Infinite series in quantum phyics
Re: Infinite series in quantum phyics
@chriwi
While it's fun to speculate, and I'm not sure I understand your idea, ultimately we'll find out when some quantum gravity theory is experimentally proven... BUT Lorentz invariance (which is experimentally very well established) does rule out a lot of things.
@ansgar:
I see what you're getting at. In fact what you are talking about is a theory on a lattice. There, positions are not continuous and so there is no derivative (but an approximation to it). But so far at least these approaches are only an approximation of the full quantum theory, useful mainly for numerical computations.
An example that maybe will help: Despite quantized energies, you still have the energytime uncertainty principle, which says that measured energy values are always going to be somewhat "fuzzy"... and quantum theory describes just this behaviour.
While it's fun to speculate, and I'm not sure I understand your idea, ultimately we'll find out when some quantum gravity theory is experimentally proven... BUT Lorentz invariance (which is experimentally very well established) does rule out a lot of things.
@ansgar:
I see what you're getting at. In fact what you are talking about is a theory on a lattice. There, positions are not continuous and so there is no derivative (but an approximation to it). But so far at least these approaches are only an approximation of the full quantum theory, useful mainly for numerical computations.
An example that maybe will help: Despite quantized energies, you still have the energytime uncertainty principle, which says that measured energy values are always going to be somewhat "fuzzy"... and quantum theory describes just this behaviour.
 chriwi
 LHCPortal Guru
 Posts: 403
 Joined: Fri Nov 20, 2009 1:19 pm
 Location: Stuttgart Germany
 Contact:
Re: Infinite series in quantum phyics
@photino
whath I mean is maybe the whole universe is one big wavefunction (it surly is), but what I mean that this one contains all possibilities of Quantumhistory (for the past and for the future) and can never collaps because ther is no outside observer to probe and thereby collaps it. And maybe the fact if a smaller part of this function is collapsed or not is not the same answer to every being or even to every particle, but just relative to the piont of view of the person or partticle which probed it or not. This would let us get rid of the question who or what has to have the knolage about a certain quantumstat of an object to collaps a wavefunction because there is no really collapsed or not collapsed wave function true for all but only relative for each partigle or whatrever.
whath I mean is maybe the whole universe is one big wavefunction (it surly is), but what I mean that this one contains all possibilities of Quantumhistory (for the past and for the future) and can never collaps because ther is no outside observer to probe and thereby collaps it. And maybe the fact if a smaller part of this function is collapsed or not is not the same answer to every being or even to every particle, but just relative to the piont of view of the person or partticle which probed it or not. This would let us get rid of the question who or what has to have the knolage about a certain quantumstat of an object to collaps a wavefunction because there is no really collapsed or not collapsed wave function true for all but only relative for each partigle or whatrever.
bye
chriwi
chriwi
Re: Infinite series in quantum phyics
Mathematics, even when used to describe reality, does not share properties of reality it is describing.ansgar wrote:In calculus it is common to add very small distances as far as I know. Distances will be quickly smaller than a Planck distance. Doesn't this mean that you cannot use calculus to compute the position of an object? Doesn't it also mean that it is wrong to say that the geometric series converges to a single value? Won't every competent physicist assert that mathematics cannot be used to precisely account for the minutia of movement?
Let's take an example similar to the Zeno paradox  but with a twist. The athlete is running 1 yard, then 1/2, then 1/3, then 1/4, then 1/5 and so on. This is known as harmonic series and it is known that its sum is infinite. But it is growing very slowly so if we stop at N=2^128 (that number is bigger than number of particles in known universe), he is still not even one mile away from start and at this point we are already adding lengths way shorter than Planck length. What would you guess where is he going to end then?
In fact, equations used for quantum mechanics have already the uncertaininty principle built into them, inability to calculate sum of some particular series with them does not create the uncertainity  in worse cases it just means you can't get any result at all, in better cases if you can find an approximation with finite error, this error only adds to the real uncertainity.
One of beauties of mathematics is that series can be often rearranged and partially summed to obtain different series that calculate the same but might be easier to calculate. So if we are adding values under planck length in one series, we may be adding yards in other, refined series. That's how the proof for harmonic series is done, it is rearranged in the manner in which it is clear that the result is bigger than the one of 1 + 1 + 1 + 1 + 1 + ... series.

 Posts: 10
 Joined: Tue Feb 23, 2010 7:39 pm
 Location: Glossop, England
Re: Infinite series in quantum phyics
Errm, if 2^128 is a bigger number than the count of particles in the universe, does it have any meaning?
If I count a flock of sheep by dropping a stone on a pile for every sheep that goes out of the fold I can count them back in by taking a stone off the pile for each returning sheep. At the end of the process I can have sheep outside with no stones left (I've collected some strays), I can have stones left but no sheep returning (some sheep are lost, so better go find them) or there can be no sheep and no stones (so I can relax 'cos all sheep are accounted for).
Numbers (integers) are just conventional labels for how many stones there are in such a pile. But the number of particles in the universe sets a top limit on how high I can count that way. Do 'bigger' numbers then have any meaning? Perhaps this works like the inverse of the planck length  you can't be too big or too small without losing contact with physical reality?
If I count a flock of sheep by dropping a stone on a pile for every sheep that goes out of the fold I can count them back in by taking a stone off the pile for each returning sheep. At the end of the process I can have sheep outside with no stones left (I've collected some strays), I can have stones left but no sheep returning (some sheep are lost, so better go find them) or there can be no sheep and no stones (so I can relax 'cos all sheep are accounted for).
Numbers (integers) are just conventional labels for how many stones there are in such a pile. But the number of particles in the universe sets a top limit on how high I can count that way. Do 'bigger' numbers then have any meaning? Perhaps this works like the inverse of the planck length  you can't be too big or too small without losing contact with physical reality?
Re: Infinite series in quantum phyics
Sure it has. In mathematics it's just a number, numbers are not limited by size of the universe.PeteKropotkin wrote:Errm, if 2^128 is a bigger number than the count of particles in the universe, does it have any meaning?
And in reality, it's (for example) number of all possible values you can store in a 128bit register. You can start using such numbers when you move from unary number base (pile of stones) to binary (128 slots with [1] or without [0] a stone).
In mathematics you may need to "go around" pretty big numbers sometimes to get later back to "normal sized" numbers in the result. For instance, if you have 128 slots and just two stones, the number of ways you may put these two stones into these slots equals 128!/(2! x (1282)!). 128! is pretty big number too and I'm pretty sure you would never get so many stones either, yet the result is "only" 8128, and it's quite possible to try them all manually.

 Posts: 10
 Joined: Tue Feb 23, 2010 7:39 pm
 Location: Glossop, England
Re: Infinite series in quantum phyics
BUT you don't need such a bignum to do that calculation. The first stone has 128 slots to choose from, the second has only 127, and there are 2 ways if choosing which is the first stone. 128*127/2 = 8128.In mathematics you may need to "go around" pretty big numbers sometimes to get later back to "normal sized" numbers in the result. For instance, if you have 128 slots and just two stones, the number of ways you may put these two stones into these slots equals 128!/(2! x (1282)!). 128! is pretty big number too and I'm pretty sure you would never get so many stones either, yet the result is "only" 8128, and it's quite possible to try them all manually.
Perhaps you mistake the formula n!/r!*(nr)! for the underlying reality of what is going on?
Last edited by PeteKropotkin on Sat Mar 20, 2010 9:18 pm, edited 1 time in total.
Re: Infinite series in quantum phyics
No, that was because my example was just a simple one. My point is that in mathematics it's irrelevant.PeteKropotkin wrote:BUT you don't need such a bignum to do that calculation. The first stone has 128 slots to choose from, the second has only 127, and there are 2 ways if choosing which is the first stone. 128*127/2 = 8128.
Perhaps you mistake the formula n!/r!*(nr)! for the underlying reality of what is going on?
If you try that with 1000 slots and 300 stones you'll have pretty hard time avoiding big numbers but if you just calculate the formula without worrying whether numbers you are just using are 'real' or 'unreal' you'll get the result pretty quickly.
It reminds me of times when I was implementing algorithms on old computers where I needed to multiply and divide numbers and needed to avoid overflow in the process. The problem here was not that number over 65535 is unreal, just that it didn't fit in a register. The algorithm was much more complicated because of that but the result was exactly the same as with mathematics which don't need to care about register size.
Re: Infinite series in quantum phyics
I would like to add a very simple example to Kasuha's argumentation that the validity of mathematical theorems is independent of physics.
c²=25 has two solutions: c=5 and c=5. Now lets take a right triangle with legs of the length 3 and 4. Pythagoras tells us that the length of the hypotenuse c is given by c²=3²+4²=25. So by only using math we expect to find a hypotenuse with the length 5 and one with the length 5. Yet in physics negative lengths don't have any meaning so we discard 5 and are left with 5 as our physical solution.
By the way: The use of complex numbers (, which are often very handy to simplify ones calculation,) follows simmilar rules. Start of somewhere meaningful (e.g. a differential equation) > go crazy with the math > interpret your result back to s.th. meaningful (discard the imaginary part).
I find it quite astonishing that a single tool such as math has been found to be that powerful when describing nature. Yet one has to acknowledge that physics is more then just applied math and that the underlying mechanism of nature (, which we only have a very faint idea of) is most likely not to be a mathematical clockwork. So whenever using math one has to always check whether ones tools and the results derived are appropriat.
If you want some further input on the topic have a look at the Feynman Messenger Lectures on the nature of physical laws: http://research.microsoft.com/apps/tools/tuva/
c²=25 has two solutions: c=5 and c=5. Now lets take a right triangle with legs of the length 3 and 4. Pythagoras tells us that the length of the hypotenuse c is given by c²=3²+4²=25. So by only using math we expect to find a hypotenuse with the length 5 and one with the length 5. Yet in physics negative lengths don't have any meaning so we discard 5 and are left with 5 as our physical solution.
By the way: The use of complex numbers (, which are often very handy to simplify ones calculation,) follows simmilar rules. Start of somewhere meaningful (e.g. a differential equation) > go crazy with the math > interpret your result back to s.th. meaningful (discard the imaginary part).
I find it quite astonishing that a single tool such as math has been found to be that powerful when describing nature. Yet one has to acknowledge that physics is more then just applied math and that the underlying mechanism of nature (, which we only have a very faint idea of) is most likely not to be a mathematical clockwork. So whenever using math one has to always check whether ones tools and the results derived are appropriat.
If you want some further input on the topic have a look at the Feynman Messenger Lectures on the nature of physical laws: http://research.microsoft.com/apps/tools/tuva/

 Posts: 10
 Joined: Tue Feb 23, 2010 7:39 pm
 Location: Glossop, England
Re: Infinite series in quantum phyics
If you try that with 1000 slots and 300 stones you'll have a long job ahead of you! there are 542825004640614064815358503892902599588060075560435179852301016412253602009800031872232761420804306539976220810204913677796961128392686442868524741815732892024613137013599170443939815681313827516308854820419235457578544489551749630302863689773725905288736148678480Kasuha wrote: No, that was because my example was just a simple one. My point is that in mathematics it's irrelevant.
If you try that with 1000 slots and 300 stones you'll have pretty hard time avoiding big numbers but if you just calculate the formula without worrying whether numbers you are just using are 'real' or 'unreal' you'll get the result pretty quickly.
It reminds me of times when I was implementing algorithms on old computers where I needed to multiply and divide numbers and needed to avoid overflow in the process. The problem here was not that number over 65535 is unreal, just that it didn't fit in a register. The algorithm was much more complicated because of that but the result was exactly the same as with mathematics which don't need to care about register size.
different permutations!
At 1 per second that is 1.72010864147e+256 years
That is about 1.255e+246 times the age of the universe
I think you will agree that the count is unfeasable!
Once upon a time, a very long time ago, when I was the new programmer on the team we got talking about difficulttoprogram jobs we had done. The #2 guy explained he'd had a ****ish time with a particular compoundinterest problem. He' been expanding n!/((nr)!*r!) naively, using floats to hold the (very large) intermediate results and scaling things up and down to retain precision. It had taken him several months to get the calculations to work. I did the job it 10 minutes! How? Example  Perm 300 from 1000 is 1000!/(700!*300!): that is (...(1000*999)/2 * 998) /3... *701 /300 )
All stages have integer result, full accuracy is held, not much danger of overflow on 'ordinary' sized problems.
(BTW, I cheated above  Python is VERY GOOD at bignums! But I did check the answer...)
Conclusion: You don't need universeoverflow scales to calculate ordinary results!
Last edited by PeteKropotkin on Mon Mar 22, 2010 6:52 pm, edited 1 time in total.

 Posts: 10
 Joined: Tue Feb 23, 2010 7:39 pm
 Location: Glossop, England
Re: Infinite series in quantum phyics
But.. Pythagoras's theorem is on the geometric squares constucted on the sides of a rightangled triangle. If the triangle is ABC, with the rightangle at A, it doesn't matter if you take the hypotenuse as BC (length 5) or CB (length 5). he constructed square is the same either way. So the theorem expresses the physical reality that direction is irrelevant when constructing squares.c²=25 has two solutions: c=5 and c=5. Now lets take a right triangle with legs of the length 3 and 4. Pythagoras tells us that the length of the hypotenuse c is given by c²=3²+4²=25. So by only using math we expect to find a hypotenuse with the length 5 and one with the length 5. Yet in physics negative lengths don't have any meaning so we discard 5 and are left with 5 as our physical solution.
I seem to recall that the'other' solution to the wave equation used to be regarded as irrelevant, not physical. Then some genius started wondering about present events being influenced by future results....
Discard the 'imaginary' results and avoid the prize  figure out what the result means and you may discover a new interpretation of reality  a.k.a 'doing physics'!
Re: Infinite series in quantum phyics
Length is defined as the norm of a vector. This vector can of cause have negative compontens but the length is always nonnegative!PeteKropotkin wrote:If the triangle is ABC, with the rightangle at A, it doesn't matter if you take the hypotenuse as BC (length 5) or CB (length 5).

 Posts: 10
 Joined: Tue Feb 23, 2010 7:39 pm
 Location: Glossop, England
Re: Infinite series in quantum phyics
Err, haven't you just redefined the problem? I'm talking geometry (Euclidean, even  though we can be Cartesian if you prefer) and you're talking vector analysis.Length is defined as the norm of a vector. This vector can of cause have negative compontens but the length is always nonnegative!
I wish I could redefine banking theory so that my current account balance was always nonnegative!
Re: Infinite series in quantum phyics
Geometry is realy just vector analysis that avoids the word vector so to simplify. All definitons from vector notation also apply to geometry.
When you take a ruler to measure the length of BC or CB it will in both cases be 5! You can even turn the ruler around .it doesn't matter if you take the hypotenuse as BC (length 5) or CB (length 5)

 Posts: 10
 Joined: Tue Feb 23, 2010 7:39 pm
 Location: Glossop, England
Re: Infinite series in quantum phyics
This is futile! Whatever I say you can redefine away. You redefine geometry as vector analysis. You gain some simplicity. Maybe you lose something, too? Like the distinction of lines headed in different directions?oxodoes wrote:Geometry is realy just vector analysis that avoids the word vector so to simplify. All definitons from vector notation also apply to geometry.
When you take a ruler to measure the length of BC or CB it will in both cases be 5! You can even turn the ruler around .it doesn't matter if you take the hypotenuse as BC (length 5) or CB (length 5)
However, have you ever heard of the number line? Starts WAY to the left, marked as negative numbers. Zero is in the middle. Positive numbers go WAY to the right. On that line I can construct squares. For example, I mark off 10 to 15. Then I erect a square on that part of the number line. And I can mark the line from 15 down to 10 and erect an equivalent square. Maybe that is a square on a line headed in the negative direction? That is, it can be defined to be a square on a negative length?
Re: Infinite series in quantum phyics
Is there a good book to start studying quantum physics? I'd like to get a good understanding of quantum physics. Which books should I read, and in what order? I don't study physics, but I have an excellent understanding of high school physics, if that's any help. If there are any prerequisites to studying quantum physics, please list what they are.