Mathematical Expression of Heisenberg's Uncertainty Principle

Question

In some situations the mathematical expression of the principle is $$\Delta x \Delta p \ge \hbar$$ But sometimes it is written as $$\Delta x \Delta p \ge \frac {\hbar}{2}$$ Why such a difference? And are both the relations correct for any pair of canonically conjugate variables?

Answer 1

The second inequality is the correct inequality. The first inequality is a relationship conjectured by Heisenberg. As we can see the conjecture was inaccurate by a factor of $2$. Cf. a book like Griffiths's Quantum Mechanics for a derivation, the following of which should make it clear where the factor of $2$ comes from.

Answer 2

The real reason behind the different expressions is that there are two different things under the name of Heinsenberg's uncertainty principle: the original principle proposed in a heuristic way by Heisenberg as a principle at the basis of Quantum Mechanics and related to the uncertainty of position and momentum measurements on a single system, and a theorem, proved by Kennard and other people, referring to the statistical spread of independent position and momentum measurements on equally prepared systems.

The relation derived as a theorem contains the 1/2 factor.