$$ plane that lie above the (blue) minimal uncertainty curve corresponding to the Heisenberg inequality also lie above the (red and green) minimal uncertainty curves corresponding to the other two inequalities.