What is the Correct Form of the Uncertainty Principle?

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SUMMARY

The correct form of the Uncertainty Principle is given by the equation \(\Delta x\Delta p_{x} \ge \frac{\hbar}{2}\), as confirmed by multiple sources including Wikipedia and a math instructor. The version presented in Young & Freedman's "University Physics," which states \(\Delta x\Delta p_{x} \ge \hbar\), is outdated and lacks the precision provided by the factor of 2. While some textbooks may present variations, the modern interpretation is a mathematically rigorous theorem that emphasizes the significance of this factor in quantum mechanics.

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Denver Dang
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Hi...

I have a little question I've been wondering.
In my physics book (Young & Freedman - University Physics) the "Uncertainty Principle" is given as:

\[\Delta x\Delta {{p}_{x}}\ge \hbar \]

But on Wikipedia, and my math-instructor tells me the same, it's given by:

\[\Delta x\Delta {{p}_{x}}\ge \frac{\hbar }{2}\]

The difference being the division by 2.
So, what is the correct one ? And does it even mean anything at all ?
Because, by dividing by 2, aren't you able to determine one of the things even more precisely, or...?

Well, I'm a bit confused, so I hope anyone can tell me the truth :)Regards
 
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I am not sure about the truth being hbar/2 or hbar or hbar/4...

My perspective will be a little different here: It really doesn't matter.

That number is already very small and a factor of two or four at that scale doesn't really change much.

From a theoretical point of view, everything holds just as good; so that's why different authors are being a little sloppy about the exact uncertainty relationship.

I have been seeing different versions in different textbooks as well but I was ignoring the difference.
 
It's divided by 2. Your math instructor is right.
 
I agree with both xepma and sokrates. Post #8 here might clarify the situation a bit. Some of the arguments you could use to derive the older uncertainty principle might give you the "wrong" result by a factor of 2, but it doesn't matter since it's supposed to be an order-of-magnitude estimate anyway.

The modern uncertainty principle on the other hand, is a mathematical theorem, and it's about a different kind of uncertainty.
 
Huh. I've got a copy of Young & Freedman that someone left in my bookshelf (but who stole my copy of Landau-Lifschitz? Hardly a fair exchange!), and it doesn't have the uncertainty principle or any QM at all in it (just some relativity). Tenth edition. Seems they're up to Twelve now. Guess it might've been a typo in what was new material.
 
The inclusion of the factor 1/2 is correct. It arises from a rigorous derivation of the product of the 2 uncertainties.
 
Thanks :)
 

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