Thursday, February 21, 2008

Quantum Mechanics And Classical Physics: The Limit Relation

I am not qualified to discuss neither classical physics nor modern physics (aka quantum mechanics) but I have some knowledge that I would like to share that should make things easier to understand to make things clearer to an "average Joe".

In short: Classical Physics is the limit to infinity of Quanta in Quantum Mechanics. Thats in a nutshell.

For people who are not familiar with the mathematical concept of limits consider this example:
Say, F(X)=1-5*10^-X (one minus five times ten to the power of minus X). In this example:
F(0)=-4
F(1)=0.5
F(2)=0.95
F(3)=0.995
F(4)=0.9995
F(infinity)=1

This is the concept of a limit, notice that for small values of X, say between 0 and 3 the value of the function changes dramatically. But from the values of 4 and above the change is negligible. Consider how much F(4) is close to one!!

So in Quantum Mechanics the formulas are more complex than Classical Physics, but when the number of Quanta is large, certain parts of the equation can be discarded since they are negligible. Just like how in "F(X)=1-5*10^-X" we can discard the "-5*10^-X" since its almost zero for large values of X. The concept of limits apply when moving from Quantum Mechanics to Classical Physics.

Another hot topic is the probabilistic model of Quantum Mechanics. While we cannot be certain of the outcome of Quanta in overly simplistic scenarios, this does not apply to large numbers of Quanta. Its the same concept of limits.

Consider the following scenario:
Say X is a variable that initially has the value of 0. We throw a die, and we add the value of the die to X. So lets see what happens to X after 3 throws.
Say, the outcome of the die was: 3 / 3 / 6
X=3+3+6=12
Another experiment has this outcome: 4 / 1 / 2
X=4+1+2=7
Yet a third with this outcome: 5 / 6 / 4
X=5+6+4=15


It seems that X is different for each of these trials... This is an analogy to what happens with Quanta. Each Quanta produces a different stochastic (random) behavior. And the behavior of a system of Quanta is the sum of all the behavior of Quanta at that moment.

Consider now the above example with a billion throws. While the outcome of 3 trials might have large differences in each experiment, as the number of trials increases the error margin decreases. The the most basic property of probability. The more the trials the more stable results you will get.

In the above example, I can say that the average outcome of the die is (1+2+3+4+5+6)/6 = 3.5
So in one billion trials, the outcome should be very close to 3.5 billions.

The conclusion is that although when we consider Quanta in isolation they might produce diverse results, studying large numbers of Quanta is much easier and easily predictable with high confidence.

It is worth noting that one proton contains: 2.26*10^23 Quanta!! As you can see, even a single subatomic part contains a huge number of Quanta!! So even for a single atom assuming that the number of Quanta in it is infinite is a reasonable approximation in limit equations!

11 comments:

lost within said...

Awesome!
Quantum physics rocks! although i did it for only one semester in school !

Qwaider قويدر said...

The first example has some interesting results if X was negative. And even more interesting results if X had i dimension in it. Not the bravest of mathematicians would dare to say that it converges absolutely at infinity.

For the second example. It's really not all that great of an example of probability. Because of every throw of the die we have a chance of 6, two dice, chance of six, three dice, chance of six. The probability remains static, 1 of 6. Combining them side by side doesn't mean you increased the randomness. If you throw the dice 1 billion times, you get a number between 6 billion, and 36billion. With a bill curve shape of higher probability around 21billion. Do it a trillion times. Same result around 21 trillion. A quadrillion times same result around 21 Quadrillion. You know as a fact that you can not go below 6xnumber of throws or 6 above. So there's a good nice bell-shaped curve there :)

And Lost Within,
Quantum mechanics SUCKS, It's the ugliest form of science known to man. With equations (based on probability) as long as 30 or 40 book pages. Compare that to the elegance of E=MC².

Funniest thing about Quantum mechanic was.. If only a handful of people understood relativity correctly, then NO ONE understands Quantum mechanics :)

I'm more of an elegant universe person. Although Quantum mechanics might prove to be a good step on the long way of finding the right (aka elegant) equations

(Elegant, not to be confused with intelligent design)

Devil's Mind said...

Sorry, but thats mathematically wrong Qwaider. Its true that throwing a die 1 million times theoretically gives a value between 6 millions and 36 millions. But the expected value is NOT 21 millions, thats simply mathematically wrong!! The expected value is 3.5 millions.

And no, its NOT bell-shaped [The outcome is NOT a normal distribution - X is not even a continuous value, it is a discreet (integer) value - So it is rejected without even have to think about it!! Also the normal distribution goes from negative infinity to positive infinity. Obviously X only extends to a limited boundary, thats another conclusive indication that X is not normal! X has Binomial distribution]

Not the bravest of mathematicians would dare to say that it converges absolutely at infinity. Braveness has nothing to do with the subject at hand. Limits is an established science and its results are established facts.

Back to the die experiment. I don't want to go in the deepest depths of probability but I will explain to you why your over-simplistic vision of the center value is wrong.

Consider only TWO dies.
First question is: How many different outcomes are possible?! The answer is 36.
Now how many ways can the sum add up to 2?! One: (1+1)
How many ways can you get a 12?! One: (6+6)
How many ways can you get a 7?! Six: (1+6) (2+5) (3+4) (4+3) (5+2) (6+1)

So the outcome of 7 is six times more likely than the outcome of 2!!

Apply same logic on 3, 4, and more...

I used a computer program to produce six results for a 1000 trials, here is the output:
3572; 3496; 3530; 3526; 3490; 3424
as you can see the outcome is too close to the theoretical 3500. The maximum deviation was with 3424 with a difference of 76.
76/3500*100%=2% ... the deviation was only 2% for 1000 trials!!

I used the same program but for a 1000000 trials, this is the outcome:
3501128; 3498648; 3499590; 3502530; 3500180; 3495977
The theoretical outcome is 3500000
Here the maximum deviation is with 3495977 with a difference of 4023.
4023/3500000*100%=0.1%

As you can see, the error margin in expectations for large numbers of trials decreases dramatically. That is 2% for 1000, while only 0.1% for 1000000.

Quantum mechanics SUCKS. - Science does not suck - Period.

The Observer said...

DM, have i told u that i love u before? :)

THANK YOU! I am really an average joe here, and you did a great job of explaining quanta to me. I understood your post fully! hehehe

I am not sure if my following assumption is right or not:

Can we consider the probability of measurments of quanta to be due to our lack of proper definition of it or having the tools to do so due to its very tiny size?

Tala said...

The concept of limits apply when moving from Quantum Mechanics to Classical Physics

in theory yes, but its not seen yet. there is a gap between both classic Physics and Quantum physics and its still looking for common grounds

Devil's Mind said...

Man, you're welcome. Anyway I can help :)

Can we consider the probability of measurments of quanta to be due to our lack of proper definition of it or having the tools to do so due to its very tiny size? - Scientists and philosophers are working on this problem. A few scientists (including Einstein) suggest that the answer is yes. Although I don't deny that the answer can be "no", I think it is too hasty to say "no"! Quantum Mechanics is still relatively a new science, and breakthroughs are still expected to happen.

Many scientists have diverse hypothesizes about how to interpret their results. In No angel's article he talked about the numerous interpretations. So as you can see, there is still a lot of debate on the subject, and we have to wait until more evidence to come up so that we can say this or that with confidence.

Devil's Mind said...

Tala, who said that?

It is a necessary conclusion that Quantum Mechanics leads to Classical Physics!! Science cannot contradict itself!

Tala said...

Yes true, but according to scientists classical physics out of quantum theory has not been entirely achieved yet by anybody

Tala said...
This comment has been removed by the author.
Devil's Mind said...

If this is true, then the logical conclusion is that Quantum Mechanics in its current form is not entirely accurate. So if either one has to be true, it is more logical to say that Classical Physics is correct and Quantum Mechanics is inaccurate, because Classical Physics is a well-established science, but Quantum Mechanics is the one that needs criticism.

Classical Physics has more credibility because there are already applications that use its theories, and all are tangible results... So Classical Physics has followed the scientific method closely. But Quantum Mechanics studies things that scientists can really detect or measure with high accuracy... It might work in certain areas, but would fail in other areas!

No_Angel said...

It might work in certain areas, but would fail in other areas!
that can be applied to anything, thats why special relativity was deemed as more accurate than classical which in turn was over thrown by general relativity since it handled more cases than all the others... so it's only natural to view classical as a limited model :D

As for your example ... hmm i won't even go there so we wont be stuck in another 50+ post