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?
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2$\begingroup$ This question has already been answered in physics.stackexchange.com/questions/69604/… and physics.stackexchange.com/questions/559268/… although neither link contains the whole story. $\endgroup$– GiorgioP-DoomsdayClockIsAt-90Commented May 21, 2023 at 4:46
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$\begingroup$ Keep in mind that the left side has the dimension of an angular momentum, the minimum fluctuation is the spin-1/2, so $\hbar/2$ $\endgroup$– M06-2xCommented May 21, 2023 at 10:25
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2 Answers
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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.
answered May 21, 2023 at 4:57
GiorgioP-DoomsdayClockIsAt-90GiorgioP-DoomsdayClockIsAt-90
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1$\begingroup$ @FlatterMann If units are such that the fermionic spin is an integer, fermions will be characterized by odd spin and bosons by even spin. No significant change in the underlying physics. $\endgroup$ Commented May 21, 2023 at 17:52
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1$\begingroup$ For a historical perspective, see Tomonaga’s “The Story Of Spin.” $\endgroup$– rob ♦Commented May 22, 2023 at 16:11
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$\begingroup$ That was my point. Characterizing spins with integers rather than half-integers seems more rational. I understand the historic reasons and that the physics won't change, I am just wondering if we would do it differently in hindsight, $\endgroup$ Commented May 24, 2023 at 16:50
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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.
answered May 21, 2023 at 4:44
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$\begingroup$ actually the version with $\approx \hbar$ on the RHS arises naturally from Fourier analysis so it wasn’t so much “conjectured” by Heisenberg as adapted by him to the quantum case. $\endgroup$ Commented May 21, 2023 at 12:36