# Uncertainty Principle

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#### PmbPhy

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##### Uncertainty Principle
« on: 27/02/2015 01:44:05 »
I thought that most members here would be interested in the following facts I've learned while studying the Heisenberg Uncertainty Principle (HUP) which relates the uncertainty between two non-commuting  observables, A and B. The general uncertainty principle is expressed as

$$\Delta A \Delta B \geq \frac{1}{2}|<\Psi|[A, B]|\Psi>|$$

There are two versions of the HUP:

Version #1: In this version the HUP states that if A and B don't commute, i.e. [A,B] is not zero, then A and B cannot be measured simultaneously in a single measurement, i.e. it applies to single events. In this version uncertainty, $$\Delta A$$ is defined as the error in the measurement. Some call this the "individual uncertainty principle" (IUP).

Version #2: In this version  the HUP states that if A and B don't commute, i.e. [A,B] is not zero, then A and B cannot be measured simultaneously in a single measurement since it applies to a large number of identically prepared systems and measurements. In this version uncertainty, $$\Delta A$$ is defined as the standard deviation of all of those measurements. In this version both A and B can be measured to arbitrary accuracy and the HUP does not forbid simultaneous measurements.

For more on this please see: http://statintquant.net/siq/siqse2.html

Stanford has a nice accounting of the uncertainty principle. See http://plato.stanford.edu/entries/qt-uncertainty/

I'd love some input as to what the folks here who are actually skilled in QM give me some feedback on this. That includes ALL of my friends.

#### jeffreyH

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##### Re: Uncertainty Principle
« Reply #1 on: 27/02/2015 04:25:05 »
The preparation and measurement debate is interesting but I doubt it makes that much difference in the end.
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#### alancalverd

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##### Re: Uncertainty Principle
« Reply #2 on: 27/02/2015 08:13:23 »
I think it is important to distinguish between uncertainty, which is the estimated sum of all the random and systematic errors in a practical (mesoscopic) measurement, and indeterminacy, which is the fundamental property of matter that gives the hydrogen atom a diameter greater than that of a neutron.

Many naive correspondents confuse the two, and it's not helped by Heisenberg's original thought experiment of trying to measure the position of a particle by bouncing a photon off it.
helping to stem the tide of ignorance

#### PmbPhy

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##### Re: Uncertainty Principle
« Reply #3 on: 27/02/2015 15:23:17 »
Quote from: alancalverd
I think it is important to distinguish between uncertainty, which is the estimated sum of all the random and systematic errors in a practical (mesoscopic) measurement,
As explained above, that is only one definition of uncertainty. And its not the one used in quantum mechanics textbooks.

#### alancalverd

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##### Re: Uncertainty Principle
« Reply #4 on: 27/02/2015 17:54:26 »
And that's the source of many misunderstandings. My education (the best bits of which were delivered by people who actually knew Heisenberg) has always distinguished between experimental uncertainty and quantum-level indeterminacy, so I've never had a problem with either. Unfortunately Heisenberg's seminal paper apparently used genauigkeit, which usually translates to  "accuracy" or "precision" and I can't find a German word for "indeterminacy" (in a mathematical rather than legal sense).

If you want to make a really important contribution, I'd advise starting from what we know about quantum mechanics and, eventually, showing how it is consistent with classical physics for large objects (i.e. where the uncertainty of an ensemble is orders of magnitude larger than the indeterminacy of its members). That way you avoid all the mystical nonsense about observers and consciousness, and end up with useful, robust predictions and explanations. And at some point in the development you can explain why classical and quantum mechanics do not commute!
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#### PmbPhy

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##### Re: Uncertainty Principle
« Reply #5 on: 27/02/2015 18:12:47 »
And that's the source of many misunderstandings. My education (the best bits of which were delivered by people who actually knew Heisenberg) has always distinguished between experimental uncertainty and quantum-level indeterminacy, so I've never had a problem with either. Unfortunately Heisenberg's seminal paper apparently used genauigkeit, which usually translates to  "accuracy" or "precision" and I can't find a German word for "indeterminacy" (in a mathematical rather than legal sense).
Heisenberg's definition of uncertainty is not the one that is used in modern quantum mechanics textbooks. That's the point of this thread, i.e. to point this fact out and make it very clear. Take a look at Leonard Susskin's new book Quantum Mechanics and you'll see that's the way it is there too. It's also the way it is in Griffiths text and Cohen-Tannoudji et el, Sakurri etc. But in none of them to then discuss Heisenbergs use of accuracy in measurement. Not one peep about it in fact. However the fact quantum mechanics is still indeterminate hasn't changed.

The uncertainty principle used now is the one by H.P. Robertson.