Undergrad Quantum mechanics is random in nature?

[entropy1]

I'm not sure I get it. In what way, in the entanglement setup, is there nonlocality?

For instance: if the photons get in a (weighted) superposition of passing and not passing their polarizer, can this superposition of states (or its information) persist and propagate all the way to the top where the information is joined and the outcome is produced by that? Is there any nonlocality that way?

 

[The Bill]

You're making things difficult for yourself. With entanglement experiments, we're just measuring a property that the two entangled particles have been forced to share in common. It's not much more mysterious than cutting two slivers off a block of metal, transporting them to different locations, and discovering that they're made of the same type of metal as each other.

I suggest you watch some of Leonard Susskind's lectures on entanglement on Youtube, study the problem for a while in your textbooks, then come back and see if you understand it a bit better.

 

[entropy1]

I suggest you watch some of Leonard Susskind's lectures on entanglement on Youtube, study the problem for a while in your textbooks, then come back and see if you understand it a bit better.

Ok. You're in good company saying a thing like that to me. Apparently I was not as clear as I hoped to be. Maybe I do not understand this matter. Thanks for the insight.

 

[StevieTNZ]

The correlation of photons both passing their polarizers would be 100%, right? Of course we can't predict if the photons are going to pass, but we know that if one has done so, the other will do too! (in this setup)

I, and most, would hope so. But this is inductive reasoning. [see pgs 189 - 191, "Quantum Enigma" by Bruce Rosenblum and Fred Kuttner (2nd edition)]

 

[entropy1]

Why does nobody in the physics world care about the random/non-random question?

 

[vanhees71]

Because, it's pretty obvious that this is not a physics question. To the overwhelming high-accuracy evidence the behavior of matter is described by quantum theory, and the physical part of its interpretation, i.e., the minimal interpretation, linking the elements of the mathematical formalism ($C^*$ algebra on Hilbert space, to say it in an quite abstract way) to the real-world observables (cross sections of scattering processes, atomic, molecular and nuclear spectra, condensed-matter phenomena,...) is probabilistic. Further, thanks to Bell's work it is also a physical question, whether you can mimic this probabilistic behavior with a local deterministic hidden-variable theory, and the answer is a clear no. Again the overwhelmling high-accuracy evidence shows that the corresponding Bell inequality is violated precisely in the way as predicted by quantum theory, and since there is no consistent non-local hidden variable theory compatible with Einstein causality from a physicist's (who is not spoiled by thinkgin about socalled "deep philosophical problems" ;-)) point of view it's a clear case that according to present overwhelming evidence the world is intrinsically and irreducibly probabilistic.

 

[entropy1]

@vanhees71 Is QM random in different, unrelated ways? Then I would see it is a fundamental property of the quantumworld.

I still got no answer to this, of which I would very much like some respons from someone if possible.

I'd like to understand! It is not so obvious to me why I should just shut up and calculate. That just doesn't seem to be so much fun to me, though I would probably get to it if I knew why I should!

 

[vanhees71]

What do you mean by "random in different, unrelated ways"? It's random in the usual sense of random. According to quantum theory an observable of a quantum system have either determined values due to a preparation in a corresponding quantum state or this observable has no determined value, and the state tells you the probability for measuring a certain value of this observable and nothing else.

Entanglement means that you have a quantum system with parts that can be far appart with each other showing very strong correlations, stronger correlations than possible than in any local deterministic theory. This is the content of the fact that Bell's inequality is violated with high precision, and it is precisely violated in the way as predicted by quantum theory.

In physics (or the natural sciences) "to understand" means to be able to explain a phenomenon from what has been determined to be fundamental laws of nature. These laws can not be further "understood" in the sense that you can derive them from even more fundamental laws. Of course, what you call a fundamental law may change when one finds new evidence, like in the early 1900s, where it became very clear that classical electrodynamics cannot describe the spectrum of thermal radiation ("black-body spectrum"), and Planck discovered quantum theory (in a very rough preliminary form, which was developed further to modern quantum theory in 1925/26). In this sense you can only take the fundamental laws as formulated in modern quantum theory as a condensed form of our knowledge about how nature behaves. There's no "deeper understanding" to it. We can just learn the formalism and how to apply it to new experiments and observations to test its validity in more and more detail. Perhaps one day one finds a discrepancy, and then one has to find some new even more fundamental theory adapted to this new evidence. That's how science works, and Nature is not there to give you fun. It just is as it is (in my opinion there's still a lot of fun in learning about it using mathematics and the natural sciences).

 

[DrChinese]

I'd like to understand! It is not so obvious to me why I should just shut up and calculate.

What other choice is there on this point? You are welcome to look for the underlying cause*. Look anywhere you like, and let us know when you find one.*As if no one thought of that previously, and already came up short - and you were already told this.

 

[entropy1]

Ok. I think I get it now. However, I remain that it is extremely unsatifying to me that QM has no philosophical interpretation*. However, I am not the one to find it if it would exist. So I think I just have to throw the towel in.

* Having equally qualified candidates that will never be resolved is the same thing to me.

 

[vanhees71]

There are tons of philosophical interpretations of QT. Theyr are only not of any relevance from a natural-science point of view.

 

[Nugatory]

What does one understand if anything after studying QM? Is there some answer to that?

No, but there's nothing special about QM in that regard.

Consider what happens with classical gravitation. We start by investigating such diverse phenomena as falling objects, thrown stones, the atmospheric pressure, the motion of the planets. We don't understand any of them, we follow many false paths, and eventually Isaac Newton discovers the law of gravity. One equation, $F=Gm_1m_2/r^2$, explains everything and we understand gravity in all of its varied manifestations...
Except that then some clever high school kid who should be doing her exercises and calculating the speed of a dropped object interrupts her teacher to ask "What makes the objects want to move together? What's going on that makes $F=Gm_1m_2/r^2$ work so well? What's the reason the math works?". The answer is going to be some variant of of "It works. Experiments prove it. Shut up and calculate, you still haven't finished your exercises".

What's different here is that classical gravitation fits in well enough with our common sense that once we see how well it works we tend to accept it without digging deeper. QM, on the other hand, is counterintuitive enough to provoke that "yes, but why?" question, and a feeling of deep dissatisfaction when no answer is forthcoming.

 

[Zafa Pi]

Whatever the future might show, it looks hard to get rid of randomness as mathematical concept.

Randomness is not a mathematical concept. There are random variables (functions) and probabilities, but no definition of random. Sometimes people use the word random to refer to a finite valued r.v. with equal probabilities, e.g. a random coin flip, meaning heads and tails each have probability 1/2.

 

[fresh_42]

Randomness is not a mathematical concept. There are random variables (functions) and probabilities, but no definition of random. Sometimes people use the word random to refer to a finite valued r.v. with equal probabilities, e.g. a random coin flip, meaning heads and tails each have probability 1/2.

Replace it by stochastic.

 

[Zafa Pi]

Replace it by stochastic.

If you look up stochastic in an English dictionary it will say (essentially) random. There are stochastic processes like random variables, but stochastic is no more defined in math than random.
I'm not trying to be picky. Terms like random or unpredictable are intuitive, but are too nebulous to pin down mathematically. There are some that say random means there is no algorithm that gives its value. But there is no algorithm that generates the busy beaver function, but few would call it random. That physicists often use the term random I don't find problematic, anymore than when they refer to reality. But reality isn't a math term either.

 

[David Lewis]

...can this superposition of states (or its information) persist and propagate all the way to the top where the information is joined and the outcome is produced by that?

It could be that in quantum processes, some information is lost along the way. Otherwise, by running time backwards you'd be able to figure out what the initial conditions were.

 

[vanhees71]

Randomness is not a mathematical concept.

To the contrary, randomness is a very mathematical concept, called probability theory and the theory of stochastic processes. The latter are, of course, very much motivated by physics (starting with kinetic theory in the 19th century by Maxwell and Boltzmann with some preliminary work by Bernoulli).

 

[entropy1]

I thought probability was well-defined (like $P(i) = \lim_{N \rightarrow \infty} \frac{N(i)}{N}$). In this form the equation seems to suggest that one has to possesses all information conceivable to be able to determine the probability with certainty (for at any point it could start to deviate, due to whatever factors). However, if a lengthy sample is cut into smaller samples that exhibit the same probability, is this probability then more precise? You could continue to extend with small samples and measure a different probability at some instance. How large a sample should be or how small can you make it?

 

[bhobba]

However, I remain that it is extremely unsatifying to me that QM has no philosophical interpretation*.

Where have you been dude. There are tons of them, and many are simply philosophical arguments about the meaning of probability:
http://math.ucr.edu/home/baez/bayes.html

For what its worth I hold to the formal view its just the elaboration of the Kolmogorov axioms as espoused by Feller in his classic:
https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20

He explains it very well in the early chapters.

Thanks
Bill

 

[bhobba]

I thought probability was well-defined (like $P(i) = \lim_{N \rightarrow \infty} \frac{N(i)}{N}$).

As I said read Feller. Its a lot more complicated and subtle than that. The above is actually a theorem called the law of large numbers:
https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/

Thanks
Bill

 

[entropy1]

Where have you been dude. There are tons of them, and many are simply philosophical arguments about the meaning of probability:
http://math.ucr.edu/home/baez/bayes.html

That is interesting. Unfortunately, the format of the link is a series of emails and the formulas in it are skewed.

 

[bhobba]

That is interesting. Unfortunately, the format of the link is a series of emails and the formulas in it are skewed.

The key bit is at the top before the emails and will not take long to read.

And Feller is definitely a must read of the early chapters.

Thanks
Bill

 

[Zafa Pi]

To the contrary, randomness is a very mathematical concept, called probability theory and the theory of stochastic processes. The latter are, of course, very much motivated by physics (starting with kinetic theory in the 19th century by Maxwell and Boltzmann with some preliminary work by Bernoulli).

bhobba, in posts #49, #50, and #52 refers to Feller's classic two volume set. I agree that Feller is a worthy reference. On page 20 of volume 1 Feller says, "The word "random" is not well defined," and then he goes on to mention the example I gave in post #43. In fact I know of no serious book on probability that defines the word "random" or "randomness". If you think to the contrary, please cite a reference.

Probability theory is a mathematical theory that attempts to model intuitive concepts around randomness, just as QM is a mathematical theory that attempts to model the behavior of light (among other things). But QM does not define light, nor can you derive the definition of light from the axioms of QM. It is up to the physicist to assign states and observables that model the behavior he wishes to observe, or predict about light in the lab. It is up to the statistician to assign probabilities to the "random" phenomena she wishes to model with the theory. To say that "randomness is a very mathematical concept" is to confuse the map with the territory.

 

[houlahound]

Would it be accurate to say randomness in the classical physics sense is just the lack of knowledge of the details, ie the trajectory of a single particle in a gas is random but the macroscopic behavior of kajillions of particles is mundanely predictable via thermodynamics.

 

[Zafa Pi]

No, but there's nothing special about QM in that regard.

Consider what happens with classical gravitation. We start by investigating such diverse phenomena as falling objects, thrown stones, the atmospheric pressure, the motion of the planets. We don't understand any of them, we follow many false paths, and eventually Isaac Newton discovers the law of gravity. One equation, $F=Gm_1m_2/r^2$, explains everything and we understand gravity in all of its varied manifestations...
Except that then some clever high school kid who should be doing her exercises and calculating the speed of a dropped object interrupts her teacher to ask "What makes the objects want to move together? What's going on that makes $F=Gm_1m_2/r^2$ work so well? What's the reason the math works?". The answer is going to be some variant of of "It works. Experiments prove it. Shut up and calculate, you still haven't finished your exercises".

What's different here is that classical gravitation fits in well enough with our common sense that once we see how well it works we tend to accept it without digging deeper. QM, on the other hand, is counterintuitive enough to provoke that "yes, but why?" question, and a feeling of deep dissatisfaction when no answer is forthcoming.

When my kid was 4 and took a fall, he asked me why the ground kept pulling him down. I said, "I don't know why." The poor kid had lousy classical intuition.

 

[Stephen Tashi]

Randomness is not a mathematical concept. There are random variables (functions) and probabilities, but no definition of random.

I agree if we are talking about the common non-mathematical concept of "randomness" (or "probablility" or "stochastic process") as something that involves a state of "potential" or "tendency" or "degree of possibility" which then becomes an "actuality" or a "realization". When the mathematical theory of probability is applied to a specific situation, people think about randomness in that way (e.g. a fair coin has the "tendency" to land heads half the time and when thrown it "actually" lands heads or doesn't.) However, the formal development of probability based on measure theory there is no axiomatization of the concept of a "probability" transforming to an "actuality".

In the formal theory, events have probabilities, but there is no mathematical definition for an event with probability becoming an "actual" event. The closest one gets to the concept of a transition from "probable" to "actual" is in the definition of conditional probability. However, that definition is quite abstract and it merely defines one probability measure in terms of other probability measures. The fact that the conditional probability "given event E" can be defined does not entail any axiom that event E can exist in two states - a "probable" state and an "actual" state.

It would be hard to develop a concept of "actual occurrence" that is consistent with both intuitive idea of "actual" and the theory of probability. For example, we can't say an event with probability 0 will not "actually" occur if we admit the procedure of taking a random sample form a normally distributed random variable. Any specific value we realize from such a distribution has probability zero of occurring. One may side step paradox in practical applications by saying that we cannot "actually" take a random sample from a normal distribution, we can only obtain a value that has finite precision. However, this leaves the theoretical problem of whether some "actual" value was realized (with zero probability) and then was measured with finite precision.

Mathematical probability theory avoids any axiomatic treatment of how events that have probabilities become "actual" or don't. The theory uses suggestive terminology like "almost surely" to suggest how probability can be applied, but there is nothing in the measure theoretic definition of probability that asserts an event can transform from a state of having a probability to a state of being "actual'.

 

[Zafa Pi]

Would it be accurate to say randomness in the classical physics sense is just the lack of knowledge of the details, ie the trajectory of a single particle in a gas is random but the macroscopic behavior of kajillions of particles is mundanely predictable via thermodynamics.

The 1st half of your sentence, up to the comma, is generally considered correct.
The ie (sic) in your sentence is inappropriate. The 2nd half of your sentence is not a retelling or refinement of the 1st half.

 

[bhobba]

If you think to the contrary, please cite a reference.

Sorry if I wasn't clear but that was my point. I was trying to get people to read it rather than simply regurgitate what it says and in doing so wasn't 100% clear in what I was trying to get across.

In speaking of Feller I was talking of the introduction - The Nature Of Probability Theory.

The whole chapter needs to be read. If I was to post about it would simply repeat that chapter and I urge anyone interested in applying probability theory, and indeed applied math in general, to read it.

However a quote from page 3 sums it up:
'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of mass and energy or the geometer discusses the nature of a point. Indeed we shall prove theroms and see how they are applied'

In particular the Kolmogorov axioms leave the notion of event undefined. Applying it requires the mapping of that concept to 'things' out there which only comes from practice and experience. Its similar to the concept of point in geometry. In Euclid's axioms a point has position and no size. Such of course do not exist, but from drawing diagrams and seeing how theorems are proved you get an intuitive feel how to map it to things about there such pegs (for points) and strings (for lines) surveyors use. It's usually so obvious no one explicitly states it. The same in probability - by seeing how events are mapped to things like coin tosses etc etc we gradually build up an understanding.

In QM the primitive is observation. The usual formalism no more attempts to define precisely what that is than probability does event. It can be done - see the Geometry of Quantum Theory by Varadarajan - but like other highly abstract approaches to physics such as Symplectic geometry and mechanics its mathematical beauty is sublime - applying it however is another matter - which is why physicists in general take a different route. Its like Hilbert's axioms of Euclidean geometry. Mathematically it defines exactly what Euclidean geometry is, but from an applied viewpoint Euclid's axioms are used because how to apply it is much clearer - but from the viewpoint of pure math has issues - which is why Hilbert came up with his axioms.

An interesting thing about probability is the same Kolomogorov axioms are used for both pure and applied approaches.

Thanks
Bill

 

[bhobba]

Would it be accurate to say randomness in the classical physics sense is just the lack of knowledge of the details, ie the trajectory of a single particle in a gas is random but the macroscopic behavior of kajillions of particles is mundanely predictable via thermodynamics.

The truth is in the Kolmogorov axioms.

What it means is left up in the air - how you map its undefined concept of event is its content. As I posted above its similar to point in geometry. No one worries about exactly what that is, you simply see how its applied.

Of course philosophers can and do argue about such things, but they generally get nowhere in the sense no one agrees on anything. Because of that mathematicians and physicts avoid it and we by forum rules don't generally discuss it here. There was a very famous example of this with the great philosopher Kant and perhaps even greater mathematician Gauss. Kant thought he knew what Euclidean geometry was thinking it was a priori. Gauss however knew differently, having discovered non euclidean geometry that was just as consistent as euclidean geometry but due to Kant's prestige didn't publish it. It was a lesson well learned and nowadays mathematicians and physicists generally don't worry about such things. Axioms are freely chosen, its meaning is purely in how the undefined concepts of those axioms are mapped to whatever you apply them to. Or you can go the pure math route and don't actually do that mapping and simply prove the consequences of the axioms.

Thanks
Bill

 

[zonde]

No, but there's nothing special about QM in that regard.

Consider what happens with classical gravitation. We start by investigating such diverse phenomena as falling objects, thrown stones, the atmospheric pressure, the motion of the planets. We don't understand any of them, we follow many false paths, and eventually Isaac Newton discovers the law of gravity. One equation, $F=Gm_1m_2/r^2$, explains everything and we understand gravity in all of its varied manifestations...
Except that then some clever high school kid who should be doing her exercises and calculating the speed of a dropped object interrupts her teacher to ask "What makes the objects want to move together? What's going on that makes $F=Gm_1m_2/r^2$ work so well? What's the reason the math works?". The answer is going to be some variant of of "It works. Experiments prove it. Shut up and calculate, you still haven't finished your exercises".

What's different here is that classical gravitation fits in well enough with our common sense that once we see how well it works we tend to accept it without digging deeper. QM, on the other hand, is counterintuitive enough to provoke that "yes, but why?" question, and a feeling of deep dissatisfaction when no answer is forthcoming.

I would say that in classical gravitation the concept of "field" tries to answer the "why?" question.