Certainty Gain by Uncertainty Power?

1. May 10, 2004

Antonio Lao

One form of the Uncertainty Principle is given by $\Delta p \Delta q \geq h$ where $\Delta p$ is the change in momentum and $\Delta q$ is the change in position and h is Planck's constant. If both side of the inequality is raise to the power n, does the new inequality expresses more certainty? Can we say that certainty is directly proportional to n? As n increases so is the certainty.

2. May 13, 2004

Antonio Lao

The reverse can also be true, as n decreases, the uncertainty increases.

3. May 13, 2004

sol2

http://230nsc1.phy-astr.gsu.edu/hbase/imgmod2/unccon.gif

This is a very significant physical result because it tells us that the energy of a system described by a harmonic oscillator potential cannot have zero energy. Physical systems such as atoms in a solid lattice or in polyatomic molecules in a gas cannot have zero energy even at absolute zero temperature. The energy of the ground vibrational state is often referred to as "zero point vibration". The zero point energy is sufficient to prevent liquid helium-4 from freezing at atmospheric pressure, no matter how low the temperature.

http://230nsc1.phy-astr.gsu.edu/hbase/quantum/hosc4.html#c1

Is there another way to look at this? It made me think of something else here.

it turns out that within string theory ... there is actually an identification, we believe, between the very tiny and the very huge. So it turns out that if you, for instance, take a dimension - imagine its in a circle, imagine its really huge - and then you make it smaller and smaller and smaller, the equations tell us that if you make it smaller than a certain length (its about 10-33 centimeters, the so called 'Planck Length') ... its exactly identical, from the point of view of physical properties, as making the circle larger. So you're trying to squeeze it smaller, but actually in reality your efforts are being turned around by the theory and you're actually making the dimension larger. So in some sense, if you try to squeeze it all the way down to zero size, it would be the same as making it infinitely big. ... (CSPAN Archives Videotape #125054)

Last edited: May 13, 2004
4. May 15, 2004

Antonio Lao

The other form of the Uncertainty Principle is given by $\Delta E \Delta t \geq h$ where $\Delta E$ is the change in energy and $\Delta t$ is the change in time. So by the power of uncertainty, the square of energy is more deterministic than just plain energy.

5. May 15, 2004

Antonio Lao

But what is the physical meaning of the square of $\Delta t$? This seems to be a component of spacetime ($dt^2$) of $ds^2 = dx^2+dy^2+dz^2 - c^2dt^2$

Last edited: May 15, 2004
6. May 15, 2004

baffledMatt

It is very important not to misunderstand Heisenberg's uncertainty principle. From a quick google search I see that this was possibly something even the great man himself was guilty of:

http://www.tardyon.de/heisen.htm

The most important bit from that page is:
The Heisenberg uncertainty principle is a very strong statement about the mathematical properties of wavefunctions which are a direct consequence of the QM postulates.

The uncertainty relation is a strict lower bound of the product of observables $\Delta p$ and $\Delta x$ which are defined by

$\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2}$

where $\langle A\rangle$ is the expectation value of the observable A.

$\langle A \rangle = \langle \phi | A | \phi \rangle$

So you must think very carefully before interpreting it.

Matt

7. May 15, 2004

Antonio Lao

Matt,

Why is macroscopic phenomena much more deterministic than its microscopic counterparts? Both momentum and position (energy and time) can be accurately detected as in the predictions of planetary orbits.

8. May 15, 2004

baffledMatt

It's a question of accuracy. The sort of scales where you really notice the Heisenberg uncertainty principle are very much smaller than anything considered when predicting planetary orbits. Thanks to quantum coherence (or rather the lack of) we see very little quantum 'weirdness' at macroscopic scales.

And then again, sometimes you do. I have seen a very handwavy argument saying that the heisenberg principle can be used to understand why materials have a refractive index larger than 1. Of course I don't really believe it as it is a very poor argument, but it does indicate how these odd effects manifest themselves into our macroscopic world. In fact, the reason glass is transparent at all (at certain frequencies) is purely quantum mecanical.

Matt

9. May 15, 2004

Antonio Lao

baffledMatt,

The uncertainty is removed by the mere act of taking the power of it. The products of conjugate variables is more certain than just the conjugate variables by themselves. In statistics, the square of the variance can make a lot of sense for finding the mean and standard deviation of all measurements.

Fermat's theorem and the Pythagorean theorem works great for square of numbers. The power of square of something cannot be taken lightly. It must be taken vigorously and agressively. The power of area principle as in the theory of surfaces is what makes mathematics such a useful knowledge for understanding many physical realities.

10. May 15, 2004

Antonio Lao

Matrices are a way of defining what a surface is and is based on the power of two (dimensions) from the area priniple.

11. May 15, 2004

baffledMatt

Sorry, could you define what you mean by this? And also explain exactly how this removes the uncertainty, because I'm not sure I follow.

Well, the square root of the variance is the standard deviation so I would imagine calculating it is very useful indeed. I'm not sure how it gives you the mean though.

Now this I definitely was unaware of. Do you have any simple examples, or know of any good (simple!!) references?

This too.

BTW, those last two comments are not meant as criticisms. I am genuinly interested.

Matt

12. May 15, 2004

Antonio Lao

I have to postpone because of previously scheduled engagement. I'll get back to you.

13. May 15, 2004

sol2

Watching from the sidelines and learning:)

14. May 15, 2004

matt grime

once more antonio, you aren't looking at it in any meaningful sense. so what if raising some number to some power makes it smaller. what on earth does that have to fdo with anything meaningul.

the uncertainty principle is a simple consequance of certain analytic properties of integrals, nothing more important than that.

15. May 16, 2004

Antonio Lao

Gentlemen,

I am still looking for the subtle connection between integral and the area principle (sum of products of a function and an infinitesimal). If I can find this link then it can clearly show you the power of 2.

The area principle is the product of two quantities, say, A and B. A and B can be functions, numbers, complex number, hypercomplex numbers.

16. May 16, 2004

Antonio Lao

baffledMatt,

Which come first, the chicken or the egg? the variance or the mean?

The variance $\sigma^2$ is given by the expectation of the square of the difference between a random variable X and the mean $\mu$.

$$\sigma^2=E(X- \mu )^2 = E(X^2) - \mu^2$$

$$\mu^2= E(X^2) - \sigma^2$$

Here,again, we can notice the power of two doing its miracle!

Last edited: May 16, 2004
17. May 16, 2004

Antonio Lao

matt grime,

I am searching for the physical meaning. When I find it, I'll let you know. If you can convince me that this is a lost cause, I will stopped my search.

18. May 16, 2004

baffledMatt

Ok, fair point.

To quote the Dude:
"Well, that's just, like, your opinion man".
What you have described could also be interpreted as the fact that we can rearrange the equations for cumulants of a distribution ($\sigma^2$ is the second cumulant). However, far be it from me to begrudge you an admiration of the power two - as to some extent I agree with you - so I won't labour the point.

Matt

19. May 16, 2004

Antonio Lao

baffledMatt,

Maybe what follow are not good ways for me in trying to convince myself or other about the power of two but just to show you what I'm doing by the use of some algebraic expressions.

$$x^2 - 1^2 = (x - 1)(x + 1)$$

$$x^2 + 1^2 = 2$$

20. May 16, 2004

Janitor

Is this relevant?

You could choose physical units such that h>1. Then raising it to a power greater than one would result in a larger number rather than a smaller number.