# Proof of uncertainty principle

1. Dec 6, 2011

### ShayanJ

Is there a mathematical or physical proof for uncertainty priciple other than the one that says if you wanna observe a particle you should collide with it,a photon and blah blah blah ?

I can't accept that as a proof.Because It falls if you find another way of measuring the position of the particle.Like having a scale sensitive to the mass of that particle.

You know,I mean it does depend on your devices and doesn't suggest that the uncertainty is fundamental to the system.

thanks

2. Dec 6, 2011

### DrChinese

Well, a good proof of your position seems to be missing as well! On the other hand, after 80 years of trying, no one has succeeded in finding a chink in the armor of the Uncertainty Principle.

But to be more specific: read the EPR paper (1935) and Bell's response to it (1965). These demonstrated that either the Uncertainty Principle applies to entangled particles, or Quantum Mechanics is wrong. Tests showed that QM is not wrong, and that even if you measure the uncertainty on a system of 2 particles, the Uncertainty Principle applies. So obviously, measuring apparatus is not the issue as both particles can be measured on different bases (such as momentum, position) to unlimited precision. And yet there is no violation.

So it really comes down to what you define as proof. By normal scientific standards, it is proven by experiment (Aspect, 1981 and many more).

3. Dec 6, 2011

### Dal902

There is a mathematical "proof" but I don't know exactly what it it. It has to do with objects being represented as waves instead of particles. To represent the wave in a small area you have to use the superposition of many waves and this is where the uncertainty principle comes from. Its more of a mathematical design constraint than anything and is completely unrelated to how a property is measured.

4. Dec 6, 2011

### DrChinese

Yes, there is an actual proof that derives from the fundamental axioms of Quantum Theory. However, how do you prove those axioms to the OP? So what I was pointing to was what I consider experimental proof that particles do not have well defined attributes independent of the act of observation. Which is essentially a restatement of the HUP, as per EPR.

5. Dec 6, 2011

### vanhees71

The uncertainty relations for (non-compatible) observables follows quite easily from the very basics of quantum theory, namely from the positive definiteness of the scalar product of state vectors which, by assumption, build a Hilbert space. You find this proof in any good quantum-mechanics textbook, e.g., Sakurai, Modern Quantum Mechanics, Addison-Wesley.

6. Dec 6, 2011

### Fredrik

Staff Emeritus
For Shyan: I posted that proof here. If you haven't studied QM, or enough linear algebra to be familiar with the Cauchy-Bunyakovsky-Schwarz inequality, you won't understand it, but I thought you might still want to see what it looks like. As Vanhees71 said, you can also find a proof in almost any textbook. There's also a proof in the Wikipedia article. It's essentially the same as mine.

7. Dec 6, 2011

8. Dec 6, 2011

### bbbeard

You should follow the link in Fredrik's post, he did a nice job of reproducing the proof.

At the risk of pointing out the obvious, I would mention that the product of the uncertainties in two observables is only non-zero if the observables don't commute. If you have a pair of observables that commute, for example, that x-momentum and y-momentum operators, then both measurements can be made to arbitrarily good precision (although other uncertainties have to be respected, such as with x and px!).

9. Dec 6, 2011

### ShayanJ

Although I was looking for sth like what fredrik posted(So thanks fredrik) but the points that DrChinese mentioned were helpful too so thank you too.
But there is a point Dr.Whatever we do,physics,math or philosophy ,we need to first accept sth without proof and the results of some chosen experiments are good candidates,Like the two postulates of relativity.I think for QM,its the wave-particle duality.

10. Dec 7, 2011

### 256bits

Syyan
I you look farther down on the same post that Fredrik refers to regarding his proof, you will see another reference to the uncertainty principle titled 'Fourier Transforms and Uncertainty Relations' posted by jtbell, similar to what I had linked to previously.

From reading both links you should begin to realize how basic the uncertainty principle really is, and that it is not based of faith but can be explained by mathematical analysis. Read especially the last two paragraphs that explain an association between QM and SR - ie h and c.

11. Dec 7, 2011