Certainty Gain by Uncertainty Power?

[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.

 

[Antonio Lao]

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

 

[sol2]

Just trying to follow your thinking...


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)

 

[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.

 

[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

 

[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:

Before going into Bohr's analysis of the uncertainty principle, two points should be made clear. First, the mathematical formalism expressing relationships encompassed by the "uncertainty principle" are straight forward deductive consequences of the quantum postulate. All too often, discussions of the principle begin with a series of thought experiments intended to demonstrate that observations which determine the value of one observable within a certain range require physical conditions which preclude determining its canonically conjugate observable within a range that would contradict the uncertainty principle.

It is then easy to get the mistaken impression that the principle expresses an empirical generalization derived from analyzing physical situations and that such a presumed empirical discovery is then injected as a postulate into the theoretical formalism. However, this is not the case. Heisenberg first developed his formalism for theoretical representation of the atomic system processes, and then showed that the consequence of his formalism was that the system could not be characterized by a state which was defined in terms of precise values for both canonically conjugate parameters.

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

<br /> \Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2}<br />

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

<br /> \langle A \rangle = \langle \phi | A | \phi \rangle<br />

So you must think very carefully before interpreting it.

Matt

 

[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.

 

[baffledMatt]

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.

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

 

[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 aggressively. The power of area principle as in the theory of surfaces is what makes mathematics such a useful knowledge for understanding many physical realities.

 

[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.

 

[baffledMatt]

The products of conjugate variables is more certain

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.

In statistics, the square of the variance can make a lot of sense for finding the mean and standard deviation of all measurements.

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.

The power of area principle as in the theory of surfaces is what makes mathematics such a useful knowledge for understanding many physical realities.

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

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

This too.

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

Matt

 

[Antonio Lao]

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

 

[sol2]

Watching from the sidelines and learning:)

 

[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.

 

[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.

 

[Antonio Lao]

baffledMatt,

I'm not sure how it gives you the mean though.

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!

 

[Antonio Lao]

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.

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.

 

[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

Ok, fair point.

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

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

 

[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

 

[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.

 

[PFanalog57]

Squaring a function is basically a type of integration, finding the area of a function, etc.

derivative[e^x] = e^x

antiderivative[e^x] = e^x + C

If C = 0, then derivative[e^x] = integral/antiderivative[e^x]

So is it possible to express the Heisenberg Uncertainty Principle in terms of infinitely differentiable functions, and could it tell us anything ?

 

[PFanalog57]

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

Interesting...

Which is more fundamental? Product or sum?

A + B = C is really a product.

A + B = [A^(1/2)+i*B^(1/2)]*[A^(1/2)-i*B^(1/2)] = C

 

[matt grime]

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.

but you are evidently an uneducated crank, if you can even begin to explain what the hell iti s you mean to understand you'll be half way to gettin an answer

 

[Antonio Lao]

matt grime,

Crank or not, I want to understand the true physical meaning of mass and electric charge. Can you convince me this is a lost cause?

 

[Antonio Lao]

Janitor,

I really don't think physical constants of nature (e.g. Planck's constant) can be chosen at will. You knew there are a lot of these constants according to the Anthropic Cosmological Principle.

If you change one, you have to change them all!

 

[Antonio Lao]

Is

x^2 - 1^2 \equiv 1^2 - x^2?

for all value of x between neg and pos infinity.

 

[matt grime]

Is

x^2 - 1^2 \equiv 1^2 - x^2?

for all value of x between neg and pos infinity.

as one is minus the other, they are only equivalent mod 2, though presumably you are working in the reals, when they are definitely not equivalent as letting x be any other number than +/-1 shows.

earlier you also wrote X^2+1=2, ie x=+/-1 again.

 

[Antonio Lao]

they are only equivalent mod 2

Thanks. Could you explain further the above?

 

[Janitor]

Hello Antonio.

You seemed to be saying that squaring h leads to something smaller, if I understood you. In units of joule-seconds, h=6.626 x 10^-34.

In units of femtojoule-femtoseconds (femto being 10^-15), h=6.626 x 10^-4.

If I knew Greek (?) better, I could tell you what the prefix is for 10^-18. Since I don't know what it is, allow me to call it a 'gomer.' Then in units of gomerjoule-gomerseconds, h=626. Obviously if you square 626, you get a larger number, 391,876, where now you are in units of (gomerjoule-gomerseconds) squared.

My apologies if this has nothing to do with your line of thought.

 

[Antonio Lao]

Janitor,

I am looking for the physical meaning of h^2. We agree that the number is smaller but can it means also that the uncertainty is lesser? If h is raised to the fractional power then the result is a bigger number than h.

Oh another thing, how would you make a dimensionless number such as the fine structure constant (\frac {1}{137}) bigger or smaller?

Maybe what you did above is called rescaling. I think, there are things in physics that are not scalable. I can not give an example at the moment.

 

[Janitor]

We agree that the number is smaller but can it means also that the uncertainty is lesser? - Antonio

Actually, my point is that the number is only smaller if you choose units in which h<1. But there is nothing profound about choice of units, is there? Units in which h>1 are just as acceptable.

 

[Antonio Lao]

If your choice of unit is the same as a scale factor then there is big different. Given an area A and a scale factor s, the new area is given by s^2A. If s is smaller (s < 1) the new area is smaller. Bigger s (s > 1), bigger area.

The unit of h is actually angular momentum. So by squaring h we are actually gettting something square of angular momentum. What I want to know is what's the physical meaning of square of angular momentum? Somebody, maybe it was Russell, said that product is the same as addition (integration) if some logarithmic (exponential) function is involved like e^x.

 

[Janitor]

Antonio,

I don't have any useful thoughts at the moment on the issue of "how would you make a dimensionless number such as the fine structure constant bigger or smaller?" or "what's the physical meaning of square of angular momentum?"

To keep hammering away at my point, how would you critique this discussion between Peter and Paul?

Peter: "The mass of the Earth is really huge. It is 6 x 10^24 kg. When you square that, you get an even bigger number."

Paul: "On the contrary, the mass of the Earth is small. It is only 7 x 10^-7 in units of solar mass. When you square that, you get an even tinier number."

It seems to me that Peter and Paul are both misguided in thinking that the mass of the Earth can be fundamentally classified as either "big" or as "small."

 

[Antonio Lao]

Janitor,

You got a good point. Yours looks like reasoning behind the relativity of number among fractions and whole numbers.

A physical law with this kind of relativeness is the Law of Universal Gravitation.

F = G \frac{m_1 m_2}{R^2}

and Coulomb law

F = K \frac{q_1 q_2}{R^2}

This gravity force is effective, I think, whenever there is disparity in the mass (m1 >>> m2 , vice versa) or charge.

When m1=m2 such as the structure of a positronium, I cannot understand how the forces of gravity and electricity keep the electron and postiron apart (without crushing into each other)? What is it that keeps them apart? What force?
__________

The other is relativeness of viewing distances. The moon appears the size of my fist when I extend toward it because the fist is much nearer to me than the moon.

 

[matt grime]

re mod 2

Thanks. Could you explain further the above?

you wish fo A to be equivalent to -A, presuming you mean in some arithmetical sense, that implies 2A=0, thus you are in a field of characeristic 2, or similar

 

[Antonio Lao]

matt grime,

Thank you very much.