Ballentine's Ensemble Interpretation Of QM

[Jano L.]

According to the proponents of the ensemble interpretation, the wave is associated not with the individual atom but with the imaginary collection of copies of the atom. In different members of this collection, the electrons have different positions. Thus, if you were to observe the hydrogen atom, the result would be as if you had picked out an atom at random from this imaginary collection. The wave gives the probabilities of finding the electron in all those different places.

I had liked this idea for a long time, but all of a sudden it seemed totally crazy. How could an imaginary collection of atoms influence a measurement made on one real atom?

The imaginary collection does not influence measurement on an atom. It is the act of measurement which influences the atom !

It's clear in all interpretations that the world cannot be classically real and existing at all times. It stopped bothering me when I first learned that electrons around the nucleus are not moving(and that if they moved classically-like, they would soon lose their energy and spiral onto the nucleaus)

I am afraid here you go too far. The idea that in classical EM theory the atom necessarily has to collapse due to radiation of its nucleus and electron, and the calculation of the rate at which this happens, is actually based on very implausible assumption:

The atom is the only one in the universe and no external EM fields act on it, hence the radiated energy has to come from its internal energy.

It is very easy to make this unwarranted assumption. But in any realistic model of stability of atoms, the atoms are under action of external EM fields, if only the thermal EM field, and perhaps also the zero-point radiation. In such situation, the classical theory predicts that the electron will be maintained in chaotic motion around the nucleus. See the paper

Daniel C. Cole, Yi Zou, Quantum Mechanical Ground State of Hydrogen Obtained from Classical Electrodynamics, 2003

http://arxiv.org/abs/quant-ph/0307154

 

[Ken G]

Lee Smolin wrote a paper in arxiv about it that is shared in the other thread about this. I'd like to know if his sudden critique on ballentine ensemble interpretation is sound and if you agree too.

It sounds to me like he is asking a different question than Ballentine. Smolin seems to be asking the question "what is going on for a single real particle, such that it can behave like it is a random member of a large ensemble of real particles?" And he concludes, maybe that's because it is a random member of a large ensemble of real particles. In other words, he takes a literal interpretation of the ensemble, to achieve a more realist outcome. Ballentine, on the other hand, seems to be asking, "how can I conceptualize what is going on for a single real particle, such that I can understand what it means to say it behaves statistically?" The answer there is, we can give meaning to our words about the single particle by embedding it in a much larger imaginary collection, and use that abstraction to justify using the laws of probability. The difference is that for Ballentine, a "law" would need to be no more than an abstract way of thinking about something such that it makes sense, whereas Smolin appears to be searching for laws that are more literally true in some kind of absolute sense.

Of course both of the viewpoints have their issues-- Ballentine has the problem that Smolin points out, it's not very satisfactory from a realist perspective because the abstract ensemble could not have real influences on a single particle. Smolin has the problem that he seems to suggest that quantum mechanics couldn't work in a much smaller universe, it requires a kind of "Mach indeterminism", where if Mach says that inertia for mass here comes from all the masses over there, Smolin says that indeterminism for a particle here comes from all the particles over there.

Personally, I have no issue with Ballentine's approach, because I think it is just perfectly demonstrably true that a law of physics is, and has always been, an abstract conceptualization used by physicists to give a more formal or mathematically precise meaning to the language they invoke to make sense of observations.

 

[kye]

The imaginary collection does not influence measurement on an atom. It is the act of measurement which influences the atom !



I am afraid here you go too far. The idea that in classical EM theory the atom necessarily has to collapse due to radiation of its nucleus and electron, and the calculation of the rate at which this happens, is actually based on very implausible assumption:

The atom is the only one in the universe and no external EM fields act on it, hence the radiated energy has to come from its internal energy.

It is very easy to make this unwarranted assumption. But in any realistic model of stability of atoms, the atoms are under action of external EM fields, if only the thermal EM field, and perhaps also the zero-point radiation. In such situation, the classical theory predicts that the electron will be maintained in chaotic motion around the nucleus. See the paper

Daniel C. Cole, Yi Zou, Quantum Mechanical Ground State of Hydrogen Obtained from Classical Electrodynamics, 2003

http://arxiv.org/abs/quant-ph/0307154

Months ago I was interested in Stochastic Electrodynamics but no longer interested because it can't even explain:

1. the right hyperfine splitting of the hydrogen atom
2. the anomalous magnetic moment of the electron with many decimal digits accuracy
3. lamb shift

So I was discouraged with SED.

About Ballentine's interpretation. It's not unlikely. Remember electron spin doesn't occur in real axis but in abstract space, the fundamental forces came from gauge symmetries which don't have nut and bolts correlates, even the higgs field is a result of terms in the equations so as not to destroy the equations symmetry, general covariance shows spacetime is not a thing, so it's really possible electrons or other particles don't have real paths but only looks like they have a path because of our measurement context.

 

[Ken G]

Stating that the statistical property relates to the flow of qualitative pieces of information produced by an experiment is the only true minimal position that cannot be challenged. Stating that the distribution relates to some physical system or stating that each individual piece of information relates to an individual physical system, that already goes beyond the bare minimum since it cannot be proven experimentally.

Spoken like a true empiricist! I agree that empiricism is the bedrock of science, but I would point out that other approaches to empirical constraints are possible. Many physicsts prefer a more rationalist stance, whereby observations are used to say "which theory is most right", rather than using theories to "make sense of the observations." In practical applications of the doing of science, that distinction is rather minor, which is how science works as a consensus endeavor despite these different philosophical perspectives. But in terms of what we think we are actually doing when we do science, those two perspectives are extremely different, and can create a kind of "tower of Babel" problem when people start to discuss interpretations!

It is fully legitimate that physicists don't limit themselves to developing this minimal operative view, but they must use it as a reference in order to ascertain the consequences of adding interpretative hypotheses in order to convert the minimal reading of QM (merely dealing with describing experiments) into a model - actually a simulation - of what happens in the world.

Yes, this is a vivid statement of the core of empiricism-- that which is real is what is observed, the rest is interpretive matrix. But the rationalist does not accept that stance-- they typically feel that what is real is necessarily perceived by our limited minds as something abstract, and our reliance on observation is like a crutch that sees "through the looking glass darkly", as it were.

The rationale for this systematic comparison between the interpreted model and the neutral reading of QM is that the combination of several interpretative hypotheses may induce some side-effects leading to fictitious contradictions inside the simulation of the world. These must be recognised as being fictitious in order not to trigger complex developments of the theory which are not rooted into the experimental realm (on that basis I fully support your second sentence in the above quote).

Here you might be talking about QM and GR, and the issues with unifying them. I agree, I think we take these models too seriously. But I would have to say that I tend to side with empiricists, though I also think that rationalism has its place. I kind of see them as different hats that we can put on depending on the dress code of the affair we are attending. The empiricist hat that you are describing so accurately seems to fit me best, but I can put on the other hat too, and see reasons and places where I might want to do that.

A key example of such artificial contradictions lies with the famous “measurement problem” which results, in several interpretations of QM, from the combination of three physical hypotheses:
i) the state vector is a property of one single iteration of the experiment;
ii) the state vector is a property of “a physical system” traveling through the device;
iii) the measured value of the state vector holds at the boundary between the “preparation” and the “measurement apparatus”, inside the experimental device.
Then, noting that the location of this boundary is arbitrary in case two or more measurement devices are placed in a series, many physicists concluded that the “quantum state of the physical system” changes in a discontinuous way inside the measurement device. This obviously contrasts with the non-interpreted minimal reading of the QM formalism where the state vector, which is a property of the iterative experiment, can only evolve in response to a change in the experimental set-up, I mean it does not evolve inside the experimental device but inside a configuration space which represents a family of possible experiments.

That's an interesting way to frame the measurement problem, I will cogitate on it!

 

[kye]

I reviewed this in wikipedia and read this:

"The Ensemble interpretation is not popular, and is regarded as having been decisively refuted by some physicists. John Gribbin writes:-

"There are many difficulties with the idea, but the killer blow was struck when individual quantum entities such as photons were observed behaving in experiments in line with the quantum wave function description. The Ensemble interpretation is now only of historical interest."[12]

The "12" rrefers to Q for quantum book. So if Lee Smolin is mistakened about it, Griffen can too?
Actually I have tried reading Ballentine book years ago. I thought he was referring to throwing coins in the table and getting heads up and tails up half of the time when you can see individual pieces producing the head, tail. So in Ballentine's. Do all agree with Hobba that it is basically about "The interpretation says nothing about the interpretation of the state independent of measurement context."? Because without this emphasis, it can confused generations, and confused griffin's too?

 

[bohm2]

The Ensemble interpretation is not popular, and is regarded as having been decisively refuted by some physicists.

From my understanding, there seem to be sub-interpretations of the Ensemble interpretation just like there are sub-interpretations of all the other interpretations which just leads to more confusion.

 

[bhobba]

The Ensemble interpretation is now only of historical interest.

I think someone needs to chat to Ballentine about it.

I mean calling his textbook a modern approach and what it advocates is only of historical interest.



Thanks
Bill

 

[kith]

You misunderstand. The ensemble is state and observational apparatus. The interpretation says nothing about the interpretation of the state independent of measurement context.

I don't think I agree. If we have prepared a state |ψ(t0)> the conceptual ensemble evolves in time according to the Schrödinger equation. How is the interpretation of the quantum state as an ensemble at an arbitrary later time t dependent on a (future) measurement context? I would agree that it is dependent on a preparation procedure but this is not part of the ensemble.

How could an imaginary collection of atoms influence a measurement made on one real atom?

The ensemble is accociated with a preparation procedure. If you know that your free hydrogen atom can be described by state |ψ> this implies that a process has occurred which prepared it in this state. Such processes can in principle not lead to a state where all observables have well-defined values due to the uncertainty principle. So what influences the measurement of a single atom is not the "imaginary collection" but the fact that the preparation procedure is imperfect and thus compatible with different values of the observable you want to measure.

 

[Jano L.]

Months ago I was interested in Stochastic Electrodynamics but no longer interested because it can't even explain:

1. the right hyperfine splitting of the hydrogen atom
2. the anomalous magnetic moment of the electron with many decimal digits accuracy
3. lamb shift

Kye, you probably wanted to say "did not predict" instead of "can't explain". Showing the latter would be quite an accomplishment. If it actually has been shown, could you please post a reference?

Anyway, I was answering Maui's argument based on the belief that atom cannot be stable in the classical EM theory. In light of modern developments, the original reason for it is seen as unwarranted. There is no longer any "atom-collapse" problem in the classical EM theory.

I don't think I agree. If we have prepared a state |ψ(t0)> the conceptual ensemble evolves in time according to the Schrödinger equation. How is the interpretation of the quantum state as an ensemble at an arbitrary later time t dependent on a (future) measurement context? I would agree that it is dependent on a preparation procedure but this is not part of the ensemble.

Very good point! Our choice of the Psi function $\psi(\mathbf r, 0)$ or spin vector $|S(0)\rangle$*used to describe the system at $t = 0$ may depend on what happened to the atom previously, but in the standard use of these things, there is no part in them referring to any measurement apparatus.

It may only be that the resulting derived probability may be referring to some measurements. Here comes the source of the confusion; the situation is different for particle positions, and different for spin projections.

-> For particle positions, $\psi(\mathbf r, t)$ and the ensemble we use to interpret its squared modulus indeed do not depend on any measurements.

For example, the probability for volume $\Delta V$

$$
P(\Delta V) = \int_{\Delta V} |\psi(\mathbf r, t)|^2\,dV
$$

is simply the probability that the particle is at $\Delta V$, not that "it will be measured to be in $\Delta V$". Such measurement for small $\Delta V$ is impossible to do for atoms (low wavelength light will ionize the atom), and for many-particle systems like $\text{H}_2 \text{O}$ the probability in question is for configurations of particles, which are even more unmeasurable.

-> For spins, it is very different, both mathematically and physically. In general, it is not possible to repeat the above reasoning and correctly interpret the spin ket $|S\rangle = c_1 |z+\rangle + c_2 |z-\rangle$ by saying that the spin has value either $+1/2$ or $-1/2$ irrespective of measurement, with probabilities given by $|c_1|^2,|c_2|^2$, except for special case when either $|c_1|^2 = 1$ or $|c_2|^2 = 1$. The SG-like experiments measuring spin projections along different axes refute such idea - we know that prior to the measurement, spin ket can have any orientation in space, not just those two we measure, and it is the interaction with the SG-magnet who selects one value from $+1/2, -1/2$.

So the role of measurement for continuous configurations and discrete results of SG-like measurements is different, and we should take this into consideration everytime the confusion about measurements creeps in.

 

[Maui]

I am afraid here you go too far. The idea that in classical EM theory the atom necessarily has to collapse due to radiation of its nucleus and electron, and the calculation of the rate at which this happens, is actually based on very implausible assumption:

The atom is the only one in the universe and no external EM fields act on it, hence the radiated energy has to come from its internal energy.

It is very easy to make this unwarranted assumption. But in any realistic model of stability of atoms, the atoms are under action of external EM fields, if only the thermal EM field, and perhaps also the zero-point radiation. In such situation, the classical theory predicts that the electron will be maintained in chaotic motion around the nucleus. See the paper

Daniel C. Cole, Yi Zou, Quantum Mechanical Ground State of Hydrogen Obtained from Classical Electrodynamics, 2003

http://arxiv.org/abs/quant-ph/0307154

Correct me if I am wrong but if electrons moved in trajectories around the nucleus, covalent bonds between atoms would be impossible and a big chunk of quantum chemistry would go out the window(e.g. H has just one electron and it's shared in the H2O molecule).

 

[Jano L.]

Correct me if I am wrong but if electrons moved in trajectories around the nucleus, covalent bonds between atoms would be impossible and a big chunk of quantum chemistry would go out the window(e.g. H has just one electron and it's shared in the H2O molecule).

I do not understand how you can derive such conclusion. The theory of chemical bond is based on Schroedinger's equation for point-like charged particles. It is not based on rejection of existence of trajectories. Sure, it does not use the concept, but it also does not disprove it.

Covalent bond involves sharing of electrons and the probability distribution for positions is spread all over the molecule, so the natural way to imagine this is that electrons move all around the molecule, visiting randomly all the nuclei in the vicinity. Since electrons are negatively charged and with greatest probability in between the nuclei, they attract positive nuclei and keep them in some average distance despite their mutual repulsion. If some atoms are very attractive to electrons, they can leave some other atoms without the bonding electrons for long periods of time and the glueing effect of the latter may cease - hence the dissociation of polar molecules into ions that happens in solvents. I believe this is quite standard picture adopted in theoretical chemistry (see works by John Slater, David Cook, ...) It may not calculate exact trajectories, but for sure it does not disprove them either.

 

[kith]

For example, the probability for volume $\Delta V$

$$
P(\Delta V) = \int_{\Delta V} |\psi(\mathbf r, t)|^2\,dV
$$

is simply the probability that the particle is at $\Delta V$, not that "it will be measured to be in $\Delta V$".

How could an experiment distinguish between these two statements?

The SG-like experiments measuring spin projections along different axes refute such idea [...]

Why can't we apply the same logic to a measurement sequence of position, momentum, position, ... ?

 

[Ken G]

But surely a classical picture is quite clunky, because the guts of quantum mechanics is that the classical action per mode of excitation is in some sense both quantized and bounded below at h. I'd have a hard time seeing how this clearly universal principle can arise from a picture of chaotic classical buffeting by some zero-point field. Why quantized? Why always h? Maybe it can be done, but only by bending over backward, at best. Rather than reverse-engineering a classical picture to work in a domain where it is clearly not well suited, it would seem better to embrace the new picture of wave mechanics and the straightforward path to quantization that it presents.

 

[Jano L.]

How could an experiment distinguish between these two statements?

You cannot measure configuration of electronic position vectors in an atom. If you try that by light of wavelength comparable to the size of the atom, the radiation will have great immediate impact on its configuration, and the analysis of the scattered radiation is very hard. X-ray scattering study of electronic frequency density around nuclei of some molecule crystals of many such molecules to accumulate the signal, and it provides only frequency distribution averaged over these many molecules. Currently it is not possible to measure positions of electron in an individual molecule on the scale of its typical size.

The Born rule for $|\psi(\mathbf r)|^2$ used in treatment of atoms and molecules is not a rule to predict results of measurements of their configuration. It is a way to make sense of $\psi$, independently of such measurements, a very important thing since they cannot currently be performed.

Why can't we apply the same logic to a measurement sequence of position, momentum, position, ... ?

When we measure spin projection along some axis $\mathbf o$ chosen beforehand, the atoms split into two different trajectories and we label these by 1/2, -1/2. These two trajectories depend on the axis $\mathbf o$ chosen, so also the resulting value $\sigma =1/2$ or $\sigma=-1/2$ refers to this axis; it has no meaning without it. The actually measured value of projection $\sigma$ refers to the axis $\mathbf o$ and thus could not have existed in the atom before this axis was chosen.

This kind of reasoning does not apply in the case of position or momentum, because their meaning does not refer to any variable setting of the measurement apparatus.

When we measure position of a particle, we let the particle impinge on a detector screen. We do not choose any variable analogous to $\mathbf o$, and the result of measurement $\mathbf r$ (center of the spot left by the particle, for example, a photo grain or charged CCD pixel) has direct meaning of position, independently of any choice in the setting of the detector screen.

We cannot rotate the axis $\mathbf o$ of the position detector, show that positions will be different and conclude that they too cannot exist before the orientation of the detector was chosen. Rotating position detector has no such effect.

In short, the position has always the same thing, irrespective of how we measure it, in contrast to spin projection, which depends on how we orient the axis of the measuring magnet.

 

[Maui]

Covalent bond involves sharing of electrons and the probability distribution for positions is spread all over the molecule, so the natural way to imagine this is that electrons move all around the molecule, visiting randomly all the nuclei in the vicinity. Since electrons are negatively charged and with greatest probability in between the nuclei, they attract positive nuclei and keep them in some average distance despite their mutual repulsion. If some atoms are very attractive to electrons, they can leave some other atoms without the bonding electrons for long periods of time and the glueing effect of the latter may cease - hence the dissociation of polar molecules into ions that happens in solvents. I believe this is quite standard picture adopted in theoretical chemistry (see works by John Slater, David Cook, ...) It may not calculate exact trajectories, but for sure it does not disprove them either.

You seem to misunderstand what I asked. In the H2o(water) molecule, each of the two H atoms has just 1 electron. That 1 electron is shared between the atoms and the only possible case where that would be plausible is when the electron is spread out over the volume of two atoms at the same time, i.e. when the electron does not sit on a trajectory(at least not a on a single trajectory). If the single electron in the H atom of the H2O molecule had a trajectory, the covalent bond would fall apart and water would turn to H and O.

 

[Sugdub]

Spoken like a true empiricist! ... Yes, this is a vivid statement of the core of empiricism-- ... The empiricist hat that you are describing so accurately seems to fit me best, but I can put on the other hat too, and see reasons and places where I might want to do that.

I'm not an empiricist, far from it, but I know and I explained the value of keeping the empiricist view as a reference for distinguishing what is true (experimentally verified irrespective of any kind of interpretation) from what is added value, genuine physics if you wish, but can only be seen as a metaphoric part of our simulation of the world, I mean something that is neither verifiable nor refutable experimentally, but can be assessed in terms of its efficiency.

That's an interesting way to frame the measurement problem, I will cogitate on it!

You are welcome. More inputs on this approach in #28 and #43 of this thread: Mathematically what causes wavefunction collapse?

 

[Jano L.]

Rather than reverse-engineering a classical picture to work in a domain where it is clearly not well suited, it would seem better to embrace the new picture of wave mechanics and the straightforward path to quantization that it presents.

That has been a reasonable approach, and it has been very successful, I know, but along the way much nonsense was invented and accepted in the outbreak of enthusiasm. After 90 years, we are still having these painful debates about what's going on down there.

We can do calculations of magnetic moment that fit the experimental value to 18 decimal places, but still have no reasonable conceptual scheme such calculation would fit it. Virtual particles are real objects to some people, and artefacts of perturbation theory to others. They cannot agree on whether the theory is about particles or fields. It involves spurious operations with infinities, which even after renormalization group does not appear as a consistent theory with no loose ends. Apparently one can do 18-decimal calculations without any solid theory at all. And smart philosophers muddled the problem with complementarity and wave-particle duality.

Meanwhile, people like Schroedinger, Jaynes, or Marshall and Boyer have shown that the quantization is often quite a superfluous operation; and explained that what was thought to be a "non-intuitive quantum phenomenon" is just something that can be predicted and described in simple terms, be it with Schroedinger equation or Maxwell's equation coupled with Newtonian equations of motion.

So I suspect that "the new picture of wave mechanics and the straightforward path to quantization that it presents" is only one possible philosophy to do physics. There are many more, one of which is the rational point of view of physics before 1905. They did not have "interpretation problems" in those days, which I suspect are also mainly communication problems. Maybe something worth considering.

Do not get me wrong - I do not claim classical physics from 19th century will finally explain everything from Maxwell's and Lorentz's equations, but I do believe its $standard~of~argument$ can help us to throw light on some basic questions, especially in situations where much of wild quantum fundamentalism gets in the way.

 

[Jano L.]

You seem to misunderstand what I asked. In the H2o(water) molecule, each of the two H atoms has just 1 electron. That 1 electron is shared between the atoms and the only possible case where that would be plausible is when the electron is spread out over the volume of two atoms at the same time, i.e. when the electron does not sit on a trajectory(at least not a on a single trajectory). If the single electron in the H atom of the H2O molecule had a trajectory, the covalent bond would fall apart and water would turn to H and O.

Well, hydrogens contribute with 2 electrons, oxygen with 8 electrons. They all move somewhere around the nuclei, and they all play the same role (the probability density is symmetric with respect to transposition of electrons). The covalent bond consist of strong correlation of the motion of the nuclei. It is maintained by the attractive force from the electrons, and has only as much stability as allowed by their motion. The stability of the bond in chemical bond theory based on Schr. equation is only probabilistic (tunnel effect...). The fact that the nuclei are close to each other does not break down immediately as some electron goes little bit farther from the hydrogen. The nuclei are very heavy and have great inertia with respect to the electrons; the hydrogen nucleus is 1836 times more heavier.

Of course, occasionally it may happen that this effect plus external forces lead to separation of hydrogen from the rest - there is some OH$^{-}$ in the water too, but there is no obvious reason why motion along trajectories should break all the molecules into collection of isolated atoms.

 

[kith]

This kind of reasoning does not apply in the case of position or momentum, because their meaning does not refer to any variable setting of the measurement apparatus.

What's wrong with a measurement apparatus which contains a rotatable axis with the settings "position" (e.g. a CCD) and "momentum" (e.g. an electromagnetic calorimeter)?

I think this is mainly a question of terminology and not necessarily a fundamental difference between position / momentum and spin x / spin y. It becomes fundamental only if you make the assumption that position is always well-defined while other observables are contextual. This is a matter of taste. I think I prefer Bohr's view that all observables get their meaning from the context of their measurements.

 

[Ken G]

Bernard d’Espagnat (in his book "on Physics and Philosophy") seems to forcefully describe the ensemble theory as requiring the existence of hidden variables.

I get the impression this issue relates to how the meaning of the "ensemble theory" has evolved. bhobba pointed out that Ballentine had to backtrack a bit on his original views of the ensemble theory, when it became clear that imagining the quantum system already had certain attributes prior to the interaction with the macro apparatus was untenable for his project. So perhaps we can say that the "ensemble theory" is a hidden variables theory, so sounds a lot like deBroglie-Bohm, but the "ensemble interpretation" is merely a language for attributing meaning to "the probability that preparation X leads to measurement Y," which takes the perspective that the way to attribute that meaning is to take a frequentist interpretation of the meaning of probability (as outlined in bhobba's posts). I think bhobba makes a good case that the ensemble interpretation of QM is more or less equivalent to the frequentist interpretation of a probability, the defining feature of which is that a probability really only takes on semantic meaning when considered in the context of an ensemble. Presumably this comes with an attitude that a "subjective sense" of the magnitude of a probability in an individual outcome really has no independent meaning from imagining you could repeat the experience many times, and select any individual outcome randomly from that set.

To me this has the same flavor of much of quantum statistical mechanics, in which a probability of a given outcome requires enumerating a large set of possibilities, each equally likely, from which the given outcome is selected. In that sense, the only probability that has meaning independently from an ensemble is a concept of "equally likely" (or "equally weighted" if we wish to avoid "likeliness" issues that could lead to circularity). Once that concept is obtained, which is more or less a symmetry principle, all other probability concepts are about measures within ensembles. An interesting point that arises, which we may not have considered enough in this discussion yet, is that physical ensembles are necessarily discrete, so a frequentist/ensemble interpretation requires thinking about probabilities in terms of discrete countings of equally-weighted possibilities, whereas more general probability measures do not embrace this discreteness requirement. I don't know what important theorems are different on a discrete set, but it may be worth attention.

 

[Jano L.]

What's wrong with a measurement apparatus which contains a rotatable axis with the settings "position" (e.g. a CCD) and "momentum" (e.g. an electromagnetic calorimeter)?

I think this is mainly a question of terminology and not necessarily a fundamental difference between position / momentum and spin x / spin y. It becomes fundamental only if you make the assumption that position is always well-defined while other observables are contextual. This is a matter of taste. I think I prefer Bohr's view that all observables get their meaning from the context of their measurements.

Please read my explanation more carefully, I tried hard to explain it and think I did get the main point there. The conclusion from spin projection measurements is that the spin projection does not have meaning independent of the choice of the measurement axis. In the case of position, there is no such experiment and no such conclusion.

There is one version of wave mechanics, called Bohmian mechanics, where the particles move along trajectories derived from $\psi$. It is generally acknowledged that that theory reproduces the same predictions as theory where there are no trajectories. I do not think the trajectories are that simple as those in the Bohmian mechanics, but they are viable and useful alternative to the fuzzy Copenhagen view.

Of course, if you use the word observable, you have to use the word measurement. However consider that nobody ever measured position of electron to verify Schroedinger wave functions for atoms and molecules. It is not currently possible to do such measurements. There are some simple diffraction experiments on larger scale, but neither these prove electrons acquire positions only as a result of doing the measurement. Such conception creates much confusion - if electron localized only through the measurement, how does it come about that it localizes to such a small region where it does? What mechanism collapses its wave function? And similar stuff. This should have never been introduced. Electrons are assumed point-like particles in Schr. equation, the function $\psi$ is not point-like, and does not collapse to point nor to a region smaller than the atom.

It is very tempting to throw spin and position in one bag, but it is a temptation one should resist to get at the faithful description of things.

 

[Ken G]

So I suspect that "the new picture of wave mechanics and the straightforward path to quantization that it presents" is only one possible philosophy to do physics. There are many more, one of which is the rational point of view of physics before 1905. They did not have "interpretation problems" in those days, which I suspect are also mainly communication problems.

I'm not sure they didn't have interpretation problems-- maybe they just didn't get the same press, or have been forgotten. We tend to teach our students that a trajectory is a perfectly clear and natural thing for a particle to follow, but wave mechanics is weird and open to interpretation. I don't think that's really true-- a particle following a trajectory is just as inscrutable as a particle being told what to do in a statistical way by a wave. The inscrutability of a trajectory centers on the question, how does the particle know to keep doing what it was doing before, when no forces act on it? Relativity shows us that the momentum of a particle is probably not an attribute of the particle, it is more like, rationalistically, a property of the context with which we, the physicists, are conceptualizing that particle, and empirically, a property of the reference frame of the measuring apparatus. But if the momentum of a particle is not an attribute of the particle, how can we treat it like an attribute of the particle when we do classical physics? We were always kind of lying to ourselves when we imagined that a particle can follow a trajectory, since the concept of a trajectory must be embedded in a richer interpretative matrix being imposed by the reference frame of the observer, and indeed the very conceptualization of the physicist. In brief, classical physics already had a "measurement problem," it just chose to ignore it. And like all measurement problems, it stemmed from the same source-- physics has no idea how to include the physicist.

Do not get me wrong - I do not claim classical physics from 19th century will finally explain everything from Maxwell's and Lorentz's equations, but I do believe its $standard~of~argument$ can help us to throw light on some basic questions, especially in situations where much of wild quantum fundamentalism gets in the way.

And I can agree that completely rejecting the classical picture may not be advisable either, since it does speak more closely to our daily intuition. I am a generalist, who says, let's look at the lessons that the successes and failures of any and all interpretations have to teach us. So I think it is a good addition to the conversation to talk about what classical interpretations can still do. But I don't ascribe to the view that just because something is closer to our daily experience, it presents us with a more plausible account of happenings we do not experience.

 

[Jano L.]

... Relativity shows us that the momentum of a particle is probably not an attribute of the particle, it is more like, rationalistically, a property of the context with which we, the physicists, are conceptualizing that particle, and empirically, a property of the reference frame of the measuring apparatus. But if the momentum of a particle is not an attribute of the particle, how can we treat it like an attribute of the particle when we do classical physics? We were always kind of lying to ourselves when we imagined that a particle can follow a trajectory, since the concept of a trajectory must be embedded in a richer interpretative matrix being imposed by the reference frame of the observer, and indeed the very conceptualization of the physicist.
In brief, classical physics already had a "measurement problem," it just chose to ignore it. And like all measurement problems, it stemmed from the same source-- physics has no idea how to include the physicist.

I am not sure I see the problem you refer to. The trajectory in mechanics in calculations is eventually always talked about in terms of some concrete reference body, although the latter is not always explicitly stated. In different r.f., we have different trajectories.

Planck has a good account of the role of physicist in physics, in his Columbia lectures. Initially, physics was about what our senses and organs told us. We have ears - physics of hearing, acoustics. Eyes - physics of vision and light. Heat receptors - physics of heat and temperature. Muscles - mechanics.

But as the science evolved, the physicist as basic object played lower and lower role in it. As the mathematics and theory developed, the effort of most scientists was rather in the direction of finding non-subjective knowledge, to clean science of subjective and antropomorphic aspects. And that is a good thing ! Modern description of mechanics or electromagnetic theory has no need for the concepts of observers or measurements, on the basic level. True, to explain the stuff to students, these words are useful, but the goal we strive in CM and EM courses is a formulation that is not dependent on them.

 

[kith]

Please read my explanation more carefully, I tried hard to explain it and think I did get the main point there. The conclusion from spin projection measurements is that the spin projection does not have meaning independent of the choice of the measurement axis. In the case of position, there is no such experiment and no such conclusion.

I appreciate your effort and I think I understand quite well what you are saying. But I don't agree. To me, two orthogonal spin axes correspond to two different measurement apparatuses of conjugated observables just like a position and a momentum measurement apparatus. It is not clear why the fact that we can go from spin x to spin y by a rotation of the measurement apparatus implies that position is not contextual. As I already said I can also build an apparatus where a rotation takes us from a position to a momentum measurement.

However, I think my statement that observables get their meaning only from measurements was probably too strong. If we have a cold gas, most of the atoms are in the ground state where the electrons are localized quite well near the nucleus. I agree that this makes position special in a sense. But this doesn't imply that we have to assign this special role to position via an additional assumption. It is the interactions in the system which lead to this. IIRC, Zurek related decoherence in the position basis to the Coulomb potential.

 

[kith]

But as the science evolved, the physicist as basic object played lower and lower role in it. As the mathematics and theory developed, the effort of most scientists was rather in the direction of finding non-subjective knowledge, to clean science of subjective and antropomorphic aspects.

It is not self-evident that science can remove the observer from the scene because all knowledge is derived from interactions between the observer and the system. True, it works in CM but I think QM shows exactly the limit of this approach. Interacting entities become entangled which means that the full state cannot be separated into a state of the system and a state of the observer. There are a number of reformulations of this but none of them can get around the fact that the observed outcome is not uniquely determined by the observable state of the system.

 

[bhobba]

How is the interpretation of the quantum state as an ensemble at an arbitrary later time t dependent on a (future) measurement context?

The interpretation does not address how - it just is. The state is not given an interpretation except in measurement context - that's it - that's all. At the time of observation - but at no other time - the interpretation assumes an element of a conceptual ensemble of observational outcomes and state is selected. The ensemble is state and measurement apparatus so it does not even make sense in that interpretation to give a state an interpretation otherwise.

Now we are getting into why I prefer the ensemble interpretation with decoherence - it explains why its only given meaning in measurement context, and indeed exactly what a measurement is - its when decoherence has occurred. For example a few stray photons from the CBMR is enough to decohere a dust particle and give it a definite position (yes I know its a bit more complex than that) so we can say that, and many many similar things, have properties independent of an actual measurement with a device.

Thanks
Bill

 

[Jano L.]

What's wrong with a measurement apparatus which contains a rotatable axis with the settings "position" (e.g. a CCD) and "momentum" (e.g. an electromagnetic calorimeter)?

Aha, now I see your point kith. Sorry for the quick-and-loose answer above. I'll try to explain better what I think:

In the case of spin projection, we know that for any pure spin ket $|S\rangle$ there is exactly one direction in space $\mathbf o_S$ such that when the SG magnet is oriented to measure spin projection along it, we will obtain the projection $+1/2$ with probability 1. This characterization makes it quite plausible that when prepared into spin state $|S\rangle$, the atom has inner angular momentum vector $\hbar/2 \mathbf o_S$.

If that is the case, the inner angular momentum vector cannot point along any other direction $\mathbf o'$.


Question: Could angular momentum projections along other directions $\mathbf o'$ be determined for such system before their measurement by their appropriate SG magnet ?


It seems that such determination for all directions $\mathbf o'$ would amount to description via a strange discontinuous function defined on the sphere with function values $+1/2,-1/2$, plus the ludicrous problem what does it mean that angular momentum vector is $\hbar/2 \mathbf o_S$ and yet its projections along the other axes $\mathbf o'$ are again of the same magnitude $\hbar/2$.

So I think the most reasonable answer is: no, the angular momentum projections along other axes are not part of the atomic state prior their measurement. The only angular momentum projection that allows natural presence prior its measurement is that inferred from $|S\rangle$, that is the projection along $\mathbf o_S$.

For those other directions $\mathbf o'$, surely it is the process of measurement who creates the observed scatter of values from $+1/2,-1/2$.

Now, why is this any different from the measurement of position/momentum?


The difference is, that in the case of position or momentum measurement, there is no wave function $\psi(\mathbf r)$ that would be equivalent to $|S\rangle$ in the sense that upon measurement of position(or momentum) we would get the same value $\mathbf r_\psi (\mathbf p_\psi)$ with certainty. There is no way to attach position $\mathbf r_\psi$ to the wave function $\psi$ analogously to how we attached angular momentum vector $\hbar/2 \mathbf o_S$ to the spin ket $|S\rangle$. There will always be some spread in the wave function, so we expect (and should get) scattered positions of positions / momenta for any $\psi$, due to continuous nature of quantities like position or momentum.

This prevents us to identify a position $\mathbf r_\psi$ with the wave function $\psi$ and derive non-presence of $\mathbf r'$ from the exclusive presence of $\mathbf r_{\psi}$.
​

In short: the ability to prepare definite spin states allows us to make the presence of other spin projections prior their measurement implausible. But the non-existence of definite position/momentum centred $\psi$ functions prevents us from doing the same for position/momentum.


OK, I admit the argument is bit difficult and has some debatable parts, but at least I managed to write it down:-) What do you think ?

 

[Ken G]

I am not sure I see the problem you refer to. The trajectory in mechanics in calculations is eventually always talked about in terms of some concrete reference body, although the latter is not always explicitly stated. In different r.f., we have different trajectories.

But that's the problem. No particle has "a" trajectory, nor can the laws of physics by themselves explain any trajectory we would perceive. The laws only explain part of what we observe. Classical physics only provides the mappings between some initial and final conditions, separated by some proper time, and all that information is completely observer independent, but not the initial and final conditions themselves. The evolutionary "guts" have no measurement problem, the measurement problem appears because no physical law can say why the initial conditions (and by association, final conditions) are what they are, all we can "explain" is the observer-independent piece. Since the preparation of all systems are always observer dependent (there is no observer-independent way to say how a system is prepared), preparation involves measurement in ways that appears in no physical theory whatsoever.

Similarly, in quantum mechanics, all we get to explain is the evolution of the observer-independent part, but the outcome of any particular observation is dependent on the preparation in ways we can never explain, we have to actually generate it with a fundamentally inscrutable observing apparatus to say that we have prepared it such-and-such. In a deBroglie-Bohm interpretation, that's where all the indeterminacy comes in, in other interpretations QM adds additional uncertainties in the outcome of the measurement that an observer perceives which cannot be explained within the theory. But it's not that big a deal to add unexplainable observer-dependent aspects to something which is already not explainable in an observer-independent way!

But as the science evolved, the physicist as basic object played lower and lower role in it.

Perhaps that is true of the observer's physical apparatuses, but it leaves out the most important observer-dependent function of all-- our brains. Our brains do crucial things like declare "the initial preparation is such-and-such," a step that cannot be done without in any brand of physics. Sure the preparation itself can happen without this step, but we are not talking about what happens, we are talking about how physics explains what happens. You always at least need to say "let the initial preparation be X", or you can't account for what happens next. The very act of saying that already invokes a measurement problem, and indeed deBroglie-Bohm doesn't invoke any other measurement problems (indeed my criticism of deBroglie-Bohm has always been that it does not actually get rid of the measurement problem, it just pushes it "out of view.").

Modern description of mechanics or electromagnetic theory has no need for the concepts of observers or measurements, on the basic level.

That is exactly the claim that I do not think is congruent with the facts of doing physics.

 

[kith]

The interpretation does not address how - it just is. The state is not given an interpretation except in measurement context - that's it - that's all. At the time of observation - but at no other time - the interpretation assumes an element of a conceptual ensemble of observational outcomes and state is selected. The ensemble is state and measurement apparatus so it does not even make sense in that interpretation to give a state an interpretation otherwise.

That's not how I read Ballentine's book. Sure, he says he uses the ensemble interpretation because in measurements, superpositions get amplified into the macroscopic domain which is a problem for single system interpretations. But this is just the reason why he interprets the quantum state the way he does. It doesn't limit the interpretation to measurements.

 

[kith]

Aha, now I see your point kith. Sorry for the quick-and-loose answer above.

No offense taken. ;-)

Your argument raises an interesting point. |S> is a state of definite spin wrt the spin operator in direction of o_S. For a wavepacket ψ(x), we can construct analoguous observables mathematically. If ψ(x) is an eigenfunction of a self-adjoint operator A, our system is in a definite state wrt to A.

We can for example imagine a wavepacket observable (WO) whose eigenfunctions are wavepackets with intermediate spread in both position and momentum space. We can also consider a family of such observables which reaches from a WO with very sharply localized wavepackets in position space over a WO with intermediately localized wavepackets to a WO with sharply localized wavepackets in momentum space. So we have kind of a gradual transition from the position to the momentum operator. This is very similar to the gradual rotation of the spin operator from the x- to the y-direction.

The question is, can we construct the measurement apparatuses which correspond to these WOs? The eigenstates of an observable need to be stable wrt the interactions between the system and its environment. If Zurek is right with his connection between the position basis and the Coulomb potential, it might well be the case that we can't construct these measurement apparatuses. But this difference between spin and position/momentum would not be a fundamental difference but a result of the dynamics.