Did Grandi's Series mathematically prove the superposition of particles?

[Themohawkninja]

I recently watched a Numberphile video on Grandies Series, which states that: If you calculate 1+1-1+1-... there are three possible answers: Either the answer is 1 because 1+(-1+1)+(-1+1)+..., or the answer could be 0 because (1-1)+(1-1)+..., or the answer could be 1/2 (which I didn't understand, but it has to do with setting the series to equal S, and somehow he got S = S-1, which turned into 2S-1 = 0, which turns into 2S = 1, and therefore S = 1/2).

There is a mathematical example of this, where if you do an experiment whereby you flip a light switch after one minute, than you flip it again after 1/2 of one minute, than 1/4 of one minute and so on and so forth until two minutes is reached, you will have flipped the switch an infinite amount of times. What state is the light in? Is it on, off, something inbetween, or both at the same time?

Now, in quantum physics, there is something known as the superposition of particles, which states that an electron can be in all possible states at the same time, which sounds like the real-life example of the aforementioned paradox. This makes me beg the question, was this quantum mechanic mathematically proven centuries ago without anyone realizing that it happens in reality?

[WestAir]

There is a mathematical example of this, where if you do an experiment whereby you flip a light switch after one minute, than you flip it again after 1/2 of one minute, than 1/4 of one minute and so on and so forth until two minutes is reached, you will have flipped the switch an infinite amount of times.

I have a hard time believing such an event can become "infinite". The idea that there are an infinite amount of "moments" in time, between each moment in time, is dubious at best. In your light switch experiment, we know there is a limit to the speed information can travel, and we know there is a size where nothing can get smaller, so to this I suggest that you'll flip your switch faster and faster until you've reached a point where you must move faster than the speed of light to cut your time in half any further. [Let's just pretend that the frequency of switches is equal to the time it takes a photon to move one Plank Length] At this point your half-cutting will taper off to stagnation, and simple math will be able to tell you how many times you can do this in two minutes, and what position the switch will be in when you stop.

[Themohawkninja]

I have a hard time believing such an event can become "infinite". The idea that there are an infinite amount of "moments" in time, between each moment in time, is dubious at best. In your light switch experiment, we know there is a limit to the speed information can travel, and we know there is a size where nothing can get smaller, so to this I suggest that you'll flip your switch faster and faster until you've reached a point where you must move faster than the speed of light to cut your time in half any further. [Let's just pretend that the frequency of switches is equal to the time it takes a photon to move one Plank Length] At this point your half-cutting will taper off to stagnation, and simple math will be able to tell you how many times you can do this in two minutes, and what position the switch will be in when you stop.

Mathematical paradoxes defy physics for the sake of showing the paradox as an analogy. The experiment is just an analogy to show that if you try and get to the number 2 by adding 1 + 1/2 + 1/4 +... you will get to two, but it will take an infinite number of additions to get there.

[Xeldrak]

Well, it's not a paradox. This series (-1)^n simply does not converge. (I'm seriously missing a formula editor or LaTeX support)

The solution to 1+1-1+1-1+1-.... simply does not exist, the series fails the so-called Leibnitz-Test for example.

Also, watch out when traslating thoughts like this to reality. In reality, the switch has to remain in a certain postion for a minimum amount of time, the so called Planck time (yes, time is "granular". This is also your way out of Zeno's paradoxes). Therefore the switch is in a definite postion at every point in time. No infinity here...

Also, mathematics never prove that something exists in reality. It is possible to show mathematicaly, that something may exist in reality. But the test if reality "decides to actualize" something is not in the realm of maths. With maths and physics you can show, that the Higg-Boson MIGHT exist. But we needed the LHC to see if it's really there.

Edited August 2, 2013 by Xeldrak

[Themohawkninja]

Well, it's not a paradox. This series (-1)^n simply does not converge. (I'm seriously missing a formula editor or LaTeX support)

The solution to 1+1-1+1-1+1-.... simply does not exist, the series fails the so-called Leibnitz-Test for example.

Also, watch out when traslating thoughts like this to reality. In reality, the switch has to remain in a certain postion for a minimum amount of time, the so called Planck time (yes, time is "granular". This is also your way out of Zeno's paradoxes). Therefore the switch is in a definite postion at every point in time. No infinity here...

Once again, as I stated above, mathematical paradoxes defy physics to show an analogy. I am well aware of the existence of an indivisible unit of time, and I know about Zeno's paradox, which again defies a bit of physics to show a mathematical problem. (Plus, I think that superposition might actually show up at near-Planck times, as once you try and measure movements in Planck time, would you not have to take into account other quantum mechanics?)

[Xeldrak]

Again, it's not a mathematical paradox.

And no, while I see that there are some kind of similarities in the imagination of superposition and (nonexistent) solution of Grandi's Series, there is no actual connection.

Look at this:

For every "n" it has a definite solution, wich is either 1 or 0. If we "plug in Infinite" (wich actually means we look at the limit n->infinity), this thing has no solution. It kinda breaks.

The expesion never has "two superimposed solutions"

[Themohawkninja]

Again, it's not a mathematical paradox.

And no, while I see that there are some kind of similarities in the imagination of superposition and (nonexistent) solution of Grandi's Series, there is no actual connection.

Look at this:

For every "n" it has a definite solution, wich is either 1 or 0. If we "plug in Infinite" (wich actually means we look at the limit n->infinity), this thing has no solution. It kinda breaks.

The expesion never has "two superimposed solutions"

Ah... I see what you mean now.

[Xeldrak]

So, all questions answered?

[Themohawkninja]

So, all questions answered?

Yep, you seem to have mathematically shown that the Grandi Series does not result in superposition (which can be shown mathematically), but rather that the equation just... breaks.

[Xeldrak]

Well, if stuff like this interests you, you should visit the nearest calculus lecture. The you will get a better understanding, why it breaks and what this means

[Yourself]

Well, it's not a paradox. This series (-1)^n simply does not converge. (I'm seriously missing a formula editor or LaTeX support)

The solution to 1+1-1+1-1+1-.... simply does not exist, the series fails the so-called Leibnitz-Test for example.

Oh, but it's actually so much more fun than that. There isn't just one way to interpret an infinite series. We don't have to limit ourselves to saying that an infinite series is the limit of its partial sums, we can do so many different things. This always kind of reminds me of how we came up with complex numbers. We had polynomials that had no roots, which feels like something's just kind of missing, so we invented the complex numbers to fill those out.

The same kind of thing is happening here. We have infinite sums that aren't convergent and so don't really have any sort of meaningful value. But we can extend our methods of computing infinite sums to "complete" these. Of course at this point you kind of have to ask yourself what it really means to have an infinite sum.

[Ben 9072]

Semi-metaphysical musings aside, you would probably be interested in doing some reading about the Casimir effect. It has a rather... interesting... derivation revolving around evaluating the zeta function outside the limits of where it should be evaluated, and would probably be a better analogy to mathematical paradoxes impacting physics than your idea about the lightbulb and quantum states. (Which is slightly flawed, as a superpositioned state doesn't really "oscillate" like a lightbulb turning on and off. It's literally the vector sum of the two wavefunctions existing at once. Your observation on the system forces the wavefunction to collapse to a stable state, but while undisturbed, a superpositioned state will behave as both states simultaneously.)

Anyway, you've got some interesting ideas. You would probably enjoy calculus and linear algebra a lot. MIT has very good opencourseware resources, if you're interested. (http://ocw.mit.edu/index.htm)

[Themohawkninja]

Semi-metaphysical musings aside, you would probably be interested in doing some reading about the Casimir effect. It has a rather... interesting... derivation revolving around evaluating the zeta function outside the limits of where it should be evaluated, and would probably be a better analogy to mathematical paradoxes impacting physics than your idea about the lightbulb and quantum states. (Which is slightly flawed, as a superpositioned state doesn't really "oscillate" like a lightbulb turning on and off. It's literally the vector sum of the two wavefunctions existing at once. Your observation on the system forces the wavefunction to collapse to a stable state, but while undisturbed, a superpositioned state will behave as both states simultaneously.)

Anyway, you've got some interesting ideas. You would probably enjoy calculus and linear algebra a lot. MIT has very good opencourseware resources, if you're interested. (http://ocw.mit.edu/index.htm)

The act of the light switching isn't where the comparison comes from, it's the uncertainty of the end state that is where I draw my comparison. IIRC, the Casimir effect has to do with when you put two metal sheets really close together, there is an observable vacuum?

[Ben 9072]

The act of the light switching isn't where the comparison comes from, it's the uncertainty of the end state that is where I draw my comparison. IIRC, the Casimir effect has to do with when you put two metal sheets really close together, there is an observable vacuum?

Okay, that is a more valid comparison, I was misunderstanding your analogy. And the Casimir effect is when you have two conductive plates with no net EM field between them in a vacuum that can generate a positive or negative pressure by being in close proximity to each other.

[Diche Bach]

I am baffled and beguiled . . . need another cup 'a Joe and I'll go back and start at the beginning . . .

[natias]

I am baffled and beguiled . . . need another cup 'a Joe and I'll go back and start at the beginning . . .

i saw this youtube numberphile video and thought of quantum mechanics and the idea that measuring a spin of particle would have an immediate impact on the wave function of the particle and also on another particle very far away (sorry for not being accurate, i only read about this stuff in the net). I thought about a series that continues both ways ...,-1,1-1,1,-1,1,-1,... which is a wave function of sort, so the sum of the series can be a (1,-1) dual state and once you decide where to start the sum (measure it) you cause the entire wave function to collapse into a single state.

[AntonioX]

On 8/2/2013 at 1:57 PM, Xeldrak said:

Again, it's not a mathematical paradox.

And no, while I see that there are some kind of similarities in the imagination of superposition and (nonexistent) solution of Grandi's Series, there is no actual connection.

Look at this:

For every "n" it has a definite solution, wich is either 1 or 0. If we "plug in Infinite" (wich actually means we look at the limit n->infinity), this thing has no solution. It kinda breaks.

The expesion never has "two superimposed solutions"

An observation. I'm also vouching for something for an unique realtion between Grandi's Series and the value 1/2. Beside those videos in Numberphile, I stumbled upon the following Taylor Series of 1/(1-x). Which then can let have the following series for x in (0,2):

1/x = 1 + (1-x) + (1-x)^2 + (1-x)^3 + (1-x)^4 + ...

Of course, as it is definied, x=2 is not permissible. Yet when we take the limit of x as it approaches 2, you get something very similar to Grandi's Series. I'm not quite sure if we definitely  say  

1/2 = 

But I did find this quite interesting.

[wumpus]

On 1/16/2014 at 11:21 AM, natias said:

i saw this youtube numberphile video and thought of quantum mechanics and the idea that measuring a spin of particle would have an immediate impact on the wave function of the particle and also on another particle very far away (sorry for not being accurate, i only read about this stuff in the net). I thought about a series that continues both ways ...,-1,1-1,1,-1,1,-1,... which is a wave function of sort, so the sum of the series can be a (1,-1) dual state and once you decide where to start the sum (measure it) you cause the entire wave function to collapse into a single state.

I highly recommend numberphile, but they tend to assume you've already taken calculus and would like a different look at it.  On the other hand, the last 10? or so episodes were made at the high school level (for kids in quarantine).  If this is new to you, you might want Khan academy supplemented by numberphile.

I was thinking Blue Blue Brown.  Not that numberphile isn't great, but my description is for Blue Blue Brown.

Edited June 8, 2020 by wumpus

[K^2]

Also, an aside from mathematics, superposition has nothing to do with this. The reason superposition exists is because the wave function is an object in Hilbert space, not Cartesian space. It's a very simple statement to make and much harder one to explain, but the bottom line is that when you are picturing quantum waves as something similar to waves on a surface of the pond, you're not using the correct abstraction. And unfortunately, even quantum mechanics textbooks don't do a good job explaining this, let alone anything in popularized science.

[SOXBLOX]

Absolutely. If you're looking for a good read on physics, though, run, don't walk, and get a copy of Edward Teller's Conversations on the Dark Secrets of Physics. It is both readable and accurate; that is, it uses math. I loved it, hopefully others will find it useful.

[wumpus]

4 hours ago, K^2 said:

Also, an aside from mathematics, superposition has nothing to do with this. The reason superposition exists is because the wave function is an object in Hilbert space, not Cartesian space. It's a very simple statement to make and much harder one to explain, but the bottom line is that when you are picturing quantum waves as something similar to waves on a surface of the pond, you're not using the correct abstraction. And unfortunately, even quantum mechanics textbooks don't do a good job explaining this, let alone anything in popularized science.

Not sure what you are describing (in Hilbert space), but waves simply have a lot of the behavior attributed to "waveform collapse".  This was the Blue Blue Brown video I was thinking of when I got "numberphile" wrong:

https://www.youtube.com/watch?v=MBnnXbOM5S4

While I can't vouch for the physics (beyond the basics of the uncertainty principle), I am familiar with the math he is talking about and it gets weirder the deeper you go down (it commonly comes up in "windowing functions" in Fourier transforms (and related transforms) in DSP work).  Waves simply don't have both frequency and position arbitrarily defined, and it doesn't take any additional dimensions for them to [not] do so.

[K^2]

5 hours ago, wumpus said:

Waves simply don't have both frequency and position arbitrarily defined, and it doesn't take any additional dimensions for them to [not] do so.

Yeah, but uncertainty principle isn't where QM gets interesting. In general, things that a single particle does in QM are kind of boring, as it's all just normal wave behavior. And yeah, if you understand signal processing, you basically have one-particle QM covered. Shrodinger Equation might look scary at the first glance, but it's just a wave equation for particle with mass, which simply changes dispersion relation. That is, the wave group no longer has to propagate at the speed of light. Things start heating up when you add a second particle. Indeed, discussing superposition is kind of irrelevant until you have at least two particles interacting. We have simple and intuitive understanding of how polarization of light works, and we can simply think of superposition as coordinate system rotation. Indeed, for single particle, that's all it is. But the moment you have two photons whose polarizations are entangled, things get a little wild, and your classical thinking of polarization as just the orientation of electric field simply fails to work.

[magnemoe]

On 8/2/2013 at 8:33 PM, Xeldrak said:

Well, it's not a paradox. This series (-1)^n simply does not converge. (I'm seriously missing a formula editor or LaTeX support)

The solution to 1+1-1+1-1+1-.... simply does not exist, the series fails the so-called Leibnitz-Test for example.

Also, watch out when traslating thoughts like this to reality. In reality, the switch has to remain in a certain postion for a minimum amount of time, the so called Planck time (yes, time is "granular". This is also your way out of Zeno's paradoxes). Therefore the switch is in a definite postion at every point in time. No infinity here...

Also, mathematics never prove that something exists in reality. It is possible to show mathematicaly, that something may exist in reality. But the test if reality "decides to actualize" something is not in the realm of maths. With maths and physics you can show, that the Higg-Boson MIGHT exist. But we needed the LHC to see if it's really there.

Zeno's paradox is anyway pretty fake since its you who divide time into smaller and smaller factions, yes this is often done, the best example might be an rocket launch with an checklist who starts days before but steps get more numerous as you get closer to liftoff, even an lots of automated ones between ignition and liftoff. Reading the list and that part has to many steps to be read out in the two seconds between ignition and release but the launch is real time.