# Heisenburg's Uncertainty Principal and the definition of randomness

• cbetanco

#### cbetanco

I started this thread to continue a conversation from another thread. Does HUP imply randomness? What really is the definition of randomness?

I would say that HUP does not imply randomness, but just tells you the relationship between how the state vector is spread out along the basis of two non commuting operators (positions and momentum in this case). Randomness only comes in when we try to measure the value of a particular observable, that is when the state vector collapses along an eigenstate with the appropriate probability.

I was under the impression that a random event was one that is uncorrelated to previous events, and this is how I would define randomness; the inability to correlate a certain event with any past events.

I started this thread to continue a conversation from another thread. Does HUP imply randomness? What really is the definition of randomness?

I would say that HUP does not imply randomness, but just tells you the relationship between how the state vector is spread out along the basis of two non commuting operators (positions and momentum in this case). Randomness only comes in when we try to measure the value of a particular observable, that is when the state vector collapses along an eigenstate with the appropriate probability.

I was under the impression that a random event was one that is uncorrelated to previous events, and this is how I would define randomness; the inability to correlate a certain event with any past events.

I already made the comment that randomness does not necessarily imply a uniform probability distribution so the correlation with previous events / states would not necessarily be zero. That is, the outcome of QM 'random' situations often (always?) has a degree of correlation with the past. In fact, would not a deterministic model be the only model with a correlation of unity between initial and final states? (Correlation does not just have two possible values.)

The HUP makes it uncertain what the initial state of a system would be (if you had actually measured it) and, if you didn't measure its initial state, it would be uncertain what the final state was. I can't see how that could be described as deterministic so the alternative has to be random, no?

It seems to me that the issue is whether A).one 'believes' that there is an actual and precise state vector for any system but we just can't measure it exactly or B) one believes that there is not, actually, a precise value for the state vector of any system in the first place.

All this does seem to me to relate to practical QM situations. For example, take the wave function which describes an electron which is bound to an atom. Its energy level is very well defined so its 'position' is not. We would describe the likely position of an electron in such a bound state just in terms of a probability density function. There is your randomness. You could say that how the electron will interact with some external influence (say a passing electron) would depend on 'where' the electron could be considered to be in its position around the atom with respect to this incoming electron. I am deliberately using a quasi mechanical description which is nonsense, I know. But why is it nonsense? It is nonsense because there is no precise state vector so we can't have an orbiting bullet but a random distribution of possible positions.

Uncertainty and randomness seem, to me, to be almost synonymous.

I started this thread to continue a conversation from another thread. Does HUP imply randomness? What really is the definition of randomness?

I would say that HUP does not imply randomness, but just tells you the relationship between how the state vector is spread out along the basis of two non commuting operators (positions and momentum in this case). Randomness only comes in when we try to measure the value of a particular observable, that is when the state vector collapses along an eigenstate with the appropriate probability.

I was under the impression that a random event was one that is uncorrelated to previous events, and this is how I would define randomness; the inability to correlate a certain event with any past events.

"Random" simply means "unpredictable." It is a matter of opinion what degree of unpredictability you will accept.

"the inability to correlate a certain event with any past events" is called a Markov condition by probabilists.

The idea in QM is that we know the distributions of various random variables. If the distributions were unknown then the situation would be "more random." If we knew that the distributions could never be known that would be "very random," a real anarchy. But that is not the case. We can specify the distributions precisely. But they are still random variables. All you can predict is the probabilities of results.

So your point of view makes sense, it is just different from the definitions that mathematicians use.

Uncertainty and randomness seem, to me, to be almost synonymous.

Yes. Perhaps the difference is that randomness is "unpredictability," but once the event has occurred you are certain what happened. In QM we can't be certain about the past either: we can't know what happened in detail. So randomness concerns the future, uncertainty is about both the past and future.

I started this thread to continue a conversation from another thread. Does HUP imply randomness? What really is the definition of randomness?

I would say that HUP does not imply randomness, but just tells you the relationship between how the state vector is spread out along the basis of two non commuting operators (positions and momentum in this case). Randomness only comes in when we try to measure the value of a particular observable, that is when the state vector collapses along an eigenstate with the appropriate probability.

I was under the impression that a random event was one that is uncorrelated to previous events, and this is how I would define randomness; the inability to correlate a certain event with any past events.

A definition of randomness was given in the other thread. The HUP does not imply randomness. This is why we have random forces in classical formalisms (e.g., Langevin), for the which the HUP vanishes (h=0).

A definition of randomness was given in the other thread. The HUP does not imply randomness. This is why we have random forces in classical formalisms (e.g., Langevin), for the which the HUP vanishes (h=0).

Wouldn't a 'classical' formalism be ultimately deterministic, though?

Wouldn't a 'classical' formalism be ultimately deterministic, though?

Why?

Fair question. I can see that I am getting something 'not too right' here but I can't see where, just yet.
My problem is that I feel there would have to be a big 'if' with anything based on a classical model which made an assumption of randomness and it couldn't represent a 'reality'. But I don't subscribe to the existence of a 'reality' in any case. So I am in a bit of a quandry. I shall have to do some more reading and thinking.

"Random" simply means "unpredictable." It is a matter of opinion what degree of unpredictability you will accept.

"the inability to correlate a certain event with any past events" is called a Markov condition by probabilists.

The idea in QM is that we know the distributions of various random variables. If the distributions were unknown then the situation would be "more random." If we knew that the distributions could never be known that would be "very random," a real anarchy. But that is not the case. We can specify the distributions precisely. But they are still random variables. All you can predict is the probabilities of results.

So your point of view makes sense, it is just different from the definitions that mathematicians use.

I would not use that definition of a Markov process. The Markov condition really says that you can predict the future events from knowledge of current events and there is no need for knowledge of past events (usually framed as, you only need 1 transition probability, for 1 time-step, in order to predict the full probabilistic evolution of the process). It's not "impossible" to correlate a certain even with "any" past event. Certainly the autocorrelation functions of very well known Markov processes are not delta functions (see, e.g. the Wiener Process, or the Ornstein Uhlenbeck Process). Markov processes are simply defined such that they obey the Chapman-Kolmogorov equation.

I believe the process you are looking for is so called "white noise", or a "Martingale process", in which there really is no correlation with ANY past events.

With regards to randomness in classical mechanics. You certainly can impose randomness in classical mechanics, but it's kind of artificial. For example, in Langevin's equation, you simply add a "random process" to the differential equation. This has to do with the model. If you were able to keep track of EVERY particle in the ENTIRE universe (and we lived in a classical universe where you could do this), then there IS NO randomness in a classical model. However, if you don't care to do this, you may add a certain "randomness" in by hand (as Langevin did) to obtain a good probabilistic model of the process you are interested in. For example, in Brownian motion, you really don't care about the motions of the water molecules which impact on your Brownian particle. Therefore, you just model those impacts as a probabilistic event.

I have understood the single slit demonstration of uncertainty as follows:

Even if the initial state of every particle leaving the gun is identical, the angle of diffraction of any single particle CANNOT be predicted. This implies that the angle of diffraction is independent of the initial state. The angle cannot be predicted based on current, or prior information and is thus fundamentally indeterminate and random.

How is this logic faulty or insufficient?

I started this thread to continue a conversation from another thread. Does HUP imply randomness? What really is the definition of randomness?

I would say that HUP does not imply randomness, but just tells you the relationship between how the state vector is spread out along the basis of two non commuting operators (positions and momentum in this case). Randomness only comes in when we try to measure the value of a particular observable, that is when the state vector collapses along an eigenstate with the appropriate probability.

I was under the impression that a random event was one that is uncorrelated to previous events, and this is how I would define randomness; the inability to correlate a certain event with any past events.
I essentially agree with you. Randomness refers to experimental unpredictability. The uncertainty relations have to do with the relationship between canonically conjugate variables, which involves the formalism of qm as well as experimentally accumulated statistics.

I essentially agree with you. Randomness refers to experimental unpredictability. The uncertainty relations have to do with the relationship between canonically conjugate variables, which involves the formalism of qm as well as experimentally accumulated statistics.

So HUP is only 'part' of the randomness thing. Or is it the basis of it?
You see, people bandy around words like "random" and they all seem to mean different things. But wherever I see examples of randomness in QM, they seem to boil down to HUP, in some way (your "canonically conjugate variables"). And, despite the 'experimental' descriptions / explanations come with HUP, is there not something more fundamental than just the measurement thing involved? Is that not why QM is such a big step from classical Physics?

We can say, though, can't we, that no classical process with non-random input states could produce a random result.

I will agree that no classical process with non-random inputs can produce a random result. But I still do not think HUP is what is the random part of QM, but the wave function collapse into a definite eigenstate with a certain probability.

From wikipedia's Randomness page: "Applied usage in science, mathematics and statistics recognizes a lack of predictability when referring to randomness, but admits regularities in the occurrences of events whose outcomes are not certain. For example, when throwing 2 dice and counting the total, we can say 7 will randomly occur twice as often as 4. This view, where randomness simply refers to situations in which the certainty of the outcome is at issue, is the one taken when referring to concepts of chance, probability, and information entropy. In these situations randomness implies a measure of uncertainty and notions of haphazardness are irrelevant."

Which is consistent with my view that the actual measurements (outcome) are random, not the fact that two observables do not commute! But, I don't know if this is the definition of randomness that we want to agree upon (I am always so hesitant to use wikipedia for defining things exactly)

You are right. Wikipedia is not a 'discussion', so it doesn't do so well with this sort of thing.

I think it is interesting that a word like 'randomness' has so many interpretations. But I suppose, if there were to be just one word that would be problematical, then it could be 'random'.

But your suggested model of a die is not a fair comparison with QM. Either you are discussing a totally non-physical, Mathematical Die (which could have whatever properties you would choose it to have) or you are dealing with a Classical die (which would be totally predictable, given enough knowledge of initial conditions). Is there not something fundamentally different about a QM system? I am of the opinion that merely not knowing enough about the initial state of a (classical) system, through lack of the right equipment, is entirely different from those limitations imposed by HUP. Even if we bring in the idea of Chaos, I don't think that would help.

I am also not sure that the idea of a 'collapsing wave function' is really the answer. I think I have seen treatments of Young's Slits using HUP - though I can't give a reference - and I am sure that any situation could be reduced to one where HUP is, in fact the limiting factor.

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You are right. Wikipedia is not a 'discussion', so it doesn't do so well with this sort of thing.

I think it is interesting that a word like 'randomness' has so many interpretations. But I suppose, if there were to be just one word that would be problematical, then it could be 'random'.

But your suggested model of a die is not a fair comparison with QM. Either you are discussing a totally non-physical, Mathematical Die (which could have whatever properties you would choose it to have) or you are dealing with a Classical die (which would be totally predictable, given enough knowledge of initial conditions). Is there not something fundamentally different about a QM system? I am of the opinion that merely not knowing enough about the initial state of a (classical) system, through lack of the right equipment, is entirely different from those limitations imposed by HUP. Even if we bring in the idea of Chaos, I don't think that would help.

I am also not sure that the idea of a 'collapsing wave function' is really the answer. I think I have seen treatments of Young's Slits using HUP - though I can't give a reference - and I am sure that any situation could be reduced to one where HUP is, in fact the limiting factor.

"Random" is a catch-all word and different people use it differently in different contexts. There are many words like that. If you want to be more specific then you have to use words like "stochastic" or "Markov." Probabilists would stay that "collapse of the wave function" would be "instantiation of a random variable."

It's probability if you know what the distribution of the random variables are. It's statistics if you don't know the distribution and all you have is data and a possible model to explain its variation. So QFT is probability, measurement error is statistics.

My understanding is that the modern view is that we really do have a random variable that instantiates and measurement error is an entirely different issue.

Mathematicians have been studying probability since Pascal so it seems classical to me, but "classical" is another one of those words that depends on who is using it. To physicists it ain't classical.

I think it is quite likely that some in the future the human race will learn more about the processes that result in these random variables. But as Yogi Berra said, "Prediction is very hard, especially when it is about the future."

Ok let me rephrase from the previous thread:

Casual randomness= an action that has too many variables to predict the outcome.
An action with a source of energy that will react accordingly but from our frame of reference and tools we can't measure good enough to predict the outcome.

True randomness = an action without a source of energy,without a cause.
Action and reaction doesn't apply.

determinism= everything has an action and reaction.
Irregardless of our frame of reference and our tools of measurement.

Most of you imply that QM uses the "true" random way to function, which is contrast to physics and logic.

So HUP is only 'part' of the randomness thing. Or is it the basis of it?
Insofar as one accepts that the word 'random' refers to unpredictability, and insofar as HUP defines a quantitative relationship between variables limited by the assumption of the existence of a fundamental quantum, then I don't see how HUP is "'part' of the randomness thing" or "the basis" of randomness.

You see, people bandy around words like "random" and they all seem to mean different things.
Afaik, the word 'random' means 'unpredictable'.

But wherever I see examples of randomness in QM, they seem to boil down to HUP, in some way (your "canonically conjugate variables").
Wherever I see examples of randomness, whether wrt to QM or not, they seem to refer to unpredictability.

And, despite the 'experimental' descriptions / explanations come with HUP, is there not something more fundamental than just the measurement thing involved? Is that not why QM is such a big step from classical Physics?
Afaik, qm differs from classical physics because of the quantum hypothesis, which is the assumption of the existence of a fundamental quantum.

I would suppose that everybody likes to think that there's something more fundamental "than just the measurement thing involved". Unfortunately, all there is to go on is "the measurement thing", and any assumptions beyond that are based on inferences from that. Which is not to say that some assumptions aren't really good assumptions, imho.

We can say, though, can't we, that no classical process with non-random input states could produce a random result.
Not sure what you mean here. Let's say there's a card game involving two players, A and B, and a dealer. Let's further say that the dealer has looked at every card in the deck before he deals. So, the 'input state', the deal, is nonrandom as far as the dealer is concerned, but random as far as A and B are concerned.

Wrt quantum experimental phenomena, emissions are inferred to be random when detection events are observed to be random. Predictable statistical results, however, are not necessarily referred to as random. For example, wrt, say, optical Bell tests (ie., quantum entanglement), the accumulation of individual detection attributes is random, but the predictable quantitative result associated with a global measurement parameter isn't random and suggests a relationship (the qualitative nature of which is unknown) between underlying or 'hidden' quantum-level parameters.

I started this thread to continue a conversation from another thread. Does HUP imply randomness?
Afaik, no.

What really is the definition of randomness?
Afaik, it refers to unpredictability. So, what's random for one person, might not be random for another person.

I would say that HUP does not imply randomness ...
I agree.

I was under the impression that a random event was one that is uncorrelated to previous events, and this is how I would define randomness; the inability to correlate a certain event with any past events.
But can't every event be correlated to some past event? Anyway, afaik, wrt the way the word 'randomness' is used it simply means unpredictablitly.

I am taking on board what's in the previous two posts but I still have a problem:

ThomasT;3636405 Not sure what you mean here. Let's say there's a card game involving two players said:
I would say that the randomness described in the above model (a typical one used in these arguments and a good example to discuss) is a 'classical' randomness. Yes, one card player may not know what is in the other player's hand but he could set up an experiment in which he could know that information in a future game (mirrors, cctv et.). And there exists someone who knows the information.
Even if you introduce the effect of noise into the classical example, there is still the possibility of getting information out of a 'random' situation by doing the appropriate sort of measurement.
otoh, in the QM world, no-one can know the value of 'the other variable' in the HUP pair. Its value, I would say, is of a different order or class of randomness to that in the card-playing example.

Just how relevant that distinction is, may be a good question but I really think that there is one. Do you guys see the point I'm making?

I am taking on board what's in the previous two posts but I still have a problem:

I would say that the randomness described in the above model (a typical one used in these arguments and a good example to discuss) is a 'classical' randomness. Yes, one card player may not know what is in the other player's hand but he could set up an experiment in which he could know that information in a future game (mirrors, cctv et.). And there exists someone who knows the information.
Even if you introduce the effect of noise into the classical example, there is still the possibility of getting information out of a 'random' situation by doing the appropriate sort of measurement.
otoh, in the QM world, no-one can know the value of 'the other variable' in the HUP pair. Its value, I would say, is of a different order or class of randomness to that in the card-playing example.

Just how relevant that distinction is, may be a good question but I really think that there is one. Do you guys see the point I'm making?

Sure. In one case the situation is unpredictable but would be predictable given more information. The the other case such information is unavailable, as far as anyone knows.