QM: Measurement Operation & Operators Explained

[svnaras]

Like all beginners to QM I'm really confused about the measurement operation. I understand that measurement is simply a dot product with an "operator" and the result is one of the operator's eigenvalues.

Now my question is what exactly is an operator? If someone could explain what physical entity is an operator in the following situations that would help me understand this better.

1. A particle moving at some velocity hits a wall/detector. Till it hits the detector its position is described by a combination of position eigenstates. Once it hits the detector its position becomes a single eigenvalue. Is the wall an operator here?
2. A particle moves through an SG apparatus. Till it passes through the apparatus its spin state is a combination of two eigenstates (in some axis). Once it moves through the magnetic field its spin state becomes one of the eigenstates. Is the magnetic field an operator here?

 

[tiny-tim]

Welcome to PF!

Hi svnaras! Welcome to PF!

Like all beginners to QM I'm really confused about the measurement operation. …

Now my question is what exactly is an operator? If someone could explain what physical entity is an operator …
…
Is the wall an operator here?
…
Is the magnetic field an operator here?

Nooooo …

an operator is maths rather than physics.

For example, momentum is an operator, and if you make a measurement, then you convert the wave function into an eigenstate (eigenvector), and its eigenvalue is the actual momentum measured.

(in other words, the operator and the value measured tend to have the same name, which is confusing! )

I understand that measurement is simply a dot product with an "operator" and the result is one of the operator's eigenvalues.

No.

In terms of matrices and vectors, an operator is a matrix, and the eigenstate is an eigenvector.

And measurement is the ordinary action of a matrix on a vector, giving another vector (while a dot product is a combination of two vectors to give a scalar).

 

[JustSam]

This is the part of quantum mechanics that no one has worked out yet. It's called the "measurement problem" because there isn't really any precise definition of what a measurement is, when it happens, and what takes place when a measurement happens.

My personal view is that a measurement is an irreversible, macroscopic record of a property of a quantum system. Every time a measurement happens, the collapsed version of the wave function must be used to predict the result of subsequent measurements, but this is a mathematical process, not an actual, physical change to the quantum system. Thus:

1. A particle moving at some velocity hits a wall/detector. Till it hits the detector its position is described by a combination of position eigenstates. Once it hits the detector its position becomes a single eigenvalue. Is the wall an operator here?

The detector is taking a measurement here. The operator is a projection matrix of the position of the particle on the portion of the wall covered by the detector.

2. A particle moves through an SG apparatus. Till it passes through the apparatus its spin state is a combination of two eigenstates (in some axis). Once it moves through the magnetic field its spin state becomes one of the eigenstates. Is the magnetic field an operator here?

No measurement, since no irreversible, macroscopic record.

Just my 2 cents.

 

[kote]

This is the part of quantum mechanics that no one has worked out yet. It's called the "measurement problem" because there isn't really any precise definition of what a measurement is, when it happens, and what takes place when a measurement happens.

There is only a measurement problem in collapse theories. Bohr, Schrodinger, Einstein, Bohm, etc, did not have a problem with measurement.

I agree that the OP seems to be confusing math with reality. Operators are symbolic. Physical things are not operators. We model the physical world with math (realistically or instrumentally depending on who you ask), but the two are separate. At best certain theoretical values may correspond to certain real properties, but operators are not properties or values, they are pure math.

 

[Count Iblis]

But then, reality could be pure math:

http://arxiv.org/abs/0704.0646

 

[svnaras]

Thanks for the replies. That seems to clarify some of my misunderstandings.
A couple more questions. If an operator is to be associated with any measurable quantity then there should be an infinite number of operators right?

For example could we not define operators for velocity, acceleration or any combination of the fundamental units or are there operators for only some fundamental quantities like position, momentum, energy, etc?

 

[kote]

But then, reality could be pure math:

http://arxiv.org/abs/0704.0646

The paper seems to say exactly what you just said, and not much more. It takes as a hypothesis that the universe could be pure math, and draws some conclusions about what this might mean. I suppose as long as there are mathematicians there will be platonists. I am also not impressed when a mathematician tells me that supposing the universe is pure math is minimally hubristic .

Even if the universe were pure math, how would you know that our current mathematical formulations are accurate representations of it? You end up at the same place we are without a mathematical universe. The underlying structure is irrelevant. Human theories about nature can never be proven, so human formalism at best arbitrarily corresponds to nature in some aspects. We can propose "math is real" or "reality is math," but we can't ever say a physical event is addition (without redefining addition a posteriori).

Spinoza worked out many of the details of a logic based universe and came to these conclusions. Even if math is basic, our formalism is not.

 

[Fredrik]

There is only a measurement problem in collapse theories. Bohr, Schrodinger, Einstein, Bohm, etc, did not have a problem with measurement.

What are you talking about here? Are you saying that there's something in Bohmian mechanics that ensures that it doesn't have a measurement problem? (I don't know that theory, so I can't rule it out). And what alternatives to QM are you suggesting that Bohr, Schrödinger and Einstein used?

 

[kote]

What are you talking about here? Are you saying that there's something in Bohmian mechanics that ensures that it doesn't have a measurement problem? (I don't know that theory, so I can't rule it out). And what alternatives to QM are you suggesting that Bohr, Schrödinger and Einstein used?

I guess I'd have to ask you where the confusion is supposed to be in their views . I can't seem to find any confusion that any of them had with regard to measurement or observers or anything like that. It seems that von Neumann and opponents of CI brought about all of the collapse and measurement issues.

For Bohm it was easy. The world is deterministic. Nothing special happens during measurement, you're simply taking a reading. No one thought that measurement actually had a profound effect on reality. It only revealed it, with some necessary perturbation by the instrument. Measurement problems arise when we pretend that measuring actually does something special like force a superposition of alive and dead cat to become one or the other. No one used to think that a cat could be both alive and dead.

Einstein and Schrodinger thought it was absurd that something could be and not be at the same time, so they devised macroscopic thought experiments to show the absurdity. Bohr also did not believe that a cat was both alive and dead at the same time - measurement simply reveals whether or not you've got an alive cat or a dead cat, and you can't meaningfully ask about whether it was alive or dead before you looked because only observables are real.

Measurement problems require "the wave function is real and observation causes it to collapse."

 

[zenith8]

What are you talking about here? Are you saying that there's something in Bohmian mechanics that ensures that it doesn't have a measurement problem? (I don't know that theory, so I can't rule it out).

Not only does it not have a measurement problem, but as Bell said, 'all the usual quantum paradoxes are disposed of by the 1952 theory of Bohm'. The classical limit emerges out of the theory rather than having to be presupposed, one can derive the Born Rule, etc. etc.

In Bohm theory, measurements are just perfectly ordinary many-body interactions, special only because the interaction leaves the wave field in a particular state (an eigenfunction of a Hermitian operator). Hermitian operators are important not just for the usual reasons but because their eigenvalues are spacetime constants (i.e. you can know the value of the observable without knowing the position of the particles).

The usual problem is that if the wave function splits into branches following Schroedinger evolution all of these branches continue to exist, despite the fact that we only see one of them. In Bohm theory, things are made out of particles (which are 'guided' by the objectively-existing wave field represented mathematically by the wave function) and they just randomly but deterministically end up in one of the branches (with the usual quantum probabilities).

Then the establishment of correlations between the quantum system and its macroscopic environment quickly reduces the magnitude of the interference terms between the different branches. This means that once the particles end up in the support of one branch, they stay there (trajectories can't pass through nodes in the wave field). [This is just decoherence of course - nothing mysterious about it - it's just ordinary Schroedinger evolution. It was first described by Bohm - for just this purpose].

Note that decoherence alone does not solve the measurement problem - it merely provides a mechanism for the different branches of the wave field to stop interfering. All branches continue to exist. It requires something else such as the introduction of 'hidden variables' or some appropriate interpretation of the wave function to explain why one branch is what one sees. Bohm theory does this very simply.

Lecture 4 of the http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" is the thing to read here - a very good summary of both the measurement problem and its resolution in the Bohm theory..

 

[kote]

What are you talking about here? Are you saying that there's something in Bohmian mechanics that ensures that it doesn't have a measurement problem? (I don't know that theory, so I can't rule it out). And what alternatives to QM are you suggesting that Bohr, Schrödinger and Einstein used?

To put it another way, Bohr believed that when you ask the question "is the cat alive or dead?" what you really mean is "when we look in the box will there be a dead or living cat?" He never believed that reality could be in a superposition of states. He did not believe that the wave function represented reality directly. Science, for Bohr, is about experiments and real observables, and if it's theoretically not an observable, then it's not real. Superpositions are theoretically not observable, so they are not real.

The confusion comes from thinking "if it's not theoretically observable, it's not real" is equivalent to "observers create reality."

 

[zenith8]

He did not believe that the wave function represented reality directly.

Indeed, but unfortunately all the experimental evidence says otherwise. For example, the dynamical object represented by the wave function (let's call it the wave field) can interfere with itself. 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?

As I have pointed out here before, we find that in matter-wave optics experiments - for example - it is also possible to diffract, reflect, focus, and do stimulated emission with the wave field. This is clear experimental evidence for the objective existence of the wave. If the wave can be subject to and utilized in such a process, it logically follows that the wave field must exist in order to act and be acted upon.

Science, for Bohr, is about experiments and real observables, and if it's theoretically not an observable, then it's not real.

Sorry, I had forgotten Bohr's pronouncements were holy scripture. I withdraw my previous remarks, and will go and flagellate myself immediately.

 

[kote]

Sorry, I had forgotten Bohr's pronouncements were holy scripture. I withdraw my previous remarks, and will go and flagellate myself immediately.

Bohr's thoughts are scripture? Since when? I certainly didn't make that claim. Thanks for the constructive comment though ?

Bohr was also well aware of interference. The problem is that there is at least as much evidence for the objective existence of particles. QM has proven that we can't have both at the same time. One solution to this, which cannot be disproven experimentally without disproving all of QM, is to treat experiments as black boxes and only refer to the outputs as real. Trying to visualize what goes on inside the black box is meaningless and impossible. This was Bohr's view and is how he was uniquely able to restore classical entity realism.

It's not the only option, but it doesn't have a measurement problem. You'll notice (or not?) that I was the one who brought up Bohm as another option. QM is paradoxical and there are plenty of issues with any interpretation, Bohm's included. The point is that measurement doesn't have to be one of those problems, and it wasn't for many important physicists.

 

[zenith8]

You'll notice (or not?) that I was the one who brought up Bohm as another option.

Well done! In which case you should realize that your following remark is absolute nonsense:

Bohr was also well aware of interference. The problem is that there is at least as much evidence for the objective existence of particles. QM has proven that we can't have both at the same time.

Unforunately the point of the Bohm theory is that both particles and waves have an objective existence. Given that it's predictions match QM entirely (and how could it not - it is QM - one can derive the entire thing just by changing the usual meaning of a single word) then QM cannot possibly have 'proven that we can't have both at the same time'.

In fact, what on Earth are you talking about?

 

[kote]

Well done! In which case you should realize that your following remark is absolute nonsense:

Unforunately the point of the Bohm theory is that both particles and waves have an objective existence. Given that it's predictions match QM entirely (and how could it not - it is QM - one can derive the entire thing just by changing the usual meaning of a single word) then QM cannot possibly have 'proven that we can't have both at the same time'.

In fact, what on Earth are you talking about?

I'll assume that you're referring to the flavor of Bohmian Mechanics that takes positions to be the basic properties of particles. Particles with no basic properties besides position are not photons, electrons, etc. They are completely ineffectual by themselves and you wouldn't even know if they were there or not. This flavor of the Bohmian Interpretation does not have real classical entities. Bohr's interpretation does. Heisenberg sums up the problems with this version of the Bohmian Interpretation:

What does it mean to call waves in configuration space “real”? This space is a very abstract space. The word “real” goes back to the Latin word “res,” which means “thing;” but things are in the ordinary three-dimensional space, not in an abstract configuration space... Bohm considers himself able to assert: “We do not need to abandon the precise, rational, and objective description of individual systems in the realm of quantum theory.” This objective description, however, reveals itself as a kind of “ideological superstructure,” which has little to do with immediate physical reality.​

But maybe you were referring to Bohm's interpretation as of Wholeness and the Implicate Order, in which particles are given added complexity and free will and in part “determine themselves independently of the infinitely complex fluctuations inside the associated regions of space.” Self-determining particles are also not photons or electrons.

Or were you referring to the infinitely complex particles Bohm proposes that are guided by mind-like active information, which is explained by deeper physical levels, which are guided by deeper mind-like levels, ad infinitum? This can't be it though, since apparently you don't agree with Bohm when he says, "The deeper reality is something beyond either mind or matter, both of which are only aspects that serve as terms for analysis." Bohm believed that "deeper reality" was inherently unknowable and that none of his theoretical constructs were basic.

Bohm had some great thoughts and a viable view, but let's not pretend any variety of Bohm's interpretation gives us a reality that is in any way classical. Photons, electrons, and three dimensional objects do not exist as basic entities for any flavor of Bohm. He showed us a viable way to restore determinism, but Bohmian realism is not located in space-time and does not involve classical entities.

 

[kote]

One last Bohm quote for you:

...space and time are no longer the dominant factors determining the relationships of dependence or independence of different elements. Rather, an entirely different sort of basic connection of elements is possible, from which our ordinary notions of space and time, along with those of separately existent material particles, are abstracted as forms derived from the deeper order.​

 

[JustSam]

Here is how John Bell characterized one aspect of the measurement problem, from his paper Quantum Mechanics for Cosmologists:

It would seem that the theory is exclusively concerned with 'results of measurement' and has nothing to say about anything else. When the 'system' in question is the whole world where is the 'measurer' to be found? Inside, rather than outside, presumably. What exactly qualifies some subsystems to play this role? Was the world wave function waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer for some more highly qualified measurer - with a Ph.D.? If the theory is to apply to anything but idealized laboratory operations, are we not obliged to admit that more or less 'measurement-like' processes are going on more or less all the time more or less everywhere? Is there ever then a moment when there is no jumping and the Schrodinger equations applies?

The concept of 'measurement' becomes so fuzzy on reflection that it is quite surprising to have it appearing in physical theory at the most fundamental level.
...

The problem is this: quantum mechanics is fundamentally about 'observations'. It necessarily divides the world into two parts, a part which is observed and a part which does the observing. The results depend in detail on just how this division is made, but no definite prescription for it is given. All that we have is a recipe which, because of practical human limitations, is sufficiently unambiguous for practical purposes. So we may ask with Stapp: 'How can a theory which is fundamentally a procedure by which gross macroscopic creatures, such as human beings, calculate predicted probabilities of what they will observe under macroscopically specified circumstances ever be claimed to be a complete description of physical reality?'. Rosenfeld makes the point with equal eloquence: '... the human observer, whom we have been at pains to keep out of the picture, seems irresistibly to intrude into it, since after all the macroscopic character of the measuring apparatus is imposed by the macroscopic structure of the sense organs and the brain. It thus looks as if the mode of description of quantum theory would indeed fall short of ideal perfection to the extend that it is cut to the measure of man.'​

I daresay Bell did not find the "statistical/ensemble" interpretation to solve much of anything.

 

[Count Iblis]

Superpositions are theoretically not observable, so they are not real.

Which is not true. You can formally write down an observable corresponding to some macroscopic superposition. It can be shown that such observables can be measured in principle, but not in practice.

 

[zenith8]

I'll assume that you're referring to the flavor of Bohmian Mechanics that takes positions to be the basic properties of particles.

Don't get clever, mate. You know perfectly well that - unless stated otherwise - the use of phrases like "Bohm interpretation" or "Bohmian mechanics" refers to [the modern development of] his 1952 theory, which anyway in itself is just de Broglie's 1927 version with a proper theory of measurement bolted on. See Peter Holland's 1993 textbook for a full presentation of this.

Bohm's later Indian guru musings, while interesting to some people, are of little interest to physics or even metaphysics without further qualification on his part. Which is difficult, with his being dead and all.

And in the paragraph of mine that you quoted, I wasn't referring to Bohm, I was responding to your ludicrous assertion that 'QM has proved that particles and waves cannot both simultaneously exist' which, as you did not admit, is just wrong.

Particles with no basic properties besides position are not photons, electrons, etc. They are completely ineffectual by themselves and you wouldn't even know if they were there or not. This flavor of the Bohmian Interpretation does not have real classical entities.

Bohmian electrons have positions, momenta and trajectories just like classical ones. They only reason they don't behave exactly like classical particles is that they are acted upon by an additional force - the quantum force - which arises from being guided by the wave field.
In the limiting case that the quantum force is zero (i.e. the effect of the quantum wave becomes negligible) then they believe like classical particles.

[..does not have real classical entities.] Bohr's interpretation does.

No it doesn't. There are no entities at all in a Bohrian quantum system, or if there are, we are not permitted to know about them.

If you are referring to the classical measuring apparatus which he presupposes exists, then that is usually thought to be a weakness of his view, as there is then no well-defined boundary between the microscopic and the macroscopic.

In the Bohm case, the behaviour of macroscopic objects flows directly out of the theory rather than having to be presupposed, and measurements are just ordinary many-body interactions.

Heisenberg sums up the problems with this version of the Bohmian Interpretation:

What does it mean to call waves in configuration space “real”? This space is a very abstract space. The word “real” goes back to the Latin word “res,” which means “thing;” but things are in the ordinary three-dimensional space, not in an abstract configuration space... Bohm considers himself able to assert: “We do not need to abandon the precise, rational, and objective description of individual systems in the realm of quantum theory.” This objective description, however, reveals itself as a kind of “ideological superstructure,” which has little to do with immediate physical reality.​

Unfortunately almost all Heisenberg's pronouncements about 'hidden variable theories' are not only wrong but laughable. Do you remember this one:

'The idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them, is impossible'.

Your quote above is almost unique in being relatively sane. But only relatively. He's getting confused between the real thing and the mathematical object which represents it.

The use of a wave function defined on a multi-dimensional configuration space does not imply that this space exists in the same sense that the physical three-dimensional space may be said to exist. (Remember even in classical mechanics we tend to use a configuration space description).

In classical mechanics the config space representation is just a convenient summary of the positions of all the particles; in QM the situation is different because the physics is different - there is the possibility of entanglement due to non-local interactions. So a simply-connected 3d space alone cannot describe the holistic quantum connectiveness and nonlocality features of multi-particle quantum systems. Instead this is done formally by employment of the n-dimensional config space.

The problem with such a space actually existing are considerable, and include:

(1) needing at least 3 separate dimensions for every particle in the universe

(2) the total number of dimensions in the universe varying from moment to moment along with the creation and annihilation of particles.

(3) the extra dimensions always being completely unnoticeable at macroscopic scales, and

(4) a complete lack of any experimental evidence for the existence of multi-dimensional physical spaces.

An additional point is that we do not currently know the 'means' by which quantum non-local connections are actualized. This is not because of the non-relativistic context since non-locality is also present in relativistic versions of QM.

Given the strong reasons against taking multi-dimensional space as real, the vast amount of evidence in favour of physically real wave fields, and the absence of information about the 'means' of non-local connections, it is a coherent position to take the wave function to be a mathematical representation of a real field in physical space.

The notion of an n-particle system is described in Bohm theory by its trajectory which is traced out in 3n-dimensional config space. Even though this description is given using a multi-dimensional space, the motion of individual particles can be calculated since there is a natural mapping from the system's trajectory in 3n-dimensional space to trajectories in 3d space.

Maybe when we have discovered (or developed a model of) the 'means' by which quantum non-local connections are actualized then we will be able to describe the wave field in physical 3d space. That'll be the day.

Bohm had some great thoughts and a viable view, but let's not pretend any variety of Bohm's interpretation gives us a reality that is in any way classical. Photons, electrons, and three dimensional objects do not exist as basic entities for any flavor of Bohm. He showed us a viable way to restore determinism, but Bohmian realism is not located in space-time and does not involve classical entities.

Bugger Bohm. His original theory (which was de Broglie's anyway) lives without him, and is a living disproof of all the mystical nonsense that pervades quantum mechanics. By showing us that QM can be taken to be just classical statistical mechanics with an extra force (and therefore a quantum rather than a classical dynamics) he shows up the supposedly definitive pronouncements of Bohr and Heisenberg (particularly Heisenberg) as being simply due to a lack of imagination.

And as for your statement - the opposite is the truth. Ordinary Bohmian mechanics is located in space-time. And it does involve classical particles, if by classical you mean 'objectively existing' (as I think you do) rather than 'particles following classical Newtonian trajectories' (which surely you don't).

 

[JustSam]

What quantum states are "superpositions" depends on which basis one chooses for the Hilbert space. A definite |x-spin = +1/2> is just the superposition of |z-spin = -1/2> and |z-spin = +1/2> states. And surely the |x-spin = +1/2> state is physically meaningful.

 

[kote]

And as for your statement - the opposite is the truth. Ordinary Bohmian mechanics is located in space-time. And it does involve classical particles, if by classical you mean 'objectively existing' (as I think you do) rather than 'particles following classical Newtonian trajectories' (which surely you don't).

By classical particles I mean objectively existing and having the properties of position, momentum, spin, etc, as their basic and persistent properties.

From http://aps.arxiv.org/abs/quant-ph/0611032

"Just as psi is no classical field, the Bohmian particles are no classical particles. i.e., they are no bearers of properties other than position. ... Agreed, this is a radical departure from the classical particle concept."

Do you have any more accurate descriptions of modern Bohmian Mechanics so I could see what you are referring to?

 

[zenith8]

By classical particles I mean objectively existing and having the properties of position, momentum, spin, etc, as their basic and persistent properties.

From http://aps.arxiv.org/abs/quant-ph/0611032

"Just as psi is no classical field, the Bohmian particles are no classical particles. i.e., they are no bearers of properties other than position. ... Agreed, this is a radical departure from the classical particle concept."

The objectively-existing Bohmian particles have position and momentum, not just position.

They do not, however, possesses spin. Spin turns out to be a property of the wave field (part of it's angular momentum - the part which is dependent on the wave's polarization - since you ask). 'Measurement' of spin in a Stern-Gerlach apparatus - like so many measurements - just turns out to be an illusion. It is not really measuring the pre-existing properties of anything (which is what the word 'measurement' implies).

Do you have any more accurate descriptions of modern Bohmian Mechanics so I could see what you are referring to?

In addition to Peter Holland's 1993 textbook 'The Quantum Theory of Motion' that I referred to in my last post, there are also several more modern treatments including Duerr and Teufel's recent "Bohmian mechanics" book (2009), and the Cambridge University http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" by Mike Towler, also from 2009.

In his final lecture the latter guy also addresses Bohm's late nutter stuff that you were referring to - but sensibly keeps it completely separate from the standard theory and makes it clear why. Taking one's knowledge of the Bohm interpretation from 'Wholeness and the Implicate Order' or whatever is not advised.

 

[kote]

In his final lecture the latter guy also addresses Bohm's late nutter stuff that you were referring to - but sensibly keeps it completely separate from the standard theory and makes it clear why. Taking one's knowledge of the Bohm interpretation from 'Wholeness and the Implicate Order' or whatever is not advised.

Of course, the standard BI is very different from what Bohm actually talked about. I think it's a shame that his name is used while the last 40 years of his life's work is ignored in the standard theory and most discussions of it. An interview in the last slide talks about this some.

From slide 2, about BI:

"This essentially non-classical programme differs from Niels Bohr who strove to leave classical concepts intact as far as possible by restricting their applicability."

I think this point is just the one to remember. In standard BI you have this and you have problems explaining the pilot-wave force in realist terms, since you get either unexplained spooky action at a distance in violation of relativity, or you lose the separable objective existence of each particle.

You have similar problems with other interpretations too, but they don't generally claim to be totally realist.

 

[DrChinese]

They do not, however, possesses spin. Spin turns out to be a property of the wave field (part of it's angular momentum - the part which is dependent on the wave's polarization - since you ask). 'Measurement' of spin in a Stern-Gerlach apparatus - like so many measurements - just turns out to be an illusion.

I have seen the statement about BI lacking spin before; I have also seen BI criticized for just that reason too. So how is it that BI can say there is no spin, and yet there is the same effect present belonging to the pilot wave?

Sounds like having your cake and eating it too, I mean electrons have spin, photons have spin. I'm not trying to be cynical or skeptical, but how does this work? Can you add anything?

 

[ytuab]

They do not, however, possesses spin. Spin turns out to be a property of the wave field (part of it's angular momentum - the part which is dependent on the wave's polarization - since you ask). 'Measurement' of spin in a Stern-Gerlach apparatus - like so many measurements - just turns out to be an illusion. It is not really measuring the pre-existing properties of anything (which is what the word 'measurement' implies).

As you say, we can not look at "spin" directly. So we can not know whether spin is actually spinning or not.

We can only measure the (spin) magnetic moment by the penning trap.(not spin angular momentum and g-factor)
Only Stern-Gerlach experiment can not show the existence of "spin".
(Stern-Gerlach + the idea of quantum mechanics show this.)

You say that the "spin" is an illusion, which is the most important reason why we can not treat the electrons as real particles based on the Schrodinger equation, I think.

In 1920's Pauli strictly objected to the "spin". Because the electron is too small , so by equating the angular momentum of the electron to 1/2hbar, the sphere speed leads to more than 100 times the speed of light.

But at that time, there were no theories replacing the quantum mechanics, So finally he accepted the "spin".
But instead, he gave up the idea that the electron is a real particle.
(he gave up imaging the electron's motions concretely.)

 

[zenith8]

I have seen the statement about BI lacking spin before; I have also seen BI criticized for just that reason too. So how is it that BI can say there is no spin, and yet there is the same effect present belonging to the pilot wave?

Sounds like having your cake and eating it too, I mean electrons have spin, photons have spin. I'm not trying to be cynical or skeptical, but how does this work? Can you add anything?

Hi Dr. Chinese. Sure.. here's a brief summary:

The initial concept of spin had its origin in Stern-Gerlach experiments in the 1920s where beams of atoms get split into two. It was proposed at the time that this was because electrons have a magnetic dipole moment due to their being classical extended particles spinning about an axis through its centre. However we know now that what is called 'the spin of a quantum particle' cannot literally be the rotational angular momentum of a spinning particle, for at least the following four reasons:

(1) the rotation of an extended particle would not require an additional variable for its specification.

(2) the spin's vector does not depend on the particle's position and momentum

(3) angular momentum due to rotation about the centre of mass cannot take half-odd-integer values.

(4) the rate of rotation required to give results in agreement with experiment would need tangential velocities exceeding the speed of light in vacuum.

Now Pauli claimed that quantum-mechanical spin has a discreteness that is not describable in classical terms, e.g. spin for electrons has two discrete values. And so he introduced the Pauli equation whose solution is a two-component spinor wave function.

So in order to meet the need for incorporating spin into orthodox QM, lots of attention was given to developing spinor representations and spin algebra as a way of dealing with an aspect of quantum systems (spin) that was not properly understood. However, although it is the case that spinor methods have been formally successful, they are really a technical means of not addressing the underlying nature of the spin phenomenon. Note that the Pauli equation doesn't give any insight into the origin or characteristics of spin, nor is its ontology (a rotating particle) physically realizable for the reasons given above.

So what is quantum-mechanical spin? Bohm theory gives us a clue. If you look at the trajectory equation (usually p = grad S / m from the Schroedinger equation) as derived from the Pauli equation there is an additional spin-dependent term. (i.e. the quantum potential - if you know what that is - depends on the spin). One could say that Bohm and e.g. Holland and the others were a little remiss in not explicitly documenting that this would give a different answer (e.g. the electron in a H atom is no longer stationary ). The point about this is as follows:

In Bohm theory the wave function represents an objectively existing real field that propagates through space as a wave and shares characteristics found with other types of waves. Spin can therefore be seen to be a property of the wave field because the quantum potential (which represents a portion of the wave field's energy) has a spin dependence.

This isn't a new notion, since in electromagnetic waves the conclusion that 'spin' is a wave property has been around for years - spin is part of an electromagnetic wave's angular momentum, the part which is dependent on the waves polarization (see Jackson's well-known book for example). If you have a circularly polarized plane electromagnetic wave with a certain vector potential, you can write down a simple formula for the polarization dependent part of the wave's angular momentum (i.e. it's 'spin').

By analogy with this, we choose to regard quantum-mechanical spin as a circulating flow of energy, or a momentum density in the electron's wave field - similar to an electromagnetic field, wave fields will have states of polarization. One of the reasons this isn't normally realized is that in non-relativistic quantum mechanics the wave function represents a scalar wave and describes spinless quantum systems. (In actual calculations involving the Schroedinger equation spin is bolted on by e.g. imposing antisymmetrical wave function forms consisting of determinants of orbitals, and multiplying these orbitals by a simple 'spin function').

It might be objected therefore that if wave fields have states of polarization, then QM wave functions would have to represent vector waves, and this might conflict with the representation of quantum systems with spin by spinors. However - and this is the important point - there is more than one formal way to do this representation. In particular, either vector waves or scalar waves plus spinors can be used (spinors are indeed used in this way in classical wave theory).

The connection between spin and wave field polarization accounts for the empirical fact that the spin related to protons, electrons and neutrons i.e. spin one-half fermions, has a two-valued discreteness ('spin up' and 'spin down' if you like). The observed two-valued discreteness related to spin-half fermions is determined by the polarization state of their wave fields.

Note that the explanation of spin as the polarization dependent part of the wave field's angular momentum has not only not been accepted by most physicists who are aware of this explanation, it is almost universally ignored. I guess the main reason for this is the Bohrian legacy that the wave field is not considered to be an objectively existing real field, but a representation of 'our knowledge' or whatever. Which is kind of unfortunate.

Analyzing all the usual 'spin measurement' experiments from this perspective gives exactly the same answers, but with the somewhat unusual situation of understanding precisely what is going on. Weird, huh?

The Stern-Gerlach experiments are simply designed to split the wave field into two bits, and the electrons just randomly end up in one branch or the other with probability one-half (depending on their starting trajectory). So note you are not 'measuring the spin' of anything - it just looks like you are. As has been said about Bohm theory:

'Most of what can be measured is not real and most of what is real cannot be measured.'

Fun, isn't it.

I could go an and talk about the exclusion principle and antisymmetric wave functions and so on, but then this post would get seriously too long (it probably already is). As a final thought though just ask yourself - what is 'Pauli repulsion' exactly? This is a force powerful enough to stop a f***ing star from collapsing - and it is commonly said to be due to 'electron degeneracy pressure' - but nobody seems to know exactly what that is. Go on - try. There are only four fundamental forces, right? Which of the four is reponsible for 'electron degeneracy pressure'? Answer, none of them. It is just the Bohmian quantum force - the pushing of the electrons by the wave field - the fact that there are actually five fundamental forces is taking a surprisingly long time to permeate the collective sheep-like physicist consciousness.

 

[ytuab]

:

Spin can therefore be seen to be a property of the wave field because the quantum potential (which represents a portion of the wave field's energy) has a spin dependence.

This isn't a new notion, since in electromagnetic waves the conclusion that spin is a wave property has been around for years - spin is part of an electromagnetic wave's angular momentum, the part which is dependent on the waves polarization (see Jackson's well-known book for example).
By analogy with this, we choose to regard spin as a circulating flow of energy, or a momentum density in the electron's wave field - similar to an electromagnetic field, wave fields will have states of polarization.

I could go an and talk about the exclusion principle and antisymmetric wave functions and so on, but then this post would get seriously too long (it probably already is).

Sorry. more explanation, please.
Photon has spin 1, and the electron has spin 1/2. You say spin is caused by the electromagnetic waves. You mean the abnormal electromagnetic waves?

And spin g factor is 2. This means the charge and mass of one electron is separated.
If spin is caused by the waves(you mean real waves in BI)? , how do you explain this?

In BI, I heard that the electron is a point particle. you say the waves around the electron (which caused the spin in BI?) and the electron itself are different?

The electron has mass and charge. You say that the mass and charge of the electron is not related to the spin?

 

[zenith8]

Sorry. more explanation, please.
Photon has spin 1, and the electron has spin 1/2. You say spin is caused by the electromagnetic waves.

No - look, my post was written for someone who knows at least a little bit about BM (like Dr. Chinese). BM has both particles and waves - the wave in question is not the electromagnetic field but the Bohmian wave field/pilot wave - the objectively existing thing that is represented mathematically by the wave function.

 

[kote]

The electron has mass and charge. You say that the mass and charge of the electron is not related to the spin?

ytuab, in BI, electrons don't have charge. If charge were a property of particles in BI you would get the "ultraviolet catastrophe" from all of the radiation they would emit while being accelerated by the pilot wave.

 

[DrChinese]

1. Hi Dr. Chinese. Sure.. here's a brief summary:

...


2. I could go an and talk about the exclusion principle and antisymmetric wave functions and so on, but then this post would get seriously too long (it probably already is). As a final thought though just ask yourself - what is 'Pauli repulsion' exactly? This is a force powerful enough to stop a f***ing star from collapsing - and it is commonly said to be due to 'electron degeneracy pressure' - but nobody seems to know exactly what that is. Go on - try. There are only four fundamental forces, right? Which of the four is reponsible for 'electron degeneracy pressure'? Answer, none of them. It is just the Bohmian quantum force - the pushing of the electrons by the wave field - the fact that there are actually five fundamental forces is taking a surprisingly long time to permeate the collective sheep-like physicist consciousness.

1. Great explanation! I thank you so much for the time to give historical background as well.

2. This is a good point too. I have always looked at it that the electron degeneracy pressure was a strange manifestation of the electromagnetic force (essentially the standard view) much like time dilation is a strange manifestation of gravity (i.e. there are places in which the net attractive force of gravity is small but the time dilation effect is relatively more pronounced, such as at the center of a massive object).

And yet your point is pretty strong too. I don't see a lot out there written about this pressure either, which supports your statement that it is not well understood. As you mention, it takes a black hole to overcome it in magnitude. Hmmm.

 

[alexepascual]

Like all beginners to QM I'm really confused about the measurement operation. I understand that measurement is simply a dot product with an "operator" and the result is one of the operator's eigenvalues.

Now my question is what exactly is an operator? If someone could explain what physical entity is an operator in the following situations that would help me understand this better.

1. A particle moving at some velocity hits a wall/detector. Till it hits the detector its position is described by a combination of position eigenstates. Once it hits the detector its position becomes a single eigenvalue. Is the wall an operator here?
2. A particle moves through an SG apparatus. Till it passes through the apparatus its spin state is a combination of two eigenstates (in some axis). Once it moves through the magnetic field its spin state becomes one of the eigenstates. Is the magnetic field an operator here?

To all posters on this thread:
I thing svnaras has asked very simple questions and you all have failed to give a simple answer. svnaras must be even more confused than when he first started the thread.
I tried to provide an answer in two oportunities but had some problems submitting my post and lost the text in both oportunities. So I am a little frustrated about these technical dificulties and feel somewhat reluctand to try again.
The questions svnaras asked can be answered according to the orthodox interpretation concentrating more on how the math works than anything else. I'll try to give an answer even shorter (and necessarily incomplete) than my previous answers:
(1) An operator that represents an observable (which can be represented as a matrix for discreet variables) is used to obtain a weighted average of the possible values that the measurement may yield. The weights are the probabilities which are encoded in the state vector. To get the average you multiply (inner product) the matrix by the state vector on both sides. This gives you a single (real) number.
(2)An act of measurement is not described using the observable's operator. It is usually described using a projector. The projector in its simplest expression is a matrix with all zeros except one element in the diagonal that has a 1. To get the final state, you multiply the state vector by the projection operator and get the the new state vector. If you want to get the actual value (momentum, position etc.) that this state vector represents, you can use the procedure I described in (1) using the new state vector.
(3) The wall is not an operator. The collapse that happened at the wall can be represented using the projector operator.
(4) In SG the magnetic field only allows you to select a particular basis. Each orientation in space is a different basis (frame of reference). The magneti field does not collapse the state vector to one of the eigenvalues (you can re-combine the beams). See Feynman Lectures on Physics volume III. If you put an obstacle in one of the paths, then this is a measurement that can be represented using a projection operator. If both pathsare open, instead of a projector, you got an identity matrix. If you put an obstacle in the lower path, then you this kills the lower eigenstate. So you put a zero in the diagonal's lower right.
So, going back to your question, the magnetic field could be seen as a change of basis unitary operator.

 

[ytuab]

ytuab, in BI, electrons don't have charge. If charge were a property of particles in BI you would get the "ultraviolet catastrophe" from all of the radiation they would emit while being accelerated by the pilot wave.

Mmm... Thanks for telling me this thing.

So the word " illusion" was used.....

 

[Fredrik]

In addition to Peter Holland's 1993 textbook 'The Quantum Theory of Motion' that I referred to in my last post, there are also several more modern treatments including Duerr and Teufel's recent "Bohmian mechanics" book (2009)

Which one is better if you're only willing to buy one? (They're about 90 USD each, so I'd rather not buy both). Demystifier recommended Holland's book to me in February, but maybe the other one didn't even exist back then.

 

[zenith8]

Which one is better if you're only willing to buy one? (They're about 90 USD each, so I'd rather not buy both). Demystifier recommended Holland's book to me in February, but maybe the other one didn't even exist back then.

Hi Fredrik,

Holland's book is an exhaustively detailed presentation of the whole theory - essentially recalculating every result in standard QM from this new perspective. If you don't mind the excessive detail, it's great for the non-relativistic stuff. It's less good for the relativistic stuff (which wasn't that well developed back then anyway) and I would say his treatment of spin is misguided, but never mind.

Duerr and Teufel's book is alright, but there is far too much mathematics and not enough physics. And their chippiness about the fact that everyone ignores them even though they are obviously right might be annoying to some people. I know I do this too in my posts here, but it's a carefully studied pose to provoke people into a reaction :shy: - I can get away with doing this in an internet forum but they probably shouldn't do it in a book. Despite the length of the book, their choice of topics is also much more limited than Holland (they don't do any relativistic stuff at all).

So if you only buy one of them - well, neither of them are perfect but I would still recommend Holland (even though it's 16 years older).

You could also read Bohm + Hiley's 'The Undivided Universe' from the same year as Holland, but I wouldn't bother yet (they don't have the patience to bother with boring details, the style is a bit annoying, and they mix in far too much speculative nutter stuff to make it a good introductory textbook).

If you just want a decent summary, Towler's lecture course is good. Obviously he lacks the detail of a proper textbook but he manages to pack a surprising amount in (he doen't get very far into the relativistic theory either).

Antony Valentini is apparently writing a comprehensive textbook that should be out sometime in the next few years. This won't help you at the moment obviously but it will be the one to read, I'm sure. His recent historical study "Quantum Theory at the Crossroads: reconsidering the 1927 Solvay Conference" (2009) - also available online - was a revelation to me regarding the historical context.

A final decent option might be reading some of the review articles. There is a comprehensive list of Bohm/pilot-wave references with links on Towler's http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" (Click 'Further Reading' in the right hand column).

 

[Fredrik]

Thank you. I have added Holland's book to my Amazon cart, and it will be included in my next order.

That book by Valenti sounds interesting, but it doesn't sound like it will be suitable as an introductory text. This is from his web page:

His central tenet is that quantum theory is not fundamental but merely describes a statistical equilibrium state, which the universe happens to be in at the present time. He has extended the de Broglie-Bohm pilot-wave formulation of quantum theory to nonequilibrium distributions outside the domain of standard quantum physics, and is searching for evidence of the nonequilibrium breakdown of quantum mechanics in the early universe.Antony is completing a book that re-examines modern physics (including quantum gravity, black holes, cosmology, inflation, quantum information and computation) from a pilot-wave and general hidden-variables viewpoint.