A closer look at the uncertainty relation of position and momentum

TL;DR

This paper investigates the uncertainty relation between position and momentum for particles prepared by von Neumann-L"uders projection, revealing that infinite standard deviations are common and establishing a new, optimal uncertainty relation.

Contribution

It provides a necessary and sufficient condition for finite momentum standard deviations and introduces a new, proven optimal uncertainty relation.

Findings

Infinite standard deviations are typical for such particles.

A necessary and sufficient condition for finite standard deviations is established.

A new uncertainty relation is derived and shown to be optimal.

Abstract

We consider particles prepared by the von Neumann-L\"uders projection. For those particles the standard deviation of the momentum is discussed. We show that infinite standard deviations are not exceptions but rather typical. A necessary and sufficient condition for finite standard deviations is given. Finally, a new uncertainty relation is derived and it is shown that the latter cannot be improved.