Colapse of the Wave Funcion and the Schroedinger Equation

[wofsy]

If a single silver atom passes through a Stern-Gerlach apparatus, its state (which I will write in the form |position,spin>) changes from |center,s> to a superposition of |right,+> and |left,->. This is not a measurement. It's just an interaction that produces a correlation between spin eigenstates and position eigenstates*, which enables you to determine the spin by measuring the position. A measurement of the spin is an interaction which produces a correlation between the spin eigenstates and macroscopically distinguishable states of a system that for all practical purposes can be treated as classical.

*) They aren't really position eigenstates, but they are as close as you can get. They are superpositions of narrow ranges of position eigenstates.

For example, you can measure the spin state of a silver atom by letting it pass through a Stern-Gerlach apparatus and allowing it to be detected by one of two detectors that you have put in the two relevant positions. Each detector can always be described as either having detected a particle or not having detected a particle.

So what the Stern-Gerlach apparatus does isn't a measurement. It's a state preparation. This concept is more clear if we consider an ensemble of systems instead of an individual system. If we send a beam of silver atoms into the apparatus, it gets split in two. The spin state of the initial beam is represented by the statistical operator (a.k.a. density operator, density matrix or mixed state) |+><+|+|-><-|. The state of the beam that comes out on the right is represented by the pure state |+><+|. What this means, at least for those of us who think of both state vectors and statistical operators as representing the probabilities of different measurement results (rather than as representing objective properties of physical systems) is that the "reduction of the state vector", "collapse of the wavefunction", or whatever you'd like to call it, isn't even an interaction that the system is involved in. It's a selection of a subset of the ensemble, on which we indend to perform measurements.

Suppose that we only send one atom through, and that its initial state is |+>+|->. I like to think of a state vector as representing the probabilities of different results (and not as an objective property of the system), but this is equivalent to saying that the state vector represents an objective property of an ensemble of identically prepared systems. In other words, we have to imagine a large number of identical experiments being performed. The set of all silver atoms in all those experiments is an ensemble, and the "collapse of the wavefunction" is a selection of a subset of that ensemble.

By the way, what I just described here is also described in the excellent book "Lectures on quantum theory: mathematical and structural foundations" by Chris Isham, which I read about a week ago. I had not understood the importance of distinguishing between state preparation and measurement until I read it.

OK. I understand this. However if I put a barrier inside the Stern-Gerlach apparatus that blocks the left trajectory then I have collapsed the wave function of any particle that exits the apparatus. The barrier is perfectly consistent with the Shroedinger equation. So without measurement using a detector I can collapse the wave function.

 

[wofsy]

I do not feel that this thread has gone in the right direction. I understand the what the postulates of quantum mechanics say. But these postulates didn't fall from the sky. They were arrived at after soul searching reflection. It is clear that the collapse of the wave function, the salient effect of a measurement, was proved to be outside of the possible solutions of the Shroedinger equation even with arbitrarily complex potentials added in.

That was my question. This must be a purely mathematical theorem.

But also it is clear that physicists decided that total collapse was what actually happened rather than approximate collapse and it was only later that some people proposed that approximate collapse could occur spontaneously. But this is modeled as a random even independent of the shroedinger equation. So it must also be that approximate collapse is not possible in the Shroedinger equation either.

In my feeble examples I tried to construct collapses that were in fact solutions of the Shroedinger equation but those examples are stabs and it doesn't really matter if they are right or wrong.

What is important is the theorem that says that collapses and approximate collapses are not solutions of the Shroedinger equation. Any ideas?

 

[Fredrik]

OK. I understand this. However if I put a barrier inside the Stern-Gerlach apparatus that blocks the left trajectory then I have collapsed the wave function of any particle that exits the apparatus. The barrier is perfectly consistent with the Shroedinger equation. So without measurement using a detector I can collapse the wave function.

When a silver atom is sent through the SG apparatus, its state is changed according to

|\mbox{center}\rangle|s\rangle\rightarrow|\mbox{right}\rangle|+\rangle+|\mbox{left}\rangle|-\rangle

where

|s\rangle=|+\rangle+|-\rangle

This is a unitary process, meaning that the state vector on the right is just

e^{-iHt}|\mbox{center}\rangle|s\rangle

where the Hamiltonian is defined by the SG apparatus. This process is not a measurement. It's just a state preparation. A state preparation of an ensemble E is an interaction that that partitions E into smaller ensembles E_i that have the property that a subsequent measurement of some observable A on E_i would yield the result a_i with certainty. A measurement is an interaction that also entangles the states that the E_i are in with macroscopically distinguishable states of a system that's approximately classical.

The interaction that prepares the state can be a measurement, but it doesn't have to be. In the Stern-Gerlach experiment, it isn't. If it was, the position of the silver atom would be either "right" or "left". It wouldn't be a superposition of both.

I said before that the collapse of the wavefunction is just a selection of one of the E_i. I think I'm going to have to take that back. A collapse is a transition from a pure state to a mixed state, but we don't get a mixed state just by choosing one of the beams.

You're suggesting that we insert a barrier that prevents the right beam from leaving the SG apparatus. I'm not sure what the effect of the barrier would be. Does it or does it not entangle the spin eigenstates with macroscopically distinguishable states of the barrier? I don't know. If it does, then it has collapsed the wavefunction, but this isn't a collapse without a detector. In this case the barrier is the detector. It's just a really bad one, in the sense that doesn't make it easy for you to find out what the result was.

 

[Fredrik]

That was my question. This must be a purely mathematical theorem.
...
What is important is the theorem that says that collapses and approximate collapses are not solutions of the Shroedinger equation. Any ideas?

If the system has been prepared in state |\psi\rangle, the SE says that the corresponding statistical operator (density matrix) evolves according to

|\psi\rangle\langle\psi|\rightarrow e^{-iHt}|\psi\rangle\langle\psi|e^{iHt}

The operator on the right is a pure state, i.e. it's the projection operator of a 1-dimensional subspace. A measurement on the other hand, changes the state according to

|\psi\rangle\langle\psi|\rightarrow\sum_aP_a|\psi\rangle\langle\psi|P_a

where P_a=|a\rangle\langle a| is the projection operator of the 1-dimensional subspace that contains the eigenvector with eigenvalue a. Here the operator on the right isn't a pure state.

 

[zenith8]

I do not feel that this thread has gone in the right direction. I understand the what the postulates of quantum mechanics say. But these postulates didn't fall from the sky. They were arrived at after soul searching reflection. It is clear that the collapse of the wave function, the salient effect of a measurement, was proved to be outside of the possible solutions of the Shroedinger equation even with arbitrarily complex potentials added in.

That was my question. This must be a purely mathematical theorem.

But also it is clear that physicists decided that total collapse was what actually happened rather than approximate collapse and it was only later that some people proposed that approximate collapse could occur spontaneously. But this is modeled as a random even independent of the shroedinger equation. So it must also be that approximate collapse is not possible in the Shroedinger equation either.

In my feeble examples I tried to construct collapses that were in fact solutions of the Shroedinger equation but those examples are stabs and it doesn't really matter if they are right or wrong.

What is important is the theorem that says that collapses and approximate collapses are not solutions of the Shroedinger equation. Any ideas?

Let us first assume the wave function is a real field (a 'real wave' if you like). Of course you can say that it isn't and that QM is just a way of cataloguing observations from an unknown underlying 'mechanism' which one should refuse to speculate about - but (a) that doesn't get you anywhere and is a bit boring, (b) you then have no way of explaining how you get a perfectly standard interference pattern in e.g. a two-slit experiment. What exactly is interfering with what, if it isn't a real wave passing through the slits? To say that 'nothing passes through the slits' as is often done is simply silly - clearly something does and to say otherwise is merely to play with words (imperfectly quoting Deutsch).

So OK - the wave function objectively exists - then the Schroedinger evolution in time will give you a wave representing all possible outcomes of the experiment (in your example, a finite lump of wave in both branches of the SG apparatus). When this is coupled to a macroscopic apparatus (e.g. detect the position of your electron/silver atom with a phosphorescent screen) then the Schroedinger time evolution predicts a macroscopic superposition of everything that could happen, which is not what you see.

The standard way of getting round this is to say (effectively) "Ah well, when we observe it, the wave function er.. stops evolving according to the time-dependent Schroedinger equation, and er.. does something else - it collapses.". This is not really a solution - though often presented as such - it is merely stating that what you see is different from what the Schroedinger equation predicts (what do you need to do to make this happen, exactly? How is this different from just any ordinary many-body interaction?) .

At this point someone will bring in 'decoherence' to save the day - however this merely says that the different 'branches' of the wave function will tend to become orthogonal in time (i.e. they cease to have a finite overlap integral) thus you can't subsequently bring the different branches to interference. But so what? All the different branches continue to exist.

Then in desperation, someone will say "Ah well, you see, er.. every branch of the wave function then forms er.. its own separate parallel universe and disappears off into that". He ought to be carted off in a straightjacket at this point, but somehow manages through sheer bloody mindedness to convince himself that this is the only way to understand QM. Well OK, you can do this if you want, but it really should be absolutely the last resort in the absence of any simpler and much less bizarre explanation.

In reality you have - or should have - just two genuine options to explain what you see (I imperfecty quote Bell):

(1) the wave function is not all there is (--> hidden variables, with pilot-wave/Bohm theory being the only mainstream example),

or

(2) The Schroedinger equation - note the c - is wrong (--> objective collapse models - GRW etc.).

Which of these two you prefer is up to you. I like the former because it's just simple, and all the usual paradoxes disappear. Just say that particles (electrons or whatever) exist and are guided by the wave. The wave evolves according to the Schroedinger equation. The particles follow trajectories derived from the probability current (the guidance equation). The particle deterministically ends up in one branch or the other. The wave function doesn't collapse at all, but it 'effectively' does because the particles end up being guided only by one of the branches (remember decoherence has made them not overlap).

In the two-slit experiment, the particle goes through one slit, the wave goes through both. An interference pattern develops in the wave, which affects the trajectories of the particles and guides them into clumps. In the SG experiment, the wave separates into two branches, the particles ends up in one branch or the other at random (depending on its starting position) - note we're not measuring a 'property of the particle called spin' here - which is interesting.

Note that the answer to most questions that people put on the Quantum Physics forum depend on a careful understanding of the various interpretations of QM. So which ever moderator keeps shoving discussions of them into the Philosophy section (e.g. this morning's Interpretation Poll - whose originator has made a similar complaint) should in my opinion stop doing so. How does stopping people discussing the meaning of a theory contribute to its understanding?

For what it's worth, both the above options (1) and (2) make specific experimentally testable predictions (albeit with a large degree of practical difficulty) and as such should be taken to be physics - not philosophy. To say otherwise and maintain that all discussion of 'mechanism' is meaningless is merely to maintain the discredited Bohrian attitude of the 1960s and before. And you don't want to do that, do you? Keep up! Things are moving on..

 

[meopemuk]

Let us first assume the wave function is a real field (a 'real wave' if you like). Of course you can say that it isn't and that QM is just a way of cataloguing observations from an unknown underlying 'mechanism' which one should refuse to speculate about - but (a) that doesn't get you anywhere and is a bit boring, (b) you then have no way of explaining how you get a perfectly standard interference pattern in e.g. a two-slit experiment. What exactly is interfering with what, if it isn't a real wave passing through the slits? To say that 'nothing passes through the slits' as is often done is simply silly - clearly something does and to say otherwise is merely to play with words (imperfectly quoting Deutsch).

Well said. Though I disagree with it. I prefer the (boring) idea that wave function is just an abstract quantity, which is a record of our ignorance about the physical system. Wave function is not a "real" or "physical" field. I am not interested in understanding of "what interferes with what" when electron passes through the slits. The detailed mechanism of what happens in the region of slits is not observable in principle (yes, you can observe what's going on there, but then you would need to change the experimental setup, and the answers you'd get will not have any relevance to the original double-slit setup), so this mechanism should not concern physicists. It is entirely sufficient to have an abstract mathematical model of what's going on, and this model is well-known: the Hilbert space, the Hamiltonian, the wave function, the Schroedinger equation for the wave function, and the wave function collapse at the point of measurement.

 

[zenith8]

Well said. Though I disagree with it. I prefer the (boring) idea that wave function is just an abstract quantity, which is a record of our ignorance about the physical system.

Fair enough. But..

(1) Whose ignorance?

(2) Ignorance about what exactly?

(3) I repeat - how can the terms of a quantum superposition interfere with each other, producing an observable interference pattern, if such a superposition is just an expression of our ignorance? And don't say "I'm not allowed to ask the question."

and

(4) For God's sake, why? Don't you understand that girls don't like boring people?

Wave function is not a "real" or "physical" field.

Nobody knows that. Do you have some subtle reason for being certain about it? What little evidence there is points to exactly the opposite view.

I am not interested in understanding of "what interferes with what" when electron passes through the slits.

See (4) above. But - I'm genuinely curious here. Can you speculate as to why - out of all the hundred or so branches of science - only quantum physicists insist that they may absolutely not try to understand - as a fundamental point of principle - how or why their methods work? Doesn't quality of explanation count for anything? Weird.

The detailed mechanism of what happens in the region of slits is not observable in principle (yes, you can observe what's going on there, but then you would need to change the experimental setup, and the answers you'd get will not have any relevance to the original double-slit setup), so this mechanism should not concern physicists.

Yes - but its absolutely clear as to why that would be the case. If the probe is as significant as the probed then you'll have problems measuring things without disturbing them. So what? That's what you would expect. It's like trying to measure the trajectory of the space shuttle by bouncing the Starship Enterprise off it, then claiming that as a result the objects in classical mechanics can't possibly exist.. Nevertheless in hidden variables QM like pilot-wave theory there are circumstances in which experimentally testable predictions can be made (though with great difficulty).

You have totally missed the fact that logical positivism (for such is your belief) was completely discredited as a philosophical concept as far back as the 1960s. It is of course true that most physicists haven't noticed yet. There are all sorts of unobservable things which nevertheless are respectable theoretical and philosophical entities. Quarks, wave functions(!), particles in classical stat mech, blah, etc. How would you examine the approach to thermal equilibrium in classical stat mech if you refuse to acknowledge the possibility of little particles running around and bashing into each other?

It is entirely sufficient to have an abstract mathematical model of what's going on, and this model is well-known: the Hilbert space, the Hamiltonian, the wave function, the Schroedinger equation for the wave function, and the wave function collapse at the point of measurement.

But don't you find it even slightly interesting that the whole Hilbert space thing turns out to be precisely the right theoretical apparatus to catalogue (probabilistically) the dynamics of particles obeying the obvious (1927) quantum dynamics deduced from the Schroedinger equation? Not even slightly? So much so that one must never ever even think about it?

The fact is that you/we have been brainwashed (and I mean this with the greatest respect - it isn't your fault) to believe that empirical adequacy plus a formalized proof procedure is the best any theory can properly aspire to - and you've been brainwashed by a man who is increasingly acknowledged as a poor philosopher and (carefully hidden, this) a worse mathematician. I'm not going to name names, because it's probably against the forum rules. But you know who I'm talking about..

 

[meopemuk]

(3) I repeat - how can the terms of a quantum superposition interfere with each other, producing an observable interference pattern, if such a superposition is just an expression of our ignorance? And don't say "I'm not allowed to ask the question."

Any (quantum-mechanical) experiment consists roughly of three stages:

1) Preparation of the physical system
2) Evolution of the physical system
3) Measurement

These three steps are repeated many times, sufficient statistics is accumulated, and the outcome is the probability distribution for different possible measurement results. For example, in the double-slit experiment the step 1) is the emission of an electron by an electron gun; step 2) is passing of the electron through the slits; step 3) is a flash on the scintillating screen. Note that only in steps 1) and 3) our physical system (the electron in this case) can be directly observed. The goal of physics is to establish and predict correlations between experimentally measured (or measurable) results. So, ideally, a physical theory should tell us how results measured in step 3) depend on parameters (e.g., the current flowing through the electron gun) in step 1).

Quantum mechanics gives us a clear recipe for establishing such correlations:

1. Imagine 1-electron Hilbert space.
2. Construct a Hamiltonian in this Hilbert space, which takes into account the configuration of the slits and the relative positions of the electron gun, slits, and the screen.
3. Represent the scintillating screen by the Hermitian operator of position.
4. Form an initial state vector of the electron localized near the gun.
5. Calculate the time evolution of the state vector by employing the Hamiltonian found in 2.
6. Find the wave function (state vector) in the vicinity of the scintillating screen.
7. Expand this wave function in position eigenstates.
8. Squares of the expansion coefficients are probabilities for seeing flashes at particular points on the screen.

Quantum mechanics does not tell us what the electron is "actually" doing during its time evolution in step 2). It simply gives us a mathematical recipe for predicting experimental results. All parts of the QM formalism (Hilbert space, wave function, Hermitian operators, etc.) do not exist in nature. They exist only in our imagination or on paper.

Of course, you have the right to demand more. You may try to design a detailed "mechanism" of what "actually" is going on in step 2). However, the important point is that you can never check experimentally whether your suggested mechanism is right or wrong. Simply because, by definition, in step 2) the physical system is not in contact with any measuring device, and there is absolutely no way to "see" what the system is doing.

You may play smart and try to modify the experimental setup (e.g., by placing additional measuring devices near the slits). But then you have an entirely different experiment, and its quantum-mechanical description should be entirely different. The above steps 1), 2), 3) now change to 1'), 2'), 3'). You are not able to get rid of the mysterious step 2). You've simply changed it to 2'). The Hamiltonian of the system should change too. So, whatever information you obtain in this modified experimental setup maybe not relevant to the original setup.

The conclusion is that all these "mechanisms of quantum behavior" (also known as "interpretations of quantum mechanics") cannot be verified in experiments. So, if you and I stick to different interpretations, there is no objective way to resolve our dispute. This is like arguing which religion is better Islam or Buddhism?

If a question has grammatical sense, but it cannot be answered experimentally, then I argue that this question has no physical sense and should not be asked in physical context. Leave this question to theologians.

Can you speculate as to why - out of all the hundred or so branches of science - only quantum physicists insist that they may absolutely not try to understand - as a fundamental point of principle - how or why their methods work? Doesn't quality of explanation count for anything? Weird.

That's because quantum physicists are on the leading edge of science. They were first to reach the measurable limit of the physical world and to understand that there is nothing beyond that limit. It doesn't even make sense to ask what is beyond that limit. That's why quantum mechanics is the most advanced and weird creation of the human mind. (That's why girls love quantum physicists.)

The fact is that you/we have been brainwashed (and I mean this with the greatest respect - it isn't your fault) to believe that empirical adequacy plus a formalized proof procedure is the best any theory can properly aspire to - and you've been brainwashed by a man who is increasingly acknowledged as a poor philosopher and (carefully hidden, this) a worse mathematician. I'm not going to name names, because it's probably against the forum rules. But you know who I'm talking about..

Are you talking about Niels Bohr? Actually, I didn't have much respect for his ideas early in my scientific life. Only recently I've appreciated his deep philosophy. His main point is that physics is an experimental science, and the goal of quantum mechanics is not to describe the world in its completeness and complexity. Quantum mechanics is simply a mathematical tool for the analysis of specific experiments. In each experiment there is a clear separation between the physical system and the measuring apparatus (i.e., steps 2) and 3) above). So, there is absolutely no contradiction in applying different descriptions to the physical system and the measuring device.

If people want to get a "comprehensive" picture of the world, I'm afraid, quantum mechanics can't help them, and physics in general can't help them either. They should visit their nearby church/mosque/temple/sinagogue instead.

 

[Fredrik]

Can you speculate as to why - out of all the hundred or so branches of science - only quantum physicists insist that they may absolutely not try to understand - as a fundamental point of principle - how or why their methods work? Doesn't quality of explanation count for anything? Weird.

You have totally missed the fact that logical positivism (for such is your belief) was completely discredited as a philosophical concept as far back as the 1960s.

I really don't get why you say that Meopemuk's position is logical positivism. The Wikipedia article says that Karl Popper criticized logical positivism because he felt that their requirement of verifiability was too strong, and that what we should require instead is falsifiability. I think of QM as an algorithm that tells us how to calculate the probabilities of possible results of experiments, given the results of other experiments. I don't think it makes sense to think of QM as a description of our universe, or even as a description of a possible universe that resembles our own. It's just an algorithm. This sounds a lot more like Meopemuk's position than like yours. I came to this conclusion about QM, not by being brainwashed by Niels Bohr (I didn't even know that you where talking about him), but by accepting that falsifiability is the appropriate concept.

I define a "theory" as a set of statements that makes predictions about probabilities of possible results of experiments. (This set also has to be finite, logically consistent, etc., but those details aren't the issue here). Note that this is precisely the minimum requirement that we must impose to make sure that theories are falsifiable. All theories in science satsify this condition. QM just happens to be the first theory that was found that assigns non-trivial probabilities (i.e. not 0 or 1) to possible results of experiments. This is how QM is fundamentally different from all the other theories.

Let's pretend for a moment that we don't know QM already, and that we want to construct a theory that assigns non-trivial probabilities, just to make sure that such theories can exist. How would we do it? One way, possibly the easiest way, starts with the observation that in classical mechanics, the set of experimentally verifiable statements can be identified with subsets of phase space. (The statement that the energy of the system is between 9 and 10 Joules corresponds to the set of states that have an energy in that interval). Now suppose that we instead associate the experimentally verifiable statements with the subspaces of an inner product space space, and use the inner product to assign probabilities. Orthogonal means probability 0, parallel means probability 1, and all other angles correspond to non-trivial probabilities.

I think you know where this is going (which is good, because I don't know (yet) how to carry this through to completion). What we end up with is quantum mechanics. The Schrödinger equation follows from a very natural definition of the concept of "symmetry", and the requirement that the symmetries of spacetime (in particular invariance under translations in time) correspond to symmetries of the quantum theory.

But don't you find it even slightly interesting that the whole Hilbert space thing turns out to be precisely the right theoretical apparatus to catalogue (probabilistically) the dynamics of particles obeying the obvious (1927) quantum dynamics deduced from the Schroedinger equation? Not even slightly? So much so that one must never ever even think about it?

I think the Hilbert space framework is very natural if we just look at the minimum requirement that a set of statements must satisfy in order to be a "theory", but I'm amazed that it works so well, because it really looks like a toy theory that someone made up just to demonstrate that it's possible to assign non-trivial probabilities. This is one of many reasons why I think it's futile to try to interpret QM as a description of what "actually happens". A theory of what "actually happens" would be desirable, but it wouldn't look anything like QM. I think almost everyone would like to find such a theory, but no one even knows where to begin looking for it. It's also possible that such a "theory" wouldn't make any testable predictions (which means that it isn't falsifiable, and therefore not a theory). If that's the case, we have reached the limit of the scientific method.

 

[meopemuk]

Let's pretend for a moment that we don't know QM already, and that we want to construct a theory that assigns non-trivial probabilities, just to make sure that such theories can exist. How would we do it? One way, possibly the easiest way, starts with the observation that in classical mechanics, the set of experimentally verifiable statements can be identified with subsets of phase space. (The statement that the energy of the system is between 9 and 10 Joules corresponds to the set of states that have an energy in that interval). Now suppose that we instead associate the experimentally verifiable statements with the subspaces of an inner product space space, and use the inner product to assign probabilities. Orthogonal means probability 0, parallel means probability 1, and all other angles correspond to non-trivial probabilities.

I think you know where this is going (which is good, because I don't know (yet) how to carry this through to completion). What we end up with is quantum mechanics.

This is exactly the path to "deriving" the formalism of quantum mechanics adopted in the "quantum logic" approach. The derivation of the Hilbert space machinery has been "carried through to completion". All the basic ideas are contained in the beautiful classic paper

G. Birkhoff, J. von Neumann, "The logic of quantum mechanics", Ann. Math. 37 (1936), 823.

Highly recommended.

 

[Fredrik]

Yes, I recently bought a book on quantum logic, but after the first 20 hours or so of reading it, I don't even fully understand the title of the book. ("Geometry of quantum theory"). I'm going to make another effort soon, but right now I'm trying to learn some functional analysis.

By the way, the fact that you said a few words about quantum logic in some other post months ago, is one of the reasons I became curious enough to buy a book on the subject.

 

[meopemuk]

Yes, I recently bought a book on quantum logic, but after the first 20 hours or so of reading it, I don't even fully understand the title of the book. ("Geometry of quantum theory"). I'm going to make another effort soon, but right now I'm trying to learn some functional analysis.

By the way, the fact that you said a few words about quantum logic in some other post months ago, is one of the reasons I became curious enough to buy a book on the subject.

Hi Fredrik,

There are lot of papers/books written about quantum logic, but less than 5% of them are comprehensible (to me). So, it is very important to choose your reading material carefully. Otherwise you could be swamped by (IMHO) irrelevant hyper-mathematical stuff. (I have nothing to say about the book you chose, I haven't read it.) For starters I would suggest the cited Birkhoff - von Neumann paper. It is out-of-date, of course, but it has all basic ideas. Moreover, both authors were first-rate scientists, and they knew how to write well in those 1930's.

The next step would be

G. W. Mackey, "The mathematical foundations of quantum mechanics", (W. A. Benjamin, New York, 1963), see esp. Section 2-2.

It is not heavy on the math side, but clearly presents physical arguments. Exactly, what's needed for an introduction. Once you get familiar with the ideas, you can continue to more rigorous treatment in

C. Piron, "Foundations of Quantum Physics", (W. A. Benjamin, Reading, 1976)

 

[Fredrik]

Varadarajan's book (the one I bought) is apparently intended for "advanced graduate students", and it does contain a lot of "hyper-mathematical stuff". I'm not sure it's irrelevant though. Some of it probably is.

Thanks for the references. I'll keep them in mind. I'll probably check out at least the Birkhoff/von Neumann paper if I can find it online. The Piron book seems to be difficult to find anywhere.

 

[Pollock]

This is exactly the central philosophical message of quantum mechanics: It does not make sense to talk about how physical systems look "when no-one is looking at them". Physics is a science about observations. Things that cannot be observed even in principle (ghosts, angels, electrons in the absence of a measuring device, etc.) should be left to theologians.

I don't agree.You are confusing "existence" with "observation".If you think the moon doesn't exist when you are not looking at it then you must be a confirmed solipsist.Electrons and all other particles existed before there were humans to worry about them.The positron just didn't come into existence when Dirac predicted it.

 

[meopemuk]

I don't agree.You are confusing "existence" with "observation".If you think the moon doesn't exist when you are not looking at it then you must be a confirmed solipsist.Electrons and all other particles existed before there were humans to worry about them.The positron just didn't come into existence when Dirac predicted it.

I am not saying that the moon and electron do not exist while nobody is looking. Surely, the electron passes somehow through the double-slit. However, I am against designing physical theories about how exactly the electron passes through the slits. My point is that any physical theory must stand an experimental test. If a theory cannot be verified by experiment, then it does not belong to physics. In the double-slit setup we are explicitly banned from observing the electron during its passage through the slits. So, any theory attempting to "describe" this process is doomed to failure, according to my criterion. Quantum mechanics does not pretend to provide such a "description". It simply gives a mathematical recipe for predicting results of measurements. The actual "mechanism" remains undisclosed. This is good enough.

So, I am not a solipsist. I am an agnostic.

 

[zenith8]

meopemuk: OK - so I note you only attempted to answer 2 out of my 8 or 9 questions in #37, and one of your answers was an answer to a completely different question. From looking at your post, you imply this is because:

That's because quantum physicists are on the leading edge of science. They were first to reach the measurable limit of the physical world and to understand that there is nothing beyond that limit. It doesn't even make sense to ask what is beyond that limit. That's why quantum mechanics is the most advanced and weird creation of the human mind.

So - to translate into ordinary language - because you (and quantum physicists in general) are very clever, then it doesn't even make sense to think about the implications of the equations of quantum mechanics. Do you think that's a fair summary?

[ADOPT DEEP VOICE]

"Who is [meopemuk]?" the advert for after-shave asks. "If you are fortunate, you have known him. To assume he is uncaring or aloof is to misread him . . . he directs his actions with energy and passion, commanding respect as he walks that fine line separating arrogance from an awareness of self-worth."

I'm only teasing, but that's what it sounds like. From the hidden variables perspective, QM actually appears quite mundane - so I see your point about the girls.

I think the Hilbert space framework is very natural if we just look at the minimum requirement that a set of statements must satisfy in order to be a "theory", but I'm amazed that it works so well, because it really looks like a toy theory that someone made up just to demonstrate that it's possible to assign non-trivial probabilities. This is one of many reasons why I think it's futile to try to interpret QM as a description of what "actually happens". A theory of what "actually happens" would be desirable, but it wouldn't look anything like QM. I think almost everyone would like to find such a theory, but no one even knows where to begin looking for it. It's also possible that such a "theory" wouldn't make any testable predictions (which means that it isn't falsifiable, and therefore not a theory). If that's the case, we have reached the limit of the scientific method.

Fredrik: I'm not amazed why it works so well - if you work through it from the hidden variables perspective, then it's obvious. The original theory of "what actually happens" - pilot-wave/de Broglie-Bohm (1927) - obtained by allowing the particles to follow the trajectories suggested by the Schroedinger probability current - gives results that are precisely probabilistically catalogued by the Hilbert space formalism. Now you can believe pilot-wave theory or not - that's fine - but to say that "no one even knows where to begin looking for [a theory about what actually happens]" - well, I mean, how can you think that? It simply isn't true, and never has been.

Here's the hidden variable/pilot-wave perspective:

Humans are oblivious to trajectories; the classical limit intuition arises from appropriately narrow wave packets (and 'quantum equilibrium' ensuring the particle configuration is near the bulk packet). In the quantum case we use the (supposedly fundamental) classical language for measurement. But 'energy' and 'momentum' are really only relevant for Newtonian mechanics where they are conserved. Otherwise who's interested in e.g. m times v?

In a momentum measurement - we perform operations that classically would measure the momentum p. Quite wrongly, we assume the same operations would yield the 'value of momentum' p even for nonclassical systems. Instead the 'result' (for the ideal case) equals the eigenvalue of a linear operator \hat{P} (acting on the final wave function guiding the system) which has nothing to do with any real property of a system prior to measurement.

How do we get away with this? It's formally possible since (from the linearity of the Schroedinger equation) there is a general mathematical correspondence between linear operators and classical variables. The classical variable 'corresponding' to the operator \hat{P} arises from guidance by a narrow packet with the spectrum peaked around the eigenvalue p: such a system may, phenomenonologically, be assigned a classical variable with value p. This ensures the usual scheme works as a phenomenological book-keeping device. The solutions of a linear equation can be conveniently classified in terms of a linear vector space. Literal identification of the eigenvalues with real physical quantities turns out to be a fundamental error in quantum measurement theory.

Unfortunately quantum physicists are infallible, so they can't have made an error. So that shoots me down.

And yes the de Broglie-Bohm interpretation does make testable predictions - just not the kind you can do in freshman lab experiments! You tend to need black holes or the early universe or similar.

I really don't get why you say that Meopemuk's position is logical positivism.

Well - because it just is. Though looking at the Wikipedia site that you refer to, it's perhaps best to look at the entry on "positivism" rather than "logical positivism", as this presents the ideas in a clearer manner.

It says that positivists "hold that the only authentic knowledge is that based on actual sense experience". That's what meopemuk believes, no?

 

[f95toli]

It says that positivists "hold that the only authentic knowledge is that based on actual sense experience". That's what meopemuk believes, no?

No, I think he is saying that the the main purpose of a scientific theory is to predict the outcome of experiments. Something most scientists would agree with.

I think you are missing the main point here, the main purpose of science is not to "discover the truth" or even obtaining "authentic knowledge". Science is much more modest today than it was a few hundred years ago when Newton&co were active.
At the end of the day we judge a theory by comparing the numbers that is produces with the number we get from our experiments. Most of us would gladly use a theory that involved the interaction of the flying spaghetti monster with invisible unicorns if it was in better agreement with experiments than current theories; whether or not the theory "makes sense" is not relevant.
This is the reason why most working physicsts are not interested in different interpretation of QM etc; and those of us who actually have an opinion are usually agnostics (although I guess that is technically speaking not an opinion).

 

[zenith8]

It says that positivists "hold that the only authentic knowledge is that based on actual sense experience". That's what meopemuk believes, no?

No, I think he is saying that the the main purpose of a scientific theory is to predict the outcome of experiments. Something most scientists would agree with.

The 'actual sense experience' being 'the outcome of experiments' though i.e. positivism implies science is about what is measurable.

 

[meopemuk]

So - to translate into ordinary language - because you (and quantum physicists in general) are very clever, then it doesn't even make sense to think about the implications of the equations of quantum mechanics. Do you think that's a fair summary?

No, it is not fair. I didn't say that quantum physicists are smarter than others. I simply said that they work on the leading edge of human knowledge. Their job description requires resolving fundamental problems of nature. So, it is not surprising that being as dumb as they are, quantum physicists were first to encounter and resolve some deep enigmas.

I cannot forbid you to think about "implications of the equations of quantum mechanics". I would be very interested if you find something there. However, in my personal opinion, replacing ordinary quantum mechanics with "hidden variables", "pilot waves", "many worlds", etc is not going to lead anywhere. These kinds of attempts are as old as quantum mechanics itself, and in 80+ years they haven't produced a single verifiable prediction that is different from QM. So, I doubt that they ever will. For myself I decided to ignore this "philosophical" noise and focus on something productive instead. You are welcome to disagree.

 

[f95toli]

The 'actual sense experience' being 'the outcome of experiments' though i.e. positivism implies science is about what is measurable.

Indeed, but my point was that science does not claim to produce "authentic knowledge" so it was the first part of the sentence I disagreed with; our theories might very well be completely "wrong" in the sense that they might not have anything to do with an "objective reality" (assuming such a thing exists); but that is strictly speaking irrelevant as long as our theories produce numbers that agree with experiments.
As far as I know this can't be considered positivism since the latter generally assumes that we can actually say something "absolute" about the "real world" by measuring/experiencing it. What I am saying that this is -in my view- beyond the scope of science.

 

[Hurkyl]

These kinds of attempts are as old as quantum mechanics itself, and in 80+ years they haven't produced a single verifiable prediction that is different from QM.

So what? The possibility of discovering new physics is only a minor point of the question of interpretation. Among the more important aspects is how to go about learning and understanding quantum mechanics.

And besides, weren't things like decoherence, or the more complex ramifications of entanglement, discovered precisely because people were thinking about interpretations?

For myself I decided to ignore this "philosophical" noise and focus on something productive instead.

Fine, go do something more "productive". Just be aware that you are reaping the benefits of the fact that other people decided to pay attention to "philosophical" "noise".

 

[zenith8]

and in 80+ years they haven't produced a single verifiable prediction that is different from QM. So, I doubt that they ever will.

Look, I've stated about 5 times in this thread already that pilot-wave (and others) make testable predictions.

Now the way it goes is you're supposed to say indignantly "Please provide cited references for your dubious claims!". Go on, over to you..

 

[meopemuk]

Look, I've stated about 5 times in this thread already that pilot-wave (and others) make testable predictions.

Now the way it goes is you're supposed to say indignantly "Please provide cited references for your dubious claims!". Go on, over to you..

I re-read your posts again and found something about experimental predictions:

"And yes the de Broglie-Bohm interpretation does make testable predictions - just not the kind you can do in freshman lab experiments! You tend to need black holes or the early universe or similar."

Black holes? Early universe? OK. I rest my case.

 

[zenith8]

These kinds of attempts are as old as quantum mechanics itself, and in 80+ years they haven't produced a single verifiable prediction that is different from QM.

So what? The possibility of discovering new physics is only a minor point of the question of interpretation. Among the more important aspects is how to go about learning and understanding quantum mechanics.

And besides, weren't things like decoherence, or the more complex ramifications of entanglement, discovered precisely because people were thinking about interpretations?

I completely agree with Hurkyl:

* Decoherence - effectively invented by Bohm in his first papers about pilot-wave theory in 1952 (the only thing added by him to de Broglie's 1927 effort).

* Bell's theorem etc. - invented by Bell as a result of thinking about how to test whether the pilot-wave theory was wrong.

* The modern concepts arising from the hidden-variables perspective regarding quantum non-equiibrium and emergent relativity are all now minor industries.

etc.

The point Hurkyl makes about learning is equally important. Almost all questions about QM on this forum by people new to the subject (before they've been intimidated by meopemuk et al. into understanding they mustn't ask) are about things - such as Schroedinger's cat, measurement, Heisenberg uncertainty, double-slit, wave-particle duality etc. etc. - which are incomprehensible in standard QM but which have an obvious and enlightening meaning in the de Broglie-Bohm interpretation.

You don't have to make some sweeping philosophical statement about how nature 'is' if you don't wish to. Just say 'it acts as if' there are particles and waves, or whatever - then these questions no longer need to be asked. QM does not have to be weird. This is what people do in all other branches of physics - I repeat the example of the approach to equilibrium in classical statistical mechanics. How do you rationalize what is happening unless you postulate the existence of swarms of particles banging into each other? Otherwise it just sounds like Mach and the "energeticists" in the early 1900s violently asserting that atoms don't exist because we can't see them - in their world view the problem of thermal equilibrium is insoluble. Then you just say the system "acts as if it were made up of lots of little atoms" and grand vistas open up. I repeat, to deny the possibility of doing this is the result of brainwashing (see my earlier posts).

 

[zenith8]

Black holes? Early universe? OK. I rest my case.

How exactly?

Ah - of course.. You've never seen a black hole or the early universe, so you don't believe they exist, right?

 

[meopemuk]

The point Hurkyl makes about learning is equally important. Almost all questions about QM on this forum by people new to the subject (before they've been intimidated by meopemuk et al. into understanding they mustn't ask) are about things - such as Schroedinger's cat, measurement, Heisenberg uncertainty, double-slit, wave-particle duality etc. etc. - which are incomprehensible in standard QM but which have an obvious and enlightening meaning in the de Broglie-Bohm interpretation.

I think all these questions are perfectly comprehensible within standard QM, if you clearly separate observable and non-observable things and refuse to answer questions about non-observable things. There would be much less confusion if people learned this simple trick.

 

[meopemuk]

I repeat the example of the approach to equilibrium in classical statistical mechanics. How do you rationalize what is happening unless you postulate the existence of swarms of particles banging into each other? Otherwise it just sounds like Mach and the "energeticists" in the early 1900s violently asserting that atoms don't exist because we can't see them - in their world view the problem of thermal equilibrium is insoluble. Then you just say the system "acts as if it were made up of lots of little atoms" and grand vistas open up.

Well, Mach and others were wrong 100 years ago. This has been proven by experiments. But this does not mean that their ways of thinking are not applicable in new circumstances (i.e., in quantum mechanics).

 

[meopemuk]

You've never seen a black hole or the early universe, so you don't believe they exist, right?

I've told you: I'm not a believer, I'm an agnostic.

If the experimental proof of the de Broglie-Bohm theory requires such exotic objects, then quantum mechanics is safe, at least for my lifetime.

 

[Fredrik]

Well - because it just is. Though looking at the Wikipedia site that you refer to, it's perhaps best to look at the entry on "positivism" rather than "logical positivism", as this presents the ideas in a clearer manner.

Neither of those articles presents the ideas in a clear manner, but the article on logical positivism claims that logical positivists require theories to be verifiable (rather than just falsifiable). I haven't seen anything that suggests that Meopemuk holds that view. To require more than falsifiability is a huge blunder in my opinion.