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My book on quantum mechanics introduces the notation $\mathcal O(1)$ as follows:

We represent it by the formula $\Delta x \Delta k \gtrsim \mathcal O(1)$ where $\Delta x$ and $\Delta k$ are the "widths" of the two distributions, and we imply by $\mathcal O(1)$ that this is a number that may depend on the functions that we are dealing with, but is not signifiantly smaller than 1.

This seems to differ from how the big-O notation is normally used. Is it related, or simply another function?

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    This is an incorrect use of big-O notation. They want something like big-Omega: http://en.wikipedia.org/wiki/Big_O_notation#Related_asymptotic_notations2011-05-08

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