## Infinite series in quantum phyics

### Infinite series in quantum phyics

I've asked a similar question about the mathematics of QM before, but I still need some clarification from the experts. So bear with me.

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?

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?

- chriwi
- LHCPortal Guru
**Posts:**403**Joined:**Fri Nov 20, 2009 1:19 pm**Location:**Stuttgart Germany-
**Contact:**

### Re: Infinite series in quantum phyics

I guess if it was not true what you write there we could not get any macroscopic prove of quantum-physical effects at all, but we get lik the collaps of the wave pattern on the screen only by observation, what is not predictet only by usin calculus, so you must be right.

bye

chriwi

chriwi

### Re: Infinite series in quantum phyics

I don't quite get the answer. Was it meant to be ironic?

- chriwi
- LHCPortal Guru
**Posts:**403**Joined:**Fri Nov 20, 2009 1:19 pm**Location:**Stuttgart Germany-
**Contact:**

### Re: Infinite series in quantum phyics

was not exactly meant to be ironic, but only means that it is true that pure calculus especially infinitesimal calculus for the very small doesn't work for quantum physical effects.

Actually this is the reason why Einstein got the first ide wich later lead others to the quantum theory. There was a problem in calculating the phatoelectrical effect which would lead to infinit high energies when you use normal calculus and assume that energy can be devided int smaller and smaller portions without any lower limit. Since reality showed that the energy is not indefinitely high Einstein did not give up admitting that he just could not find the solution, but rather started to question his assumtions (of the continuousely dividable energy), actually he did this in the same way as later for his theory of relativity where he could not find any solution without giving up the asumption that the flow of time and the length of objects muet be equal regardless of the point ov view of the observer.

Actually this is the reason why Einstein got the first ide wich later lead others to the quantum theory. There was a problem in calculating the phatoelectrical effect which would lead to infinit high energies when you use normal calculus and assume that energy can be devided int smaller and smaller portions without any lower limit. Since reality showed that the energy is not indefinitely high Einstein did not give up admitting that he just could not find the solution, but rather started to question his assumtions (of the continuousely dividable energy), actually he did this in the same way as later for his theory of relativity where he could not find any solution without giving up the asumption that the flow of time and the length of objects muet be equal regardless of the point ov view of the observer.

bye

chriwi

chriwi

### Re: Infinite series in quantum phyics

So this is the reason why no competent physicist will use calculus, and why the theorems of calculus are essentially, deep down, wrong. Thanks. I just wanted to know whether you can use mathematics to describe the motion of objects, and you seem to confirm that you cannot.

- chriwi
- LHCPortal Guru
**Posts:**403**Joined:**Fri Nov 20, 2009 1:19 pm**Location:**Stuttgart Germany-
**Contact:**

### Re: Infinite series in quantum phyics

I did not say that one cannot use mathematics to describe quantum physics and also the mathematical theorems are not wrong themselfes only some of them like one quoted by you dividing everything into smaller and infinitesimal smaller portions cannot be used for Quantumphysics and produce wrong predictions when they are used there, but ther are other fields of mathematics which work fine especially for quantumphysics for example matrixcalculations which never go in infinitesimal portions but only dealing with whole numbers.

bye

chriwi

chriwi

### Re: Infinite series in quantum phyics

Doesn't this means that geometric series do not converge. Take for example Zeno paradoxes. The traditional solution assumes that the runner takes steps smaller than the smallest possible distance. It is therefore impossible to compute to which value a series converges, since it by definition get in to distances smaller than possible. Or to use calculus to determine when when two trajectories intersect. Because it fundamentally assumes a one-to-one correspondence on small scales that is just not given in reality, isn't it.

### Re: Infinite series in quantum phyics

Not sure I understand what is being asked here, but

1) Quantum mechanics is based on calculus. Of course physicists use calculus.

2) Yes, in quantum mechanics, you cannot calculate "the position" of something, but you can calculate quantities like the expectation value of the position (the average position, averaged over many measurements). This is just like in statistics.

3) If there is a smallest lengthscale (the Planck length), that does NOT mean that you don't have to (or can't) calculate with smaller lengths. It just means you won't ever be able to resolve (measure) shorter distances experimentally.

4) Regarding Zeno: It is just a fact that the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + ... is equal to 1. (This is made precise by the mathematical theory of limits)

1) Quantum mechanics is based on calculus. Of course physicists use calculus.

2) Yes, in quantum mechanics, you cannot calculate "the position" of something, but you can calculate quantities like the expectation value of the position (the average position, averaged over many measurements). This is just like in statistics.

3) If there is a smallest lengthscale (the Planck length), that does NOT mean that you don't have to (or can't) calculate with smaller lengths. It just means you won't ever be able to resolve (measure) shorter distances experimentally.

4) Regarding Zeno: It is just a fact that the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + ... is equal to 1. (This is made precise by the mathematical theory of limits)

Last edited by photino on Mon Feb 22, 2010 1:11 am, edited 1 time in total.

### Re: Infinite series in quantum phyics

May I ask if you are a real physicist or merely a mathematician, and if I can quote you on the claim that you can calculate with smaller lengths, even though you are not able to measure them.

- chriwi
- LHCPortal Guru
**Posts:**403**Joined:**Fri Nov 20, 2009 1:19 pm**Location:**Stuttgart Germany-
**Contact:**

### Re: Infinite series in quantum phyics

Numbers themselfes are not bound to quantumrestrictions and hypothetical experiments like the runner with the smaller and smaller steps are only for visualisation but would fail if they would really be carried out near planklength by measurement, evenso they are still true for numbers.

But as I said before: just theese theoremes with the smaller and smaller fractions really cannot be used for quantum physics when the smaller and smaller fractions represent length or amounts of energy. Statistics is not using theese smaller and smaller fraction theorems and thereby can be used for quantumphysicalcalculations.

Without the problem stated here Einstein would not have found the first hint leading to quantumphysics.

But as I said before: just theese theoremes with the smaller and smaller fractions really cannot be used for quantum physics when the smaller and smaller fractions represent length or amounts of energy. Statistics is not using theese smaller and smaller fraction theorems and thereby can be used for quantumphysicalcalculations.

Without the problem stated here Einstein would not have found the first hint leading to quantumphysics.

bye

chriwi

chriwi

### Re: Infinite series in quantum phyics

So it the true after all that Einstein proved that calculus is wrong.

### Re: Infinite series in quantum phyics

You put your finger on one of the "philosophical" difficulties of quantum mechanics: the state of any system is described by means of a wavefunction. This wavefunction is not directly observable. People have long argued over whether it should be considered "real"...

Regarding minimum lengthscales, I forgot to mention that there are in fact some people working on theories of quantum gravity which do impose a hard cutoff in some fashion (in a sense more like what you were suggesting). Personally I don't think these theories are likely to work (roughly speaking because in relativity, lengths are not fixed, but observer-dependent - so what looks like a large distance to one observer can appear much shorter to another), but we will see... The generic situation in quantum mechanics is certainly along the lines of what I said earlier (for instance the famous uncertainty relation, which also imposes limits on what is measurable).

Einstein did NOT prove calculus wrong! He used it all the time. Where did you get that idea??

To answer your other question, I studied physics, but ultimately you shouldn't really trust #RANDOM_GUY on a webforum... a decent popular science book on quantum mechanics would perhaps give you more understanding, now (hopefully) you know what to look for.

Regarding minimum lengthscales, I forgot to mention that there are in fact some people working on theories of quantum gravity which do impose a hard cutoff in some fashion (in a sense more like what you were suggesting). Personally I don't think these theories are likely to work (roughly speaking because in relativity, lengths are not fixed, but observer-dependent - so what looks like a large distance to one observer can appear much shorter to another), but we will see... The generic situation in quantum mechanics is certainly along the lines of what I said earlier (for instance the famous uncertainty relation, which also imposes limits on what is measurable).

Einstein did NOT prove calculus wrong! He used it all the time. Where did you get that idea??

To answer your other question, I studied physics, but ultimately you shouldn't really trust #RANDOM_GUY on a webforum... a decent popular science book on quantum mechanics would perhaps give you more understanding, now (hopefully) you know what to look for.

- chriwi
- LHCPortal Guru
**Posts:**403**Joined:**Fri Nov 20, 2009 1:19 pm**Location:**Stuttgart Germany-
**Contact:**

### Re: Infinite series in quantum phyics

Einstein did not proof calculus wrong, but only showd up an experiment which could not be explained by calculus of the smaller and smaller fractions without assumining that there is a lower limit of the energy of a portion of light what cannot be made smaller anymore without disaprearing completely.

Besides that the rows adding up the smaller and smaller are only a tiny part of calculus.

@photino:

the relativity of the plank length should not be a reason not to belive in it, maybe relativity goes even further and the quantumphysical fact if the wavefunction is collapsed or not is also relativ to the point of view of the observer like length, time, acceleration and gravity in the general relativity. That idea would also solve the problem of asking who or what must have observed a particle to colaps the wave function, because then it is always relative to the point of view and can be true or fals for every single link aof the chain, so shrödingers cat can be dead for an observer sitting in the box , but still neither dead or allive for another one outside who not yet opened the the box.

Besides that the rows adding up the smaller and smaller are only a tiny part of calculus.

@photino:

the relativity of the plank length should not be a reason not to belive in it, maybe relativity goes even further and the quantumphysical fact if the wavefunction is collapsed or not is also relativ to the point of view of the observer like length, time, acceleration and gravity in the general relativity. That idea would also solve the problem of asking who or what must have observed a particle to colaps the wave function, because then it is always relative to the point of view and can be true or fals for every single link aof the chain, so shrödingers cat can be dead for an observer sitting in the box , but still neither dead or allive for another one outside who not yet opened the the box.

bye

chriwi

chriwi

### Re: Infinite series in quantum phyics

It has been said in this tread a few times that theorems that involve smaller and smaller fractions cannot be used and are incorrect. And if I recall correctly, derivatives and integrals are limits that take smaller and smaller fractions. This could mean that it is useless, given the experimental results of quantum physics on cut-off values.photino wrote: Einstein did NOT prove calculus wrong! He used it all the time. Where did you get that idea??

- chriwi
- LHCPortal Guru
**Posts:**403**Joined:**Fri Nov 20, 2009 1:19 pm**Location:**Stuttgart Germany-
**Contact:**

### Re: Infinite series in quantum phyics

Hi ansgar,

I cannot comment on your consideration of differential and integral, at leas I agree that they use small fractions going towards 0 and might not be fully aplicable for all quantumphysical problems.

But keep in mind most of the calculation in quantum physics is done with matrix- and tensor-operations and not so much with integral.

I cannot comment on your consideration of differential and integral, at leas I agree that they use small fractions going towards 0 and might not be fully aplicable for all quantumphysical problems.

But keep in mind most of the calculation in quantum physics is done with matrix- and tensor-operations and not so much with integral.

bye

chriwi

chriwi