The 7 Basic Rules of Quantum Mechanics

[bhobba]

Sure, I am not arguing Hilbert spaces are irrelevant, on the contrary.

I also want to add strictly speaking its a Rigged Hibert space, and in fact using it you can have things like resonances that are difficult or perhaps even impossible to handle without it. Rafael Madrid did a thesis on the full technical detail, although he does not give the proof of the key Generalised Eigenvalue Theorem (also called the Nuclear Spectral Theorem);
http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf

An outline of the proof can be found here:
https://www.uni-ulm.de/fileadmin/we...SS15/qm/lnotes_mathematical_found_qm_temp.pdf

Note that the proof in the main tome on the subject by Gelfland - Generalised Functions (now - gulp I think 6 volumes) is generally considered wrong (but may now have been fixed), however correct proofs can be found in other sources. I did look up one once at a university library when I was interested in such things, but have now outgrown these sort of pedantic niceties.

Not for the beginning student, except to keep in mind as you become more advanced. For the beginning student I do HIGHLY recommend the following, not just for QM, but for any applied or pure mathematician - its worth it for its treatment of the Fourier transform alone:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

Thanks
Bill

 

[vanhees71]

Pedagogically I like Ballentine, but though many agree, not all do. And you need to work up to it - to start with I actually like Susskind's theoretical minimum book, then Griffiths, then Sakurai, then Ballentine. But having an agreed set of axioms is good - and the ones here I like.

Thanks
Bill

Skip Griffiths. It's so sloppy that it causes more confusion than it helps!

 

[martinbn]

For the beginning student I do HIGHLY recommend the following, not just for QM, but for any applied or pure mathematician - its worth it for its treatment of the Fourier transform alone:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

Thanks
Bill

I personally find this one very good.
https://www.amazon.com/GUIDE-DISTRIBUTION-THEORY-FOURIER-TRANSFORMS/dp/9812384219/oks&sr=1-1

 

[bhobba]

Skip Griffiths. It's so sloppy that it causes more confusion than it helps!

Advice from a person that actually teaches it. You can go directly from Susskind to Sakurai. I have both books and do prefer Sakurai.

Thanks
Bill

 

[andresB]

Well for that it's best to introduce Feynman's path integral approach from those axioms. I did it in a series of posts I made in the classical mechanics sub-forum:
https://www.physicsforums.com/threads/what-do-Newtons-laws-say-when-carefully-analysed.979739/

Basically classical mechanics is QM were you can cancel most paths and get the classical Principle Of Least Action.

Thanks
Bill

Path integrals are not inherently quantum either https://journals.aps.org/prd/abstract/10.1103/PhysRevD.40.3363

What I'm trying to say is that something else besides the 7 postulate presented is required to truly nail down quantum mechanics.

 

[bhobba]

Path integrals are not inherently quantum either

A implies B does not mean B implies A.

The axioms given are pretty standard. So you are saying the standard formalism of QM is wrong. Pretty strong claim - so strong I think a peer reviewed paper is in order before discussing that further.

Thanks
Bill

 

[andresB]

A implies B does not mean B implies A.

The axioms given are pretty standard. So you are saying the standard formalism of QM is wrong. Pretty strong claim - so strong I think a peer reviewed paper is in order before discussing that further.

Thanks
Bill

I was not saying that the formalism of QM is wrong. Just that the postulates used are also valid for classical mechanics. Plenty of references here https://en.wikipedia.org/wiki/Koopman–von_Neumann_classical_mechanics

Though, checking again, the last sentence of postulate 2 (that the wavefunction depends only on x) is truly quantum (classical mechanics allows the wave function to contain more information). So, I retire my objection.

 

[PeterDonis]

Just that the postulates used are also valid for classical mechanics.

But in this formulation of classical mechanics there is an additional postulate, that all of the observables commute. So you don't need to add an additional postulate to the 7 to define QM. You need to add one to define classical mechanics.

 

[andresB]

But in this formulation of classical mechanics there is an additional postulate, that all of the observables commute. So you don't need to add an additional postulate to the 7 to define QM. You need to add one to define classical mechanics.

Well, that's correct in a sense.

Though, I would prefer that postulate 2 made it clear that the reason the wave function only depends on x (ignoring spin obviously) is the non commutativity of the position and momentum operators.

 

[PeterDonis]

(Where does he use the phrase "effective rule" or equivalent? I don't see that.)

I don't know that he does. The phrase "effective rule" is from the Insights article that I referred to earlier in this thread.

 

[strangerep]

I don't know that he does. The phrase "effective rule" is from the Insights article that I referred to earlier in this thread.

(Sigh.) I see now that I should have made time to proof-read that Insights article instead of ignoring it. @A. Neumaier mentions eq(9.21) and p243f, which seem to me to be incorrect references. Although Arnold says (in the comments) that what he wrote about Ballentine was designed to be compatible with what he says in his book, I think it's a bit misleading and open to misinterpretation. But before discussing that, we need Arnold to check those references. (?)

 

[PeterDonis]

before discussing that, we need Arnold to check those references. (?)

Agreed. I had raised the possibility earlier that something might have changed between editions of Ballentine, but that turned out not to be the case. So it looks like we'll need to make some corrections to the article.

 

[PeterDonis]

Agreed.

I've sent @A. Neumaier a PM.

 

[kith]

This thread is specifically about Ballentine, as I noted in my previous post just now. And as I also noted in that post, the relevant version of any postulate for purposes of this thread is what is in the 7 Basic Rules Insights article, not any other source.

The claimed error in Ballentine is that he omits the projection postulate. The 7 Basic Rules aren't clear on whether this is possible. They state "The most general kind of quantum measurement and the resulting prepared state is described by so-called positive operator valued measures (POVMs)." What does "described" mean here? Do we need a postulate which generalizes the projection postulate or can this description be derived from the other postulates?

So I don't think that the question of whether Ballentine contains this error can be resolved by the 7 Basic Rules.

 

[PeterDonis]

The claimed error in Ballentine is that he omits the projection postulate. The 7 Basic Rules aren't clear on whether this is possible.

As far as the 7 Basic Rules are concerned, Rule 7 (the projection postulate) is a separate postulate, not derived from the others. The question I have asked for input from @A. Neumaier on concerns the statement in the article about Ballentine, where it is said he doesn't accept the postulate as "fundamental" but derives it as an "effective rule". I don't think we have general agreement yet on what that means and whether, or how, the article needs to be corrected. (At a minimum, it looks like the equation reference in the article needs to be updated.)

They state "The most general kind of quantum measurement and the resulting prepared state is described by so-called positive operator valued measures (POVMs)." What does "described" mean here? Do we need a postulate which generalizes the projection postulate or can this description be derived from the other postulates?

I think this is a separate question from the one I described above. The 7 Basic Rules, as stated, don't use the more general POVM formalism and so they are limited in application. Perhaps we need to either augment the article or do a follow-up article to cover how the rules need to be generalized to the POVM formalism. If there is interest in doing that, I'll start a separate thread on that topic (and post a link to it here).

 

[atyy]

The claimed error in Ballentine is that he omits the projection postulate. The 7 Basic Rules aren't clear on whether this is possible. They state "The most general kind of quantum measurement and the resulting prepared state is described by so-called positive operator valued measures (POVMs)." What does "described" mean here? Do we need a postulate which generalizes the projection postulate or can this description be derived from the other postulates?

I think this is a separate question from the one I described above. The 7 Basic Rules, as stated, don't use the more general POVM formalism and so they are limited in application.

An early draft of the statement "Basdevant 2016; ... and for measurements not defined by self-adjoint operators but by POVMs." in the 7 Basic Rules was suggested by me. In my original suggestion (which included many of the textbooks referenced in A. Neumaier's final version, but did not refer to Ballentine) , I intended it to mean that the projection postulate is not the most general state reduction postulate, and was thinking that rule 7 can be replaced by something like the state reduction postulate in Nielsen and Chuang. I did not intend to suggest that state reduction can be derived from the other 6 postulates alone. Nielsen and Chuang also have the interesting statement that derivations of the Born rule and state reduction postulate remain controversial, and they have therefore included both in their postulates.

 

[vanhees71]

The claimed error in Ballentine is that he omits the projection postulate. The 7 Basic Rules aren't clear on whether this is possible. They state "The most general kind of quantum measurement and the resulting prepared state is described by so-called positive operator valued measures (POVMs)." What does "described" mean here? Do we need a postulate which generalizes the projection postulate or can this description be derived from the other postulates?

So I don't think that the question of whether Ballentine contains this error can be resolved by the 7 Basic Rules.

We neither need the projection postulate nor a generalization, because what's happening to the system and its description when interacting with a measurement or filter device depends on the specific experimental setup. It's only an opinion that Ballentine's ensemble interpretation without the projection postulate of some generalization of it were incomplete. I you consider real-world experiments, you have a preparation procedure which you have to describe well enough as the initial state of the system in the quantum formalism. Then you have some Hamiltonian describing the system's dynamics and then measure it. What's predicted by QT are the probabilities for the outcome of these measurements.

If you want to know the state of the system after these measurements you must consider this again as a preparation procedure (if you cannot include the interaction with the measurement devices with sufficient accuracy in the Hamiltonian describing the time evolution of the system). Whether or not you perform a more or less well realized projection measurement (corresponding to the collapse postulate) or not depends on the setup and cannot be generally postulated.

 

[PeterDonis]

We neither need the projection postulate nor a generalization

If you want to know the state of the system after these measurements you must consider this again as a preparation procedure

Don't these two statements contradict each other? Rule 7 in the Insights article points out, correctly, that the projection in the projection postulate is a preparation procedure.

 

[vanhees71]

Yes, and in this formulation it's ok. It's not a general postulate but describes a preparation procedure, i.e., it's referring to specific experimental setups and not to a general description of the behavior of a quantum system as the "dynamical postulates" (unitary time evolution) do.

In other words, the projection postulate is the description of a specific kind of preparation procedure and not a fundamental postulate of the quantum formalism. So there's no contradiction in my statement but it's the statement!

 

[PeterDonis]

it's referring to specific experimental setups and not to a general description of the behavior of a quantum system as the "dynamical postulates" (unitary time evolution) do

According to the 7 Basic Rules as given in the Insights article, unitary time evolution only applies to an isolated quantum system. So it is also only referring to a specific experimental setup. Quantum systems in general are not isolated; you have to make a special effort to set up an isolated quantum system in the lab.

 

[atyy]

Yes, and in this formulation it's ok. It's not a general postulate but describes a preparation procedure, i.e., it's referring to specific experimental setups and not to a general description of the behavior of a quantum system as the "dynamical postulates" (unitary time evolution) do.

In other words, the projection postulate is the description of a specific kind of preparation procedure and not a fundamental postulate of the quantum formalism. So there's no contradiction in my statement but it's the statement!

Yes, it's a preparation procedure. However, it is a preparation procedure that also uses the measurement outcome to label the state prepared.

 

[vanhees71]

That's of course also right, and you also have to make a special effort to realize projection postulates.

The point is that the foundation of quantum mechanics (as the foundation of classical mechanics) refers to closed systems. The behavior of open systems then is derived with many different methods (coarse-graining a la Kadanoff, Baym et al, projection formalism a la Zwanzig et al, influence functional formalism a la Feynman, Vernon, Caldeira, Leggett et al,...).

 

[A. Neumaier]

(Sigh.) I see now that I should have made time to proof-read that Insights article instead of ignoring it. @A. Neumaier mentions eq(9.21) and p243f, which seem to me to be incorrect references. Although Arnold says (in the comments) that what he wrote about Ballentine was designed to be compatible with what he says in his book, I think it's a bit misleading and open to misinterpretation. But before discussing that, we need Arnold to check those references. (?)

You can proofread it now and post your comments here.

Agreed. I had raised the possibility earlier that something might have changed between editions of Ballentine, but that turned out not to be the case. So it looks like we'll need to make some corrections to the article.

In the Insight article, I had originally stated in the second paragraph:

[Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it in the form (9.21) as an effective rule.]

I now replaced it by the more accurate

[Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it at the bottom of p.243 as an effective rule.]

On p.241, Ballentine writes: ''Some evidence that the state vector retains its integrity, and is not subject
to any “reduction” process, is provided by [...]''. No state reduction is his basic credo that he wants to support here. He says on the next page that state reduction should produce a mixed state, (9.18), and on p.243 that in a spin recombination experiment, only the pure state (9.21) is compatible with the experimental results. This is his ''evidence''. Since there was no measurement at the point B/C of investigation - only unitary 2-state dynamics happens -, this is no surprise, anyone would agree. It is not a situation where state reduction should be invoked. Thus his ''evidence'' is bogus.

On the other hand, at the end of page 243 he says

Thus we see that the so-called “reduced” state is physically significant in certain circumstances. But it is only a phenomenological description of an effect on the system (the neutron and spectrometer) due to its environment (the cause of the noise fluctuations)''.

This is the effective rule referred to in the Insight article.

 

[A. Neumaier]

The 7 Basic Rules, as stated, don't use the more general POVM formalism and so they are limited in application. Perhaps we need to either augment the article or do a follow-up article to cover how the rules need to be generalized to the POVM formalism. If there is interest in doing that, I'll start a separate thread on that topic (and post a link to it here).

Postulate 7 in the Insight article was explicitly restricted to the special case of projective von Neumann experiments. In the formal comments to the rule, the more general case of POVM measurements is mentioned but not detailed.

Indeed, POVMs also feature state reduction under measurement, though not projective ones. Instead, the posterior state after a measurement is obtained from the prior state by the application of the POVM operator corresponding to the measurement result obtained. For a discussion of POVMs in terms of a single basic postulate see my paper Born's rule and measurement.

 

[PeterDonis]

@A. Neumaier, thanks for the clarifications!

Since there was no measurement at the point B/C of investigation - only unitary 2-state dynamics happens -, this is no surprise, anyone would agree. It is not a situation where state reduction should be invoked. Thus his ''evidence'' is bogus.

Yes, I agree with this. I think the experiment he describes is interesting because of the fact that coherence is maintained during the passage of the neutron through a solid object, but I agree it doesn't involve any measurement at B/C so it doesn't tell us anything about state reduction as a result of measurement.

 

[strangerep]

You can proofread it now and post your comments here.

I'm still seeing the old version, so I'll wait for the new version to appear and then proofread it.

 

[A. Neumaier]

I'm still seeing the old version, so I'll wait for the new version to appear and then proofread it.

Strange. The new version is online for 18 hours. Maybe you got a cached version. Note that I only edited a few words in that sentence.

 

[PeterDonis]

The new version is online for 18 hours.

I'm seeing the new version.

 

[strangerep]

You can proofread it now and post your comments here.

It's still dated May 11, 2019, but I now see your modified sentence.
[ @Greg Bernhardt: is there a way for a "last-modified" date to be automatically included in these Insights, as well as the original date?]

I now replaced it by the more accurate

Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it at the bottom of p.243 as an effective rule.

I see no such derivation at the bottom of p243. Rather, the last paragraph on that page talks about how an imperfect apparatus could give rise to the "reduced" state eq(9.18) by environmental decoherence mechanisms. This is not a "non-destructive projective measurement" of the type addressed by Rule 7. Hence it is incorrect to link the two, as you currently do.

On p.241, Ballentine writes: ''Some evidence that the state vector retains its integrity, and is not subject
to any “reduction” process, is provided by [...]''. No state reduction is his basic credo that he wants to support here. He says on the next page that state reduction should produce a mixed state, (9.18), and on p.243 that in a spin recombination experiment, only the pure state (9.21) is compatible with the experimental results. This is his ''evidence''. Since there was no measurement at the point B/C of investigation - only unitary 2-state dynamics happens -, this is no surprise, anyone would agree. It is not a situation where state reduction should be invoked. Thus his ''evidence'' is bogus.

I think you misread Ballentine's sect 9.5. As I read it, Ballentine's point (starting at the 2nd paragraph on p242) is this: IF one supposed that all coherence were lost between the wavefunctions at points B and C, then the spin state should be (9.18), i,e., $$\rho^{inc} ~=~ \frac12 \; \Big( |+\rangle \langle +| ~+~ |-\rangle \langle -|\Big).$$ But then, the spin-recombination experiment (with sufficiently good apparatus) described on the rest of p242 and over onto the top of p243, would reveal one's error.

That's what he means by "evidence" (in my humble opinion, of course, since I'm not a mind reader, though neither is anyone else around here, afaik). In other words, IF one (mistakenly) assumed reduction at points B and C, the actual experiment furnishes evidence of one's mistake.

 

[A. Neumaier]

[ @Greg Bernhardt: is there a way for a "last-modified" date to be automatically included in these Insights, as well as the original date?]

Such an addition would be nice indeed.

 

[gill1109]

Continue reading...

I miss qBism as one of the interpretations. It is certainly is presently quite popular. It is a bit more than "shut up and calculate" because it is the claim that this is all that physics ought to do, namely tell an agent what they ought to believe given what they presently know.

I also read in the description of the Quantum Mechanics forum that there was a separate physics forum for Interpretations of Quantum Mechanics, but it seems that this never materialised.

So questions can be put about foundational issues, after all?

 

[bhobba]

So questions can be put about foundational issues, after all?

Of course, Gill. There is a very active subforum on the QM forum about foundational and interpretation issues. The only rule is our general rule against purely philosophical posts. It is recognised that it will occasionally be tough to avoid such problems, so mentors will keep an eye on it to ensure it doesn't get out of hand. I want to emphasise we have the philosophy rule, not because we are anti-philosophy on this forum. We had a sub-forum on it for many years. It just became low quality, and we do not have the mentors expert to ensure it is of the appropriate standard.

Arnold has recently posted an interesting paper on his interpretation:
https://www.physicsforums.com/threads/quantum-mechanics-via-quantum-tomography.1007993/

Thanks
Bill

 

[Demystifier]

a separate physics forum for Interpretations of Quantum Mechanics, but it seems that this never materialised.

 

[vanhees71]

WHAT? We've more postings about interpretation than about the "real" QT (pun intended)!

 

[PeterDonis]

I also read in the description of the Quantum Mechanics forum that there was a separate physics forum for Interpretations of Quantum Mechanics, but it seems that this never materialised.

Yes, it did:

https://www.physicsforums.com/forums/quantum-interpretations-and-foundations.292/