The 7 Basic Rules of Quantum Mechanics

[andresB]

I know the insight is aimed at people learning QM and as that is correct and I have nothing to add. But I can't resist to say that nothing of the 7 rules postulated is inherently quantum, you can formulate classical mechanics (or at least classical statistical mechanics) in a way that incorporate all of them (With a possible exception of rule 5 that might require stating that not all self-adjoint operators are observables).

 

[bhobba]

The rules are precise formulations of corresponding statements found more loosely formulated in all textbooks cited (apart from Ballentine).

Indeed. The rules in the article are excellent.

Just to elaborate on what Ballentine does. He only uses two rules:

1. The eigenvalues of Hermitian operators, O (called observables), from some vector space, are the possible outcomes of the observation represented by the operator. Or words to that effect - I can dig up my copy for the exact wording if required.

2. The average of those outcomes, E(O), is given by E(O) = Trace (OS) where S is a positive operator of unit trace called the state of the system.

Note 2 to some extent follows from 1 by Gleason's Theorem, but that is a whole thread in itself and hinges on non-contextuality which even the great Von-Neumann got 'wrong' and Greta Herman was ignored when she pointed it out - not one of sciences finest hours.

How does he get away with 2? He is sneaky and the rest are introduced as assumptions so reasonable you do not notice it's an assumption eg his derivation of Schrödinger's equation assumes the POR and Galilean transformation but it's not stated explicitly - he just assumes probabilities are frame independent which is so 'obvious' you do not recognise, unless you think about it, it's invoking the POR. Elegant, but hides important details - it's still my favourite treatment though. Also there is another assumption not mentioned in the above that for two systems treated as a single system you take the combination of vector spaces ie the space generated from the basis vectors of both spaces, but that is hardly ever mentioned, although it is an assumption. QM is a bit quirky like that - it can be presented in a way assumptions can just seem so natural you do not recognise them as assumptions. There are probably others I haven't mentioned, and perhaps do not even realize them myself.

Thanks
Bill

 

[bhobba]

Busch et al. nowhere refer to reduction. Your use of the term is nonstandard. What is termed state reduction is a process that turns pure states into pure states. It corresponds to the Lüders operator $\rho\to P\rho P$ discussed at the end of Section II.3.1 and in II.4.

What is it Dirac calls it - I think complete set of commuting observables. Not that I recommend using Dirac as the book to base the axioms on. Everyone should eventually own a copy because of it historical significance, but I had the misfortune to use it as my first serious introduction to QM and now regret it. Nor do I recommend the next book I read - Von Neumann's classic - although serious students should also own a copy of that.

Thanks
Bill

 

[bhobba]

So I really would like to hear your opinions (both professors and students) about the pedagocical aspect.
(Sorry if I'm diverting the topic but that is an important part of it I believe...)

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

 

[bhobba]

I know the insight is aimed at people learning QM and as that is correct and I have nothing to add. But I can't resist to say that nothing of the 7 rules postulated is inherently quantum, you can formulate classical mechanics (or at least classical statistical mechanics) in a way that incorporate all of them (With a possible exception of rule 5 that might require stating that not all self-adjoint operators are observables).

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

 

[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.