Meaning of Wave Function Collapse

[ZapperZ]

I would disagree with Zapperz that the formalism came first, and also "It is the interpretation that is trying to put into words what the formalism presents!" QM effectively started with Planck, Einstein, Bohr/Sommerfeld (on a wrong track), de Broglie then Schrödinger, and we generally agree that the Schrödinger equation, in the form Schrödinger presented it, and the Uncertainty Principle (which in my opinion is actually implied by the Schrödinger equation) came first and essentially contains quantum mechanics. The formalism followed. Interpretation might involve what the formalism means in some eyes, but to me interpretation falls back to what does ψ mean? In my opinion, there are three basic interpretations, with a variety of variations to each. First there is the fundamental question, is there actually a wave or is it merely a mathematical artefact? De Broglie/Bohm, (and for that matter, me) consider that there is a wave (but that raises problems because where is it and why can't we detect it?) while most seem to say, no, there isn't, but that raises problems as to what actually causes diffraction? The probabilistic view works well mathematically, but it does not explain how the probabilities arise, or, for that matter, how they resolve. All of these issues are independent of formalism, but of course if you have used said formalism consistently, you probably feel very comfortable with it.

Sorry, you have a misunderstanding of what "formalism" means, at least in the way that I've used it. Formalism is the mathematical description. So the Schrodinger equation is part of the QM formalism. The mathematics of QM came first. Then people tried to figure out what those means.

Zz.

 

[Grinkle]

Now it appears that the issue that you have brought up is going in a different direction.

You have me confused with someone else, I think. As far as I know, I didn't bring up any issues in this thread.

 

[ZapperZ]

You have me confused with someone else, I think. As far as I know, I didn't bring up any issues in this thread.

You said:

The question of whether particles possesses specific states in between interactions/observations/measurements is not a problem with QM.

That's quantum superposition, is it not? That's not what is being asked in this thread, i.e. "collapse" of the wavefunction upon measurement.

Zz.

 

[Grinkle]

That's quantum superposition, is it not? That's not what is being asked in this thread, i.e. "collapse" of the wavefunction upon measurement.

I see. So you can understand my "B" level thinking - When I hear "collapse" I picture the possibility that pre-collapse there was no well defined state of the particle and post-collapse there is.

So the question that occurs to me is whether that pre/post collapse transition is 'real' or whether there was a well defined state all along. I didn't realize I was talking about a different thing altogether than collapse.

In any case, my point was not that I am owed an answer from QM. My point is that asking such a question is not the act of saying there is a problem with QM.

I can accept that my questions are poorly posed. Instead of wasting folks time honing my layman's phrasing, I will watch with interest any discussion on the below question.

is there actually a wave or is it merely a mathematical artefact?

 

[ZapperZ]

I see. So you can understand my "B" level thinking - When I hear "collapse" I picture the possibility that pre-collapse there was no well defined state of the particle and post-collapse there is.

So the question that occurs to me is whether that pre/post collapse transition is 'real' or whether there was a well defined state all along. I didn't realize I was talking about a different thing altogether than collapse.

In any case, my point was not that I am owed an answer from QM. My point is that asking such a question is not the act of saying there is a problem with QM.

I can accept that my questions are poorly posed. Instead of wasting folks time honing my layman's phrasing, I will watch with interest any discussion on the below question.

But if you want "answers" on superposition or whether it is "real" (it is), then you should scour the numerous threads we already have on that topic or the topic of Schrodinger cat, which is basically an attempt at illustrating the issue of quantum superposition. Do a search on Delft/Stony Brook experiments on SQUIDs and the measurement of the coherence gap, which would not be there if quantum superposition doesn't occur.

But this is not what the OP asked in this thread.

Zz.

 

[Grinkle]

But if you want "answers" on superposition or whether it is "real"

Yes, I am confident I can find a lot of good material with a thread search. It was not my intent to change the topic of this thread.

 

[bhobba]

But I think the main thing is that everyone agrees on some consistent use of terminology, whatever that is.

Right, that's all that's needed. We do not want to confuse anyone, just discuss and/or help people with physics. I am happy with any reasonable view, just as long as we all stick to it.

Thanks
Bill

 

[atyy]

Right, that's all that's needed. We do not want to confuse anyone, just discuss and/or help people with physics. I am happy with any reasonable view, just as long as we all stick to it.

Thanks
Bill

So is the projection postulate common to all interpretations?

 

[bhobba]

So is the projection postulate common to all interpretations?

Under the following definition yes:

The postulate in quantum mechanics that observation of a physical system, by determining the value of an observable, results in the transition of the quantum state of the system to a particular eigenstate corresponding to the eigenvalue of the observed quantity.

But IMHO its not a postulate - it comes from the two axioms of Ballentine.

Thanks
Bill

 

[atyy]

Under the following definition yes:

The postulate in quantum mechanics that observation of a physical system, by determining the value of an observable, results in the transition of the quantum state of the system to a particular eigenstate corresponding to the eigenvalue of the observed quantity.

But IMHO its not a postulate - it comes from the two axioms of Ballentine.

Thanks
Bill

What is the difference between "collapse" (in your language) and the projection postulate?

 

[bhobba]

What is the difference between "collapse" (in your language) and the projection postulate?

According to John Von-Neumann its basically the same thing.

Thanks
Bill

 

[PeterDonis]

The postulate in quantum mechanics that observation of a physical system, by determining the value of an observable, results in the transition of the quantum state of the system to a particular eigenstate corresponding to the eigenvalue of the observed quantity.

Note carefully, though, that as far as basic QM (i.e., without any interpretations, just calculating predictions) is concerned, "transition of the quantum state" does not mean the actual system's state necessarily changes. It's only a transition in the mathematical model--the machinery you use to make predictions about future measurement results. Whether or not that transition in the model corresponds to a "real" transition in the system being modeled is interpretation dependent. Unfortunately, it takes a lot of care and precision in language to keep this distinction clear, since the usual ordinary language we use to talk about this stuff leaves it ambiguous.

 

[zonde]

It's only a transition in the mathematical model--the machinery you use to make predictions about future measurement results. Whether or not that transition in the model corresponds to a "real" transition in the system being modeled is interpretation dependent.

It can't be interpretation dependent. Think of three polarizers experiment where you have two orthogonal polarizers and insert the third between. The change in observed result between two polarizers and three polarizers setups is real. Interpretation has to predict it. How would any current interpretation predict it without a physical transition in the system being modeled?

 

[rubi]

How would any current interpretation predict it without a physical transition in the system being modeled?

A polarizer isn't modeled by a projection, but by unitary evolution. A projection is only required when we obtain information. Most physicists interpret it as an update of (quantum) information, rather than a physical process.

 

[zonde]

A polarizer isn't modeled by a projection, but by unitary evolution.

Let's treat the first polarizer as state preparation so that we have pure H polarized state after the first polarizer. After the second polarizer the state is say in superposition of being +45deg and absorbed by the second polarizer + being -45deg and passed through the second polarizer. Is it ok so far? Now how do you model interaction with the third polarizer. Only the part that passed the second polarizer interacts with the third polarizer. Or no?

 

[rubi]

Let's treat the first polarizer as state preparation so that we have pure H polarized state after the first polarizer. After the second polarizer the state is say in superposition of being +45deg and absorbed by the second polarizer + being -45deg and passed through the second polarizer. Is it ok so far? Now how do you model interaction with the third polarizer. Only the part that passed the second polarizer interacts with the third polarizer. Or no?

Every polarizer $i$ has an associated unitary operator $U_i$. You start with a state $\psi$. After the first polarizer, the state it $U_1\psi$. After the second one, it is $U_2 U_1\psi$ and after the third polarizer, it is $U_3 U_2 U_1\psi$. No projections are involved. You have to insert projections only if you perform measurements in between.

 

[zonde]

Every polarizer $i$ has an associated unitary operator $U_i$. You start with a state $\psi$. After the first polarizer, the state it $U_1\psi$. After the second one, it is $U_2 U_1\psi$ and after the third polarizer, it is $U_3 U_2 U_1\psi$. No projections are involved. You have to insert projections only if you perform measurements in between.

And what are the components of the state $U_3 U_2 U_1\psi$? One component has to have amplitude who's square is 1/8 given common polarizer angles for this example (H, 45deg, V)

 

[rubi]

And what are the components of the state $U_3 U_2 U_1\psi$? One component has to have amplitude who's square is 1/8 given common polarizer angles for this example (H, 45deg, V)

Well, I'm too lazy right now to write down all the matrices and perform the calculation, but of course, if you choose $U_i$ and $\psi$ correctly and then expand the final state in the basis of your choice, you will get the correct experimental predictions.

 

[Lord Jestocost]

As defined in the standard textbook - Decoherence and the Quantum-to-Classical Transition by Maximilian A. Schopenhauer:

I think you surely meant “Schlosshauer” and not the German philosopher Arthur Schopenhauer (1788 – 1860) (https://en.wikipedia.org/wiki/Arthur_Schopenhauer).

Nevertheless, Schopenhauer’s reasoning and transcendental idealism might be of interest with respect to discussions about quantum mechanics. Schopenhauer described transcendental idealism as a "distinction between the phenomenon and the thing in itself", and a recognition that only the phenomenon is accessible to us because "we know neither ourselves nor things as they are in themselves, but merely as they appear.”

 

[Carpe Physicum]

One way to ground everything in reality is to think purely about the records of experiments that are stored in computer memory. Very often, that's a list of times at which events happened. If you think about APDs, for such devices we might run a wire (or we use a fiber optic cable, or wi-fi, ...) from the APD to the computer that records the data. On that wire, there will be a voltage that most of the time will be near zero voltage, but occasionally an "avalanche" happens, the voltage goes to non-zero (1 volt, 20 volts, whatever), then the hardware checks a clock for the time and records it in memory and then to hard disk for later analysis. The hardware also resets the APD as soon as possible so another avalanche can happen. From a computing and signal analysis point of view, what's just happened was a compression: we could have recorded the voltage picosecond by picosecond to 14-bit accuracy, but we just recorded the time when there was a signal transition from zero to not-zero.
There are certainly experiments that record continuous signals (at finite accuracy and resolution, with a fixed schedule, because it's all going into digital memory), but the analyses that you'll find detailed in physics papers are often on the hunt for discrete structure of some kind, and very often a discrete structure is there to be found.

Everything so far is classical electronics (except the last sentence, which presaged what comes next here) about events and signals. There is no mention of particles or of particle properties whatsoever. Now comes the analysis, where we will introduce the idea that particles (or, more generally, "systems", a field, thing or things that are kinda classical) explain why we see the events and signals that we see. The Correspondence Principle gives us a way, called quantization, to convert a classical dynamics for some kind of classical system (mechanics or electromagnetism) into a differential equation that describes the evolution over time of a "statevector", the Schrödinger equation. The statevector models/predicts the statistics of many different kinds of measurement results (anything that can come out of a mathematical analysis of the raw data of the previous paragraph), and, crucially, how they change over time. Some of those measurement results are "incompatible" with each other, so that properly speaking we can't talk about correlations between incompatible measurements.
The Correspondence Principle is quite tricky because it cannot be a perfect map from a classical dynamics to a quantum dynamics, but it's been a fairly decent guide for the last 90 years, so we're not going to give it up until we have something better. If we find that the quantization of a classical mechanics works well as a model for the signal analysis we do for the raw data, which has to work nicely as the statistics change over time, we pretty much say that the quantized classical system explains the raw data, except of course that we don't as much understand what we're doing when we quantize as we'd like to.
So, @Carpe Physicum, it looks as if you might have left this conversation. If you're still here, I hope you find this a little useful even though it's definitely my idiosyncratic way of thinking about the question. I've tuned the above a little to the computing world because that seems to be your sort of thing, which wasn't hard to do, however, because that's also pretty close to my sort of thing. If you reply, I can point you to the first video on my YouTube channel. I'm trying to figure out whether you really mean Carpe, or perhaps there's a little Carping in your question?

Sorry I did drop for a bit. Very interesting and informative. As a layman it always is amazing how theory and brute reality, as in where the data actually comes from and how it's used, how those two relate. As for the name, yes Carpe, Seize The Physics! (in broken latin of course :) )

 

[stevendaryl]

Every polarizer $i$ has an associated unitary operator $U_i$. You start with a state $\psi$. After the first polarizer, the state it $U_1\psi$. After the second one, it is $U_2 U_1\psi$ and after the third polarizer, it is $U_3 U_2 U_1\psi$. No projections are involved. You have to insert projections only if you perform measurements in between.

Are you talking about a special kind of polarizer, or the usual "polarizing filter" (such as is used in some sunglasses)? In the usual kind of polarizing filter, light that is polarized one way passes through unchanged, while light that is polarized perpendicular to that is absorbed by the filter. Absorption of a photon is irreversible, which usually means it is not described by a unitary transformation. Maybe I'm misunderstanding what you're saying?

I think there are devices that simply rotate the polarization of incoming light, without absorption.

 

[member 648030]

I try to explain what "collapse of the wave function" means without entering "semantic" discussions.
Take a coin and throw it. While the coin is launched, it has no value, it is, so to speak, both head and cross. Since the head and the cross are 1/2 chance to exit, we can formalize the "state" of the coin as a linear combination of "heads" and "cross": indicating with $\psi$ the wave function of the coin we can write:
$\psi = 1/2H+1/2C$
where C means cross, H, head, and 1/2 is the probability factor that exits H (or C).
Suppose the coin lands on a table and you say, "Oh, head out!"
Then the wave function, which was $\psi$ before became H.
In other words there has been a reduction (or collapse) from $\psi$ to H
Or, mathematically formalizing:
$\psi \rightarrow H$
This is the "collapse" of the wave function, neither more nor less

 

[rubi]

Are you talking about a special kind of polarizer, or the usual "polarizing filter" (such as is used in some sunglasses)? In the usual kind of polarizing filter, light that is polarized one way passes through unchanged, while light that is polarized perpendicular to that is absorbed by the filter. Absorption of a photon is irreversible, which usually means it is not described by a unitary transformation. Maybe I'm misunderstanding what you're saying?

I think there are devices that simply rotate the polarization of incoming light, without absorption.

Well, polarizers are understood quite well quantum mechanically in solid-state physics. What happens during an absorption is that photons are scattered into phonons, i.e. lattice vibrations, in the polarizer and the photon-phonon cross section is higher if the photon is polarized appropriately with respect to the lattice structure of the polarizer material. Effectively, this results in a low probability for photons with the wrong polarization to pass. Nevertheless, the whole process is unitary.

 

[Nugatory]

This is the "collapse" of the wave function, neither more nor less

That is not right.
The quantum mechanical state "superposition of A and B" is different from the quantum mechanical state "It is A or B and we don't know which yet", and there is no classical analogy for the former.

 

[stevendaryl]

Well, polarizers are understood quite well quantum mechanically in solid-state physics. What happens during an absorption is that photons are scattered into phonons, i.e. lattice vibrations, in the polarizer and the photon-phonon cross section is higher if the photon is polarized appropriately with respect to the lattice structure of the polarizer material. Effectively, this results in a low probability for photons with the wrong polarization to pass. Nevertheless, the whole process is unitary.

I'm willing to believe that the whole process is unitary if you include photons + phonons + the whole rest of the universe. But it's not a unitary transformation on photon states.

 

[rubi]

I'm willing to believe that the whole process is unitary if you include photons + phonons + the whole rest of the universe. But it's not a unitary transformation on photon states.

You don't need to include all that into the model (especially not the rest of the universe, which is not relevant anyway). For an effective (black box) model, it suffices to associate with the photon a property "absorbed/not absorbed", which allows you to include the transmission probability into the description, and ignore the physical details completely. A polarizer doesn't rotate a vertical photon into a horizontal photon, but rather a transmissible photon into an absorbed photon. The projection happens when we get to know whether the photon was absorbed or not. After all, the detector doesn't measure the polarization but rather just the presence of the photon and thus doesn't project onto the polarization states. If we don't measure that in between, then the whole process is completely unitary and I don't think that's controversial.

 

[stevendaryl]

You don't need to include all that into the model (especially not the rest of the universe, which is not relevant anyway). For an effective (black box) model, it suffices to associate with the photon a property "absorbed/not absorbed", which allows you to include the transmission probability into the description, and ignore the physical details completely. A polarizer doesn't rotate a vertical photon into a horizontal photon, but rather a transmissible photon into an absorbed photon. The projection happens when we get to know whether the photon was absorbed or not. After all, the detector doesn't measure the polarization but rather just the presence of the photon and thus doesn't project onto the polarization states. If we don't measure that in between, then the whole process is completely unitary and I don't think that's controversial.

I think you might be mixing up two different things. As described in the Wikipedia article on photon polarization:

A linear filter transmits one component of a plane wave and absorbs the perpendicular component...

An ideal birefringent crystal transforms the polarization state of an electromagnetic wave without loss of wave energy...A birefringent crystal is a material that has an optic axis with the property that the light has a different index of refraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis...

It's possible that a polarizing filter can be understood in terms of a birefringent crystal, but absorption is not a unitary transformation on the photon state. Passage through a birefringent crystal leaves the energy of the beam unchanged, while passage through a filter attenuates the energy.

 

[Carpe Physicum]

That is not right.
The quantum mechanical state "superposition of A and B" is different from the quantum mechanical state "It is A or B and we don't know which yet", and there is no classical analogy for the former.

Can you explain how they're different without resorting to math (I pre-appreciate your patience)? Seems like you're just mincing words. He could have said while it's in the air the coin IS heads and tails at the same time. And once it hits the table it collapses to Heads (for ex). Are you maybe saying there is no A and B really, and just some single A/B amalgamated state so to speak? (Which if this is the case then I can see the point that just talking about states really misrepresents the discussion.)

 

[rubi]

I think you might be mixing up two different things. As described in the Wikipedia article on photon polarization:
It's possible that a polarizing filter can be understood in terms of a birefringent crystal, but absorption is not a unitary transformation on the photon state. Passage through a birefringent crystal leaves the energy of the beam unchanged, while passage through a filter attenuates the energy.

No, I'm talking about a linear polarizer. You can model the absorption in the following way (ignoring all the physical details):
$\mathcal H_1 = \mathrm{span}\{h,v\},\, \mathcal H_2 = \mathrm{span}\{n,a\},\, \mathcal H = \mathcal H_1\otimes \mathcal H_2$
$U h\otimes n = h\otimes n,\, U v\otimes n = v \otimes a,\, U h \otimes a = h \otimes a,\, U v \otimes a = v \otimes n$
The Hilbert space $\mathcal H_2$ marks photons as absorbed ($a$) or not absorbed ($n$) and the unitary matrix $U$ models a linear polarizer aligned along the horizontal axis. In a more realistic model, $\mathcal H_2$ would be the Hilbert space of the phonons and the unitary matrix would be determined by some Hamiltonian that describes the photon-phonon interaction. But like I said, a black box description suffices and it's very far from a many-worlds model.

 

[DarMM]

Can you explain how they're different without resorting to math (I pre-appreciate your patience)? Seems like you're just mincing words.

No it's an important difference not word mincing. I'll try to be nontechnical.

Superposed states of spin up and down have different statistics for certain observables compared to just "Spin up or Spin down".

For example "Spin up or Spin down" has an average value when measuring Spin left/right of 0, i.e. comes up Left and Right 50:50.
"Spin up superposed with Spin down" will always come out Spin right, not 50:50 left/right.

 

[zonde]

A polarizer isn't modeled by a projection, but by unitary evolution.

Well, I'm too lazy right now to write down all the matrices and perform the calculation, but of course, if you choose $U_i$ and $\psi$ correctly and then expand the final state in the basis of your choice, you will get the correct experimental predictions.

No, I'm talking about a linear polarizer. You can model the absorption in the following way (ignoring all the physical details):
$\mathcal H_1 = \mathrm{span}\{h,v\},\, \mathcal H_2 = \mathrm{span}\{n,a\},\, \mathcal H = \mathcal H_1\otimes \mathcal H_2$
$U h\otimes n = h\otimes n,\, U v\otimes n = v \otimes a,\, U x \otimes a = x \otimes n$
The Hilbert space $\mathcal H_2$ marks photons as absorbed ($a$) or not absorbed ($n$) and the unitary matrix $U$ models a linear polarizer aligned along the horizontal axis. In a more realistic model, $\mathcal H_2$ would be the Hilbert space of the phonons and the unitary matrix would be determined by some Hamiltonian that describes the photon-phonon interaction. But like I said, a black box description suffices and it's very far from a many-worlds model.

You gave a non-mainstream idea. You refuse to show how it reproduces experimental predictions of mainstream approach. Please provide a reference otherwise it's your personal theory (not interpretation).

 

[atyy]

To give an example of something similar to what rubi is talking about in posts #49, #51 and #64, a unitary description of a polarizing beam splitter is given in Eq 1.22 of http://copilot.caltech.edu/documents/278-weihs_zeillinger_photon_statistics_at_beamsplitters_qip.pdf.

It is true that a polarizer can be modeled by a projection, but it is equally true that it can be modeled by unitary evolution in a larger Hilbert space as long as no measurement is performed.

 

[rubi]

You gave a non-mainstream idea. You refuse to show how it reproduces experimental predictions of mainstream approach. Please provide a reference otherwise it's your personal theory (not interpretation).

Physical models of polarizers are absolutely mainstream and standard solid-state physics. Modeling them by projections may suffice for some simple applications, but is not physically realistic. There is no doubt in the physics community, that the realistic situation is governed by purely unitary evolution. You even need to do that in order to get correct predictions for transmission efficiencies for example, because they follow from concrete calculations of photon-phonon scattering amplitudes. The inner workings of polarizers are part of introductory courses in optics, so it's clearly mainstream physics. If you want a reference, you can look at @atyy's article. The matrix (1.22) is exactly the matrix $U$ of my post #64.

Anyway, my point is that quantum systems are always governed by unitary evolution. Projections are only inserted when we acquire information through measurements. This is not the case in your example in post #48.

 

[stevendaryl]

To give an example of something similar to what rubi is talking about in posts #49, #51 and #64, a unitary description of a polarizing beam splitter is given in Eq 1.22 of http://copilot.caltech.edu/documents/278-weihs_zeillinger_photon_statistics_at_beamsplitters_qip.pdf.

It is true that a polarizer can be modeled by a projection, but it is equally true that it can be modeled by unitary evolution in a larger Hilbert space as long as no measurement is performed.

As I said in my post, I certainly see that some types of polarizers can be modeled by a unitary transformation. But in the case of a standard polarizing filter that absorbs light of one polarization and passes light of the orthogonal polarization, I don't see how that can be a unitary transformation on the state of the light. Maybe you can think of such a filter as a polarizing beam splitter together with an absorber: one of the two beams is directed into the absorber?

 

[zonde]

It is true that a polarizer can be modeled by a projection, but it is equally true that it can be modeled by unitary evolution in a larger Hilbert space as long as no measurement is performed.

It is easy to convince me. Calculate correct prediction for three polarizers experiment's outcome with usual angles (0, 45, 90 deg.,) using unitary evolution.