Unitary Processes and Completely Identical Quantum States

[ObjectivelyRational]

EDIT: Questions have been revised below, those immediately following are for reference, jambaugh's kind reply was in direct response to these original questions.

Could a completely unitary (QM) process act on a set of particles in "completely identical quantum states" to cause them to time evolve differently from each other absent any measurement or any collapse of the quantum state wave-functions?

Equivalently, could a completely unitary (QM) process act on a set of particles in different quantum states to cause them to time evolve into "completely identical quantum states" absent any measurement or any collapse of the quantum state wave-functions?For full disclosure, this subject arises because of some communication difficulties I had with formulating the questions in a the previous related thread:

https://www.physicsforums.com/threa...tely-identical-particles.960230/#post-6091283

REVISED QUESTIONS:
Could a completely quantum mechanical process, absent any measurement or any collapse of the quantum state wave-functions, act on a set of particles in "completely identical quantum states" to cause them to time evolve differently from each other?

Equivalently, could a completely quantum mechanical process, absent any measurement or any collapse of the quantum state wave-functions, act on a set of particles in different quantum states to cause them to time evolve into "completely identical quantum states"?

 

[jambaugh]

There are some subtle issues with your question that I would address. Your question as formulated uses the classical language of system being in actual states of reality. This gets "iffy" in quantum mechanics. To clarify it fully one needs to formulate it in the mathematical language of QM (then the question can be answered and one can argue over interpretations of what the question AND answer mean).

I'm of the school of thought (CI) that the "wave function" or more generally the Hilbert space vector, or even more generally the density operator, for a quantum system is, in point of fact, the state of our knowledge about how the system will behave under acts of measurement. That knowledge is probabilistic and for sharply measured systems that knowledge is maximal (no further statements about the system can be operationally meaningful.)

Thus, even before you invoke this unitary quantum process, I can measure two identically prepared quantum particles with identical wave functions or state vectors and I can get distinct outcomes. Say for example measuring the x-spin component of two electrons, both in the spin-z up "state" I will get a 50-50 distribution of outcomes and thus for a given pair, a 50% probability of distinct outcomes.

Does that mean "They were really in different states"? That requires we make assertions which are fundamentally beyond what we can measure (one is asking what is real beyond what is observable). Sometimes it is possible to do this consistently as we do at the classical level but with such things as Bell inequality violation in EPR type experiments, doing so becomes problematic at the quantum level.

So given this is an issue for the basic "states" (with unitary operator 1) it is likewise an issue with your more general question.

 

[ObjectivelyRational]

Perhaps unitary is not the not the proper term... I have no reservations about posting another thread as necessary. Perhaps this thread could be used to more carefully formulate the question I am asking so that it uses the proper "accepted" jargon.

I can measure two identically prepared quantum particles with identical wave functions or state vectors and I can get distinct outcomes. Say for example measuring the x-spin component of two electrons, both in the spin-z up "state" I will get a 50-50 distribution of outcomes and thus for a given pair, a 50% probability of distinct outcomes.

I completely accept for the purposes of the questions that "measurement" or "collapse" results in the particle being measured as up or down according to the probability of the respective basis states making up the whole state vector.

To elaborate: Assuming that the science is aimed at getting the expression of the QM state right, and that the CI interpretation is absolutely correct (along with the implicit assumption nothing "more" may be possibly said about the particle in the quantum state), then if we were to entertain the idea that there were any actual differences having a causative effect on the result, we should attempt to make them part of the quantum state. We have not heretofore "written them" into the state because we have not discovered them yet. In such a case our experiment has some flaw which is operating on some other "aspect" of the state we are not taking into account. Taking CI seriously, if there is nothing further about the particle which causes the up or down, (no difference between particles) then the purely probabilistic interpretation of the "mechanism of measurement" (assuming also that the processes of measurement are also identical) would be appropriate.I have heard two things, although one or both of them may be in error.

1. That quantum mechanical time evolution is absolutely deterministic (absent any collapse or measurement etc). That both our models and reality reflect the fact that although a state may be expressed as a vector being a sum of different basis vectors each with respective coefficients, those coefficients and those states, using a time operator (our model) and in actuality (confirmed by measurement) quantum particles evolve purely deterministically, at least until measurement and collapse (whatever they are). Probability only enters at the point that measurement and collapse are involved.

2. A quantum state, is all there is to the existence of a particle. There is no other aspect, hidden variable, attribute, or property to the existence of the thing. In principle it is described fully by its quantum state. IF not we would have discovered other aspects needing to go along with the description of the particle and QM would not be a full explanation. So far it appears that the quantum state is all there is to a particle.

From the above two, the idea of particles in "completely identical quantum states" as a result of completely quantum mechanical processes becomes a little dicey as the application of basic logic reveals. How could they ever get into a completely identical quantum state absent measurement or collapse, and if we started with them being completely identical how could they ever be separated other than through measurement or collapse?

 

[ObjectivelyRational]

Any insightful response would be greatly appreciated.Has not anyone else ever wondered about this?

 

[PeterDonis]

quantum mechanical time evolution is absolutely deterministic (absent any collapse or measurement etc)

This is correct.

A quantum state, is all there is to the existence of a particle.

This depends on what interpretation of QM you adopt. Note that the version of the Copenhagen interpretation you are using (based on what @jambaugh said) does not say what you say in this quote. It only says that the quantum state is all that we can know about the particle. It makes no claims whatever about what the "real" state of the particle is, only about what we can know about it.

To avoid these dicey interpretational issues, I would reframe your question purely in terms of the basic math of QM, something like this: suppose we have two quantum systems that we describe using the same quantum state. (For example, two electrons that are in identical energy levels in identical potential wells, with no external fields.) Can unitary evolution ever make their quantum states different? The answer to that is no; since both systems start with the same state vector and have the same Hamiltonian, their time evolution will be identical.

Similarly, we can reframe your second question as: suppose we have two quantum systems that we describe using different quantum states. (For example, two electrons that are in different energy levels in identical potential wells, with no external fields.) Can unitary evolution ever make their states the same? Again, the answer is no: since the two systems have different state vectors but the same Hamiltonian, time evolution will preserve the difference between them.

Note that I used "unitary evolution" above to mean "unitary time evolution according to the Hamiltonian of the system". There are, of course, other unitary operations that you could apply to one of the systems in the first case, where they start in the same state, to make them different, or to one of the systems in the second case, where they start in different states, to make them the same. But those unitary operations correspond to "doing something" to the systems, physically: turning on a magnetic field or shining laser light at them or something like that. I don't know whether you intended to include such cases in your question or not, but it seems to me that including them makes the question trivial: of course there will be things you can do to systems to change their states. The interesting question, it seems to me, is what the possibilities are if you do nothing, and that's what unitary evolution according to the Hamiltonian of the system describes.

 

[PeterDonis]

Btw, it's also worth noting that I included "having the same Hamiltonian" in the definition of "being in the same state". That's because the Hamiltonian determines the Hilbert space of the system, and the concept of two systems "being in the same state" only makes sense to begin with if the Hilbert spaces of both systems are the same.

 

[ObjectivelyRational]

Thank you! You answered my questions with clarity and provided some insight into the dicey interpretation issues...

To clarify my statement about "existence". It comes from a perspective that for anything to exist, for any property or attribute of a thing to exist, means for that thing to interact and be causatively consequential to anything observable in the universe. If there is no interaction, no causative effect whatever, it has no meaningful existence...

So, there is "something" about reality which the quantum state is meant to describe, even if only couched in terms of observations (and only probabilities at that). Of course we are fallible, but the aim is to identify the "reality of the state of X is such that" we expect these probabilities if measurements are made with respect to this basis vector etc. My point is that the QM state PSI stands for all that is knowable about it... by any kind of direct or indirect, measurement,
with the assistance of any instrumentation and deductive investigation. Assuming what is "knowable" coincides with what can cause any measurable effect on the observable universe, however small (some might fancifully describe neutrinos as "barely" there...), then in a sense everything there is to a particle which matters (anything not completely meaningless) is represented by the QM state PSI in principle, although we may not know currently how to "write all of it down" or might get it wrong from time to time.

The interesting question, it seems to me, is what the possibilities are if you do nothing, and that's what unitary evolution according to the Hamiltonian of the system describes.

This is very interesting, although distinguishing between "measurement" and "doing nothing" has always seemed somewhat of an artificial distinction, given that all constituents of nature are not supernatural, and any instrumentation we set up is simply another context for nature to play out the way it does.

So I suppose, since someone brought up Bose-Einstein condensation, do we have a non-quantum process there producing completely identical quantum states? or are we dealing with natural processes more akin to measurement or causing collapse? or are the resulting states PSI of each particle in the condensate not actually completely identical? I assume that the condensate once heated up diverges into different states... although complicated I see heating as purely natural ... (doing nothing??)

If I were to indulge in the logical implications of completely identical states, other than (objectively) probabilistic collapse (the infamous dice a certain Omnipotency plays with), the states would in principle be inseparable by any external means whatever, since, they, each being identical, would be subject to identical external influence. [and by identical I do not mean any percentage of "similarity" no matter how close to unity - I mean exactly completely identical]Please bear with my imprecision, as you can tell the concept "measurement" and its sharp distinction from natural processes is not particularly precise to me.

 

[PeroK]

To clarify my statement about "existence". It comes from a perspective that for anything to exist, for any property or attribute of a thing to exist, means for that thing to interact and be causatively consequential to anything observable in the universe. If there is no interaction, no causative effect whatever, it has no meaningful existence...

That is philosophy, not physics.

 

[PeterDonis]

everything there is to a particle which matters (anything not completely meaningless) is represented by the QM state PSI in principle

But that's obviously not true, since ##\psi## can only tell us the probabilities of observing various measurement results; it can't tell us why the particular result we observed, was observed.

distinguishing between "measurement" and "doing nothing" has always seemed somewhat of an artificial distinction

It isn't. To measure something, you have to interact with it. "Doing nothing" means not interacting.

since someone brought up Bose-Einstein condensation, do we have a non-quantum process there producing completely identical quantum states?

There is no such thing as a "non-quantum process". If you're using QM as your theory, you're using QM, and you describe all processes using it.

As for Bose-Einstein condensation, that happens when you lower the temperature of a quantum system consisting of multiple bosons. The condensate is a state of the system, not a bunch of individual bosons "all being in the same state" (though pop science treatments often describe it, incorrectly, that way).

I assume that the condensate once heated up diverges into different states

No, it means the state of the system as a whole changes. See above.

If I were to indulge in the logical implications of completely identical states

First you need to be clear about what "completely identical states" means, mathematically. I already described that in a previous post and gave an example. Note that my example was not anything like a Bose-Einstein condensate.

 

[ObjectivelyRational]

The condensate is a state of the system, not a bunch of individual bosons "all being in the same state" (though pop science treatments often describe it, incorrectly, that way).

Thank you.