The Knowable and The Unknowable in quantum mechanics

this is a reasonable way to understand the nature of the wavefunction, especially for operators with continuous spectra. it s not clear to me that this squeeze notion can be used for operators with discrete spectra, like spin.

in any event, physically reasonable or not, this is not the orthodox position of QM. the axioms of quantum mechanics describe a complete collapse of the wave function, after an ideal measurement.

there are a lot of people who don t believe in the complete collapse of the wavefunction, like these many-worlds interpretation people.

if you want to propose a looser interpretation than the orthodox one, it would be nice if you could come up with a sety of axioms, but i don t imagine that sort of thing is too easy...

in any event, the point is that the orthodox position with it s wavefunction collapse is sometimes a bit hard to interpret.

 

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Excellent point.

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Actually my view agrees quite well with the orthodoxy. No real measurements are ideal, so no real measurements totally collapse a wavefunction. Real measurements just restrict a wavefunction to some (possible very small) portion of the Hilbert space -- they don't nail it down to a point.

And those people are nuts. :bugeye: Just kidding.

Again, I don't think my view is that unorthodox, except perhaps in the case of discrete quantities like, say, parity or spin. Perhaps in that sense my view is not orthodox. Hmmmm....

Quite true.

- Warren

 

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Hi guys. Sorry to be so out of step with the discussion, but I find time when I can...

My problem with QM has always been the assumed equivalence of measurement and structure. Thus, wavefunctions describe the probability densities of measurements. Yet, these probability densities are being made altogether equivalent to the very entities being measured. Granted, experimentally this makes no difference since the theory is itself experimentally verified -- thus all it ever has to do is faithfully predict experimental outcomes. However, I hold that science must aspire to higher goals than mere prediction of observable events. It must strive to decipher the mechanisms that are responsible for the observed behavior.

For example it might be sufficient in a utilitarian sort of way, to desribe the behavior of a resistor with a simple formula relating the current flowing through it to the voltage drop across it. This is a nice predictive model of experimental outcomes. But it fails to capture the internal dynamics of the resistor -- the actual movement of charge carriers through its molecular structure and all the complex nonlinear interactions that occur. Granted, it's not very practical to make such detailed models, and they'd be unwieldy to apply. However, such a model would make a resistor into something more than just an assemblage of mathematical abstractions on paper; it would describe the actual tangible structure of the entity in question. It would also give much deeper insight into what we are dealing with.

Taking the EPR paradox (as formulated by lethe in terms of pion decay on the first page of this thread), I would claim that the two electrons have definite spins from the very beginning (definite, but unknown to us until we measure them.)

However, our measurements of the spin do not necessarily reflect the underlying entity being measured. This is the distinction I make: a measurement outcome is not informationally equivalent to the underlying cause of the outcome. Hence, I dispute the following statement from lethe's 11-12-02 05:24 PM post:

I dispute this on the grounds that it mixes together the notions of measurement and the actual entity being measured, using these notions interchangeably. I do not think they are interchangeable.

While I believe that angular momentum must be conserved, I do not believe that our measurements must necessarily reflect that fact. In other words, the entity or phenomenon that we call "spin" might indeed be complementary in the two electrons in such a way that angular momentum is conserved. But the way it manifests itself in our particular measurement setup does not necessarily have to reflect that conservation in the way we expect based on the assumed complexity equivalence between objects and probability distributions describing the observable interactions between the objects. If we are to interpret the meaning of our measurements using "hidden variables", we must include those hidden variables as well as the meaning of our experimental conditions into that interpretation.

For example, lethe states:

(where A(u,x) and B(u,x) are models of measurement.) I do not see how this is necessary. For example, assume that x is a directional variable (it entails an internal "spin" along a definite axis in space.) Thus if u is precisely orthogonal to x's directional component, then due to the symmetry of the situation A and B can no longer be viewed as deterministic functions as they would each give two simultaneous, distinct predictions vis. u and -u with a 50% probability assigned to each outcome. Hence, under this condition it may indeed turn out that A(u,x) = B(u,x) just as often as A(u,x) = -B(u,x). And this is but a very simple case with only one hidden variable that has a very simple geometrical relationship with u. Behaviors of more complex hidden-variable systems could yield even stranger observations.

Given such considerations, I do not see how Bell's derivation is the only necessary outcome of a local hidden variable assumption. Rather, his result arises out of a number of unspoken assumptions in addition to the axyoms he explicitly enunciates, and it is those hidden assumptions in particular that I find objectionable.

 

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Overdoze,

Welcome back!!

Everyone was beginning to gang-up on me

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. Now they have someone else to pick on, as well.

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Tom

 

overdoze, i don t quite follow your explanation about what you find objectionable in the foundations of quantum mechanics.

 

lethe,

Hope you can help me to understand better:

1) Can a quantum system interact with another quantum system. What happens if they do? Can it be considered that they "measure" each other and therefore, collapses each other's wavefunction?

2) What is the generally accepted explaination in the scientific community as to why quantum superposition does not manifest itself in the macro-physical world or in our everyday life?

Thanks.

 

A few things, but I'll try to take it one step at a time.

For example, take this chain of reasoning:

1) Angular momentum must be conserved
2) therefore, when a particle with 0 angular momentum transforms into two particles with nonzero angular momentum, their individual momenta must add up to 0
3) therefore, when we measure the momentum of each particle, these two separate measurements must add up to 0.

While 2 follows form 1, 3 does not follow from 2 (remember Heisenberg's uncertainty principle?)

Do you agree so far?

If you do, you have to admit that:

While the total angular momentum of the system is conserved, the exact momenta of the particles are impossible to measure -- even though they must indeed be exact in order to completely cancel each other out at every point in time prior to the first measurement. That means each particle has an exact but hidden property called "angular momentum" -- a hidden variable if you wish.

If you don't agree, then how do you conceptualize conservation of angular momentum in absense of measurement? Would you claim that angular momentum is created and destroyed at will, just as long as nobody knows about it? But then how does the momentum magically know to return to the correct value at the instant it is being measured?

 

You have to resign yourself to the fact that you can only verify conservation of angular momentum for one component per measurement. However, if you take three identically prepared systems, and use one each to measure the x-, y-, and z-components of angular momentum after the decay, QM says that each one should add up to 0, and indeed they do.

 

Hi Tom

That's fine, but it doesn't really answer the question. The question concerns existence of an exact angular momentum in absense of measurement. As far as I understand, the classical interpretation of QM says the angular momentum has no value at all or alternatively all values at once (but forget about exact value) until it is actually measured. Am I right?

If so, how does the momentum know to acquire just the right value so as to be conserved, at the exact instant when it's measured? This magical notion doesn't sound very plausible to me (mathematically convenient though it may be.)

[edit:]

Plus, I don't quite see how you could have "identically prepared" particles in the quantum realm. If this were possible, I could prepare a particle in such a way that I know exactly both its position and its momentum throughout its trajectory. I think I hear Heisenberg objecting.

 

What QM says is that you can only measure one component of angular momentum at a time. It has no answer to the question, "What is the angular momentum when I do not measure it?"

No one knows. You have to realize what a scientific theory is, and what a scientific theory isn't.

Scientific theories are not intended to explain the universe to anyone's satisfatction. Scientific theories describe the universe. Only two things are required of a theory for it to be considered "good":

1. Internal consistency (validity)
2. Agreement with past and future experiments (soundness)

That's it! So, as long as QM is self-consistent and predicts the outcome of experiments, it holds court.

Why this lecture? Because you say that my post doesn't really answer your question. That may be, but the idea I'm nudging you toward is to abandon that question in favor of the more scientific one: What will I measure if I perform such and such an experiment?

No, there is no problem because you do not need to know the position and momentum of a particle to know the maximum available information on its angular momentum (the maximum information being a single component and the magnitude). If you really want to know the angular momentum, use a spin-0 particle. That way, you know all 3 components and the magnitude (they're all zero!).

 

So many of the "holes in physics" people come up with on this site can be explained by any introductory quantum mechanics textbook.... *sigh* Why don't they just read first, and ask questions later?

- Warren

 

Which is why it's incomplete. I, for one, am inclined to think that physical entities and properties (such as angular momentum) should not cease to exist the moment they are no longer being measured, and should not pop into existence out of nothing just when a measurement is about to be performed.

But the tragedy is that you (and people like you) not only don't know, but apparently don't want to know. That's the mind-numbing effect QM seems to have on people, and that's why I've always had a gut reaction against QM (or at least this interpretation of it.)

And regarding the alleged purpose of science, that's where one of our fundamental disagreements takes root.

You might think so, except it doesn't work that way. When you observe an unfamiliar system for the first time, it is the job of theoretical science to assist you in interpreting and explaining what's going on.

For example, theoretical science can answer questions such as "why is the sky blue? why does the sun always rise in the east? what causes earthquakes?" Note the whys and the whats. Answers to such questions are not mere descriptions; they are explanations.

Explanations are needed first and foremost for practical applications. If you are an engineer, you are not interested in mere descriptions. You want functional models (and accurate ones!) of what's actually going on. Otherwise, you would never be able to put the theory to good use.

But if we think deeper, we might agree that at some level of detail descriptions and explanations become the same thing. At some point when you have a description that is detailed enough, you have an explanation.

The "descriptions" of QM are simply not detailed enough, under certain circumstances, to qualify as explanations. That doesn't mean it's the best that can be done. It simply means that QM is not good enough.

I can add a few more things. Completeness is one. Simplicity is another. Elegance, a related quality, a third. Of course, these matter most if there are competing theories that are equally consistent and in agreement with experiments. But even if a theory is uncontested, these criteria still matter. Call it esthetics; a basic rejection of the inelegant, a distaste for kludges, omissions, and fudge factors. Science, after all, is an art as much as if not more than it is a protocol.

And the other side of the coin is: How do I explain/interpret the outcome(s) of my experiment? This comes in especially handy when your measurements don't agree with your expectations.

I thought the Uncertainty Principle went deeper than just position vs. momentum. I thought the root of the principle was the stipulation that any attempt at measurement (no matter what you're measuring) disturbs the very quantity you're trying to measure. That means even if you're trying to measure angular momentum, you'll never get the value exactly (I'm talking about directional components, of course -- since they aren't quantized.)

For example, if you try to pin down a position of a particle, you're bombarding the particle with something (e.g. other particles) and thus disturbing the particle's position. In a similar fashion, as you try to measure the directional components of the particle's momentum, you're squeezing the particle through some directional filter, and thereby disturbing those very components you're trying to measure.

But the major point is that the actual momentum (whether angular or not) is not the same thing as your measurement of that momentum. The actual momentum is an intrinsic property of the particle. The measurement is an outcome of a particular interaction between the particle and some instrument. Equating the two is nonsensical. It's like saying that a scribble on paper is the same thing as the physical volume of a real object. Physically, they are not the same thing. A scribble is a scribble, and a volume is a volume. Similarly, an instrument readout is a readout, while the property being represented on the readout is physically distinct from that readout.

The classical interpretation of QM equates probability distributions with actual objects. That is fundamentally wrong. Probability has always been used and arose from the need to describe meta-characteristics of populations of samples. Things like mean, variance, and other various moments describe population behavior and have nothing to say about each individual sample apart from its relationship to the overall population.

Taking these ideas and twisting them in such a way that population statistics are re-assigned to individual entities is a semantic error. The correct interpretation of QM distributions is that they describe outcomes of repeated measurements on the same particle, not that the particle itself is a distribution. Otherwise, it's like me describing my instantaneous body temperature as a distribution of temperatures ranging over many years. It's a logical error to do that.

 

First let me say that this is a good question. i can t tell you to accept the orthodox quantum mechanical interpretation, until i convince you that Bell s theorem is correct.

the proof of the theorem is a proof by contradiction, and it relies on assuming that the particle has a hidden variable, and that it does have well-determined values for, say, angular momentum. so telling overdoze that his questions are wrong because the particle doesn t have well determined values is circular logic. you are telling him his questions about the proof are invalid by citing the theorem.

now. let me see if i understand your question correctly: you want to know why conservation of angular momentum implies conservation of the measured values of conservation of momentum, which is what i actually invoke in the proof. if the measurements are not necessarily the same as the real values (which do indeed exist, in the hidden variable picture. remember this is a proof by contradiction)

lemme think about this.

 

1) Yes.

2) I think the generally accepted view is the first one i put forth before. a measurement occurs when a quantum system interacts with a nonquantum one. large nonquantum systems do not exhibit superposition, because they have collapsed the wavefunction. and they do not exhibit quantum behavior, because for very large quanta, the quantum effects tend to average out to classical behaviour.

i believe that is the answer most scientists would give you if they were forced, but mostly i think it is regarded as a non-issue. a bunch of semantic arguments propogated by borderline philosphers, instead of real scientists who are only concerned with the agreement of QM with experiment. which is excellent.

 

I didn't say that physical properties cease to exist, I said that we can't say anything definite about them.

He He--that's nothing compared to the mind-numbing effect that the classical world has on people. Most of the people on this forum seem to think that anything that doesn't conform to "common sense" is automatically illogical.

The job of theoretical science is to look for patterns and abstract a concise, comprehensive model that not only matches the observations, but predicts future observations. Any "explanation" is left to philosophy.

Yeah, but they are explanations which require additional explanations, which means they are not really explanations at all. Any "why" question you can ask will always lead to another "why", and I think that they will all lead to "why is there something instead of nothing?" Theoretical science deals with what, how, where, and when--not why.

Actually, what I call a "description" is exactly what an engineer wants. I think of all mathematical models as descriptive, not explanatory.

To some degree, probably.

How do you know that you can know more about a system than QM says you can know? Is this founded only on a common-sensical notion that you should be able to know everything about a system, or is it based on something more?

Much of what you say here is too subjective to be of any use. What is "simple" or "elegant" to one person could be "complicated" and "ugly" to another. I'm not at all sure of what you mean by "completeness".

No, the uncertainty principle says that when you measure a quantity, you disturb the quantity that is conjugate to the quantity you are trying to measure.

Take a particle in some arbitrary state. If you measure any component of its angular momentum, and the magnitude, you will get definite values. But in doing that, you close yourself off from any information about the other components.

Actually, they are exactly the same thing by definition. You can't know anything about 'reality' apart from measurements. Observations on the physical universe are the only things we have access to.

No, it's not. Equating the mathematical model with the real property would be the same as equating a scribble with a physical volume. Both the model and the scribble are representations of something physical. The measurement is the real thing. If it's not, then what would you say is more 'real'?

What does this have to do with what you can and cannot know about physical systems?

Another interpretation would be to abandon the probability thing altogether and just recognize that particles are really waves.

Tom

 

lethe,

Thanks for the reply. I have 2 other questions that is going to sound ignorant, but I'll ask them anyway:

1) are there naturally existing or occuring quantum systems in the universe or can they only be produced in a laboratory?

2) During the Big Bang, was there a moment when the universe existed as a quantum system?

Any input is much appreciated. Thanks.

 

Hi Tom2,

Thanks for your detailed response. It certainly made me think.

One thing: your apparent distaste for philosphy and/or your assumed distinction between philosophy and science, I don't think is well-founded. In the old times scientists used to be called philosophers; the title PhD (doctor of philosophy) is a holdover of that. If it weren't for the intrinsic human drive to explain things, science would never have arisen as the institution it is today. And if you care to look you'll find that practically all of the greatest scientists (and mathematicians) in history were also avid philosophers.

Ironically enough, the very title and topic of this thread is philosophical as much as scientific.

Now for a few points:

But if they indeed exist regardless of measurement, then they are nothing other than the so-called hidden variables. Perhaps now you can see my difficulty with claims of logical proof against existence of hidden variables; most of the people who make such claims, when pressed, just like you start falling back on those very hidden variables to which they've been denying existence.

If you think more carefully about the process of discovering patterns, you might realize that it entails extraction of cognitive tokens from raw sensory input. This is indeed the foundation of all thought, regardless of whether you classify it as scientific, philosophical or something else. So I'm afraid this doesn't help you draw a clearer distinction between what you view as science and everything else; quite to the contrary, you've found the unifying principle for the very entities you've been trying to segregate.

Are you trying to argue that there is no such thing as an explanation at all? If so, then perhaps you should re-examine your notion of explanation.

Every explanation (and every model) must ultimately rest on a set of axioms – basic assumptions and definitions that are not questionable in principle. This is the mechanism that prevents the infinite regress you’ve suggested. I believe our contention ultimately concerns what we regard as basic and unquestionable. To me, the fundamental axioms of quantum mechanics do not appear irreducible enough to be comfortably axiomatic.

That’s an excellent point. I don’t know that I can know more about a system. I assume I can, because if I assumed otherwise, I would never try to know more and even if my assumption were in fact wrong, I’d remain stuck in a dead end of my own creation until I was willing to change it.

You don’t need to be a fan of philosophy to be aware that the elemental-atomic model of matter was first proposed by an ancient Greek philosopher. He was laughed and scorned out the gate by the scientific/philosophical establishment of the time, since his assumptions were considered untestable and superfluous to the theories of the day. This is exactly the analogue of the modern reaction to even the concept of hidden variables. Yet in both cases the goal is manifest: to search for the fine structure of the nontrivial phenomena that are currently assumed to be fundamental.

Once again, you have an excellent point concerning subjectivity. True enough, that. By completeness I mean description (if you will) of not only some narrow subset of existence, but at least the ability to nicely tie into the rest of it. As it stands, quantum mechanics is irreconcilable with general relativity (which is in equal agreement with experiment.) That is an obvious sign of incompleteness. Generally (and subjectively) speaking, quantum mechanics is too specialized, and too narrow in scope, to even appear complete.

As for judgements of elegance, consider that the concept of probability is connected to populations and sampling, not intrinsic properties of individual objects. Consider that the concept of a wave historically arises from complex multi-component systems, as do the related effects of interference, refraction/diffraction, resonance, and propagation. Yet all of these concepts, despite their conceptual and informational sophistication, are taken as fundamental axioms by quantum mechanics. I feel that since there’s smoke, there must be fire. The “fundamental” entities of quantum mechanics look far too complex to be fundamental. To me, it’s much too compelling to view these phenomena as reducible, collective manifestations of something finer.

I see. Thanks for the clarification. But still, are you saying that any physical quantity can be known exactly? I thought the uncertainty principle had a bearing on that. For example, you could never precisely measure position because then you’d be imparting a potentially infinite momentum. Seems to me that the uncertainty principle also puts practical energetic constraints on precision, regardless of whether we’re talking about conjugate properties or the very properties we’re trying to measure.

Agreed. However, it is foolhardy to assume that the contemporary set of measurements is representative of all measurements that are possible in principle (that is, above and beyond what is currently possible in practice due to limited knowledge and technology.)

This is an observer-centric argument. Indeed to an observer, measurement is the only reality. However, I don’t subscribe to such an egocentric outlook.

I tend to think that the universe exists independently of my own existence as an observer. That is, I believe – quite unjustifiably, mind you – that were I to be killed this instant the universe would go on just as it has always done. Call it faith, as indeed I have no justification for such a belief – other than perhaps Occam’s razor.

This assumption then leads to the conclusion that physical entities exist and have states and properties independently of whether I know about these parameters or even whether I am in principle capable of knowing.

Ok, that’s it for now. Thanks for a most stimulating discussion.

 

1) In principle, all systems are quantum mechanical in nature. every atom in your body is a quantum system, without the quantum levels, no atom would be stable. because quantum systems are very small, we only observe their quantum natures when we have carefully prepared experiments in the lab.

2) Yes. right now, the universe is cold enough and large enough that gravity is what binds it together, and it follows the rules of classical mechanics mostly, with some GR too. but we expect that closer to big bang, quantum effects would have dominated. the problem is that there is no quantum description of gravity, so there s a bit of trouble describing the moments near big bang.

but yes, in principle, in the beginning, the universe behaved as a quantum system.

 

No, the heisenberg uncertainty principle places no restrictions on knowing a particular quantity. for example, it allows you to know the position of a particle with arbitrarily high precision, provided you don t mind letting the uncertainty in the momentum approach infinity.

certain quantities, like say, the magnitude of the angular momentum, and position can both be known as well as you like.

the heisenberg uncertainty principle is completely independant of any energetic constraints

 

I'm beginning to feel a little stupid here. Like I'm being told something obvious for the third time and still not getting it.

Tell me, is it not true that if the uncertainty in momentum approaches infinity, then the uncertainty in kinetic energy of the particle also approaches infinity?

Would this not imply that your experiment can impart energies approaching infinity in the process of conducting measurement?

Unless you have discovered a source of infinite energy, does this not place a practical constraint on the precision of all experiments? Say, how about putting the practical limit at somewhere under the total mass/energy content of the Milky Way galaxy?

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