What is the Correct Form of the Uncertainty Principle?

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Discussion Overview

The discussion centers around the correct formulation of the Uncertainty Principle in quantum mechanics, comparing different representations found in textbooks and other sources. Participants explore the implications of the variations in the principle's expression, particularly the presence or absence of the factor of 1/2.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant notes a discrepancy between their physics book, which states the Uncertainty Principle as \(\Delta x\Delta {{p}_{x}}\ge \hbar\), and other sources, including Wikipedia and their math instructor, which state it as \(\Delta x\Delta {{p}_{x}}\ge \frac{\hbar }{2}\).
  • Another participant expresses uncertainty about the correct form, suggesting that the exact factor may not significantly impact the understanding of the principle since the values involved are very small.
  • A participant asserts that the version with the factor of 1/2 is correct, indicating it comes from a rigorous derivation.
  • One participant mentions that different textbooks present various forms of the principle, implying that the differences may not be critical for practical applications.
  • Another participant shares their experience with a specific edition of a textbook that does not include the Uncertainty Principle, suggesting potential discrepancies in educational materials.

Areas of Agreement / Disagreement

Participants express differing views on the correct formulation of the Uncertainty Principle, with some supporting the inclusion of the factor of 1/2 and others suggesting that the exact factor may not be crucial. No consensus is reached on which version is definitively correct.

Contextual Notes

Participants highlight that the differences in the formulations may stem from varying derivations and interpretations of the principle, and that the significance of the factor may diminish at the scales typically considered in quantum mechanics.

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:

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

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

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

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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