Errors in Ballentine (QM Textbook)?

[PeterDonis]

This thread is to get more specifics and start discussion regarding two recent claims in other QM forum threads, in a separate thread to avoid clutter in other discussions:

Ballentine lacks a clear statement of collapse or state reduction. He misrepresents the Copenhagen interpretation, and suggests that the Copenhagen interpretation is in conflict with experiment (Chapter 9). Ballentine's lack of collapse makes him give the wrong result in conflict with experimental outcomes on the "watched pot" experiment.

I think Ballentine is plain wrong on fundamentals

@atyy, can you give specific quotes and references (chapter/page) in Ballentine that these quotes are talking about?

 

[atyy]

Section 9.5 on the spin recombination - here Ballentine suggests that experimental evidence contradicts Copenhagen (whereas it only contradicts his misunderstanding of Copenhagen)
"Some evidence that the state vector retains its integrity, and is not subject to any “reduction” process, is provided by the spin recombination experiments that are possible with the single crystal neutron interferometer (see Sec. 5.5)."

Ballentine's errors on the watched pot experiment have been discussed in https://www.physicsforums.com/threads/ballentine-on-the-quantum-zeno-paradox.546523/

Ballentine's famous review from the 1970s also has errors, where he suggests that position and momentum can be simultaneous measured. While there are special cases, in general his language is misleading, because he gives an example in Fig 3 in which the variables are not canonically conjugate. In his review he makes it clear he is attacking the "orthodox" interpretation.

Of course errors do not make a textbook bad - the wonderful Feynman lectures have errors - some deep (though at that time they were not widely well understood), some incidental - but Ballentine attacks mainstream quantum mechanics, and substitutes his own personal theory.

 

[martinbn]

The watched pot is at the end of 12.2, page 432 of the 1998 edition.

 

[Delta2]

I am interested in buying Ballentine since it is recommended by @PeterDonis but let me ask something.

I don't care that much if Ballentine has his own variation of Copenhagen interpretation, but how to say it, is the book proper for someone that wants to learn QM from the complete unknown (ok I have heard Schrodinger equation before i am not that dumb), is it taught at the undergraduate level? Or Ballentine goes wild (as like say Jackson goes wild in Classical Electrodynamics where he states as obvious, things that their derivation is three pages long).

I have finished my undergraduate studies in math long ago (back at 1998) and during those I had taken an optional course in QM that was offered. The textbook was in Greek i still have it but I don't think it is so good.
I didn't bother with QM since then.

I searched for the book at amazon, which one of these is the first or the second?
Amazon.com : ballentine quantum mechanics

 

[atyy]

I don't care that much if Ballentine has his own variation of Copenhagen interpretation, but how to say it, is the book proper for someone that wants to learn QM from the complete unknown (ok I have heard Schrodinger equation before i am not that dumb), is it taught at the undergraduate level? Or Ballentine goes wild (as like say Jackson goes wild in Classical Electrodynamics where he states as obvious, things that their derivation is three pages long).

I don't think Ballentine has his own variation of Copenhagen - I think it is simply wrong - you can find the orthodox interpretation in many textbooks - Dirac; Landau & Lishitz; Messiah; Weinberg; Cohen-Tannouji, Diu & Laloe; Shankar; Griffiths; Gasiorowicz; Nielsen & Chuang - Ballentine is not in step with them, and you may be taken in by his errors if you don't start with sound foundations.

 

[martinbn]

I am interested in buying Ballentine since it is recommended by @PeterDonis but let me ask something.

I don't care that much if Ballentine has his own variation of Copenhagen interpretation, but how to say it, is the book proper for someone that wants to learn QM from the complete unknown (ok I have heard Schrodinger equation before i am not that dumb), is it taught at the undergraduate level? Or Ballentine goes wild (as like say Jackson goes wild in Classical Electrodynamics where he states as obvious, things that their derivation is three pages long).

I have finished my undergraduate studies in math long ago (back at 1998) and during those I had taken an optional course in QM that was offered. The textbook was in Greek i still have it but I don't think it is so good.
I didn't bother with QM since then.

I searched for the book at amazon, which one of these is the first or the second?
Amazon.com : ballentine quantum mechanics

I would say that the level is graduate or advanced undergraduate, but i might be wrong. It just means that it may be that you need to put in more effort compare to other texts, but in my opinion you will benefit more. As to the "errors" discussed here, they are less than 1% of the book, so you will not even notice them.

 

[Demystifier]

is the book proper for someone that wants to learn QM from the complete unknown (ok I have heard Schrodinger equation before i am not that dumb), is it taught at the undergraduate level? Or Ballentine goes wild (as like say Jackson goes wild in Classical Electrodynamics where he states as obvious, things that their derivation is three pages long).

I would agree with @martinbn that Ballentine is an advanced book, very much like Jackson for electrodynamics.

 

[Demystifier]

I don't think Ballentine has his own variation of Copenhagen - I think it is simply wrong - you can find the orthodox interpretation in many textbooks - Dirac; Landau & Lishitz; Messiah; Weinberg; Cohen-Tannouji, Diu & Laloe; Shankar; Griffiths; Gasiorowicz; Nielsen & Chuang - Ballentine is not in step with them, and you may be taken in by his errors if you don't start with sound foundations.

Paradoxically, even though those other books do not make the fundamental error that Ballentine does, one can learn foundations much more deeply from Ballentine than from those other books. That's because Ballentine takes much more space and effort than the other books to discuss foundations at a deep level, while other books are rather superficial on that. It's much easier to make an error if you try to think deeply than if you choose to stay in the safety zone.

 

[PeroK]

I am interested in buying Ballentine since it is recommended by @PeterDonis but let me ask something.

An alternative to the standard texts is available free here:

https://physics.mq.edu.au/~jcresser/Phys304/Handouts/QuantumPhysicsNotes.pdf

Cresser puts a lot of emphasis on why, ultimately, QM is both rational and necessary for the universe we know and love.

 

[atyy]

Paradoxically, even though those other books do not make the fundamental error that Ballentine does, one can learn foundations much more deeply from Ballentine than from those other books. That's because Ballentine takes much more space and effort than the other books to discuss foundations at a deep level, while other books are rather superficial on that. It's much easier to make an error if you try to think deeply than if you choose to stay in the safety zone.

I can't agree. Copenhagen is what makes the measurement problem clear. And the measurement problem is the fundamental problem in foundations. I think it is telling that Ballentine does not give the merits of the Bohmian interpretation clearly, instead favoring his own non-solution of the Ensemble Interpretation (which is just Copenhagen without the proper postulates), because Ballentine is unaware that there is a measurement problem.

Ballentine's Ensemble Interpretation is not Einstein's ensemble intrepretation, because it lacks a clear statement of hidden variables. In contrast the old books like L&L allude pretty clearly to an absurdity, and Messiah explicitly discusses Einstein's proposal of hidden variables.

 

[Demystifier]

I can't agree. Copenhagen is what makes the measurement problem clear. And the measurement problem is the fundamental problem in foundations. I think it is telling that Ballentine does not give the merits of the Bohmian interpretation clearly, instead favoring his own non-solution of the Ensemble Interpretation (which is just Copenhagen without the proper postulates), because Ballentine is unaware that there is a measurement problem.

Ballentine's Ensemble Interpretation is not Einstein's ensemble intrepretation, because it lacks a clear statement of hidden variables. In contrast the old books like L&L allude pretty clearly to an absurdity, and Messiah explicitly discusses Einstein's proposal of hidden variables.

Yes, but I think Ballentine is great on other foundational problems that are not directly related to the measurement problem.

Even his errors are instructive, because they are presented in a form that provokes thinking.

 

[PeterDonis]

Section 9.5 on the spin recombination - here Ballentine suggests that experimental evidence contradicts Copenhagen (whereas it only contradicts his misunderstanding of Copenhagen)

The term "Copenhagen" never appears in that section, so I don't see how Ballentine is claiming that experimental evidence contradicts "Copenhagen". I do think it is true that in that section, Ballentine is using the term "reduction" in an idiosyncratic way, driven by his preference for the ensemble interpretation. However, since this thread is not in the QM interpretation forum, I don't want to get into detail here about how Ballentine deals with QM interpretation/foundations issues vs. other sources--if there is enough interest for that type of discussion I can spin off a separate thread in the interpretations/foundations subforum.

Ballentine's errors on the watched pot experiment

Hm, yes, I agree that his argument at the end of the "watched pot" subsection of 12.2 is just handwaving: no math and no explanation, just a bare assertion.

Ballentine attacks mainstream quantum mechanics, and substitutes his own personal theory.

I think this is too strong (as is the claim that Ballentine is "plain wrong on fundamentals"). To the extent Ballentine differs from other sources with regard to which interpretation of QM he prefers, and points out issues he sees with other interpretations, that is not "personal theory" unless every QM textbook that discusses interpretation at all is "personal theory". The fact is that there are multiple interpretations of QM, and with our current state of knowledge, none of them are "wrong" (or "right", for that matter). I think a better description of this aspect (and, as noted above, if more detailed discussion is desired I can spin off a separate thread in the appropriate subforum for that) would be that Ballentine's preferred interpretation is not shared by most other sources, and that one should not use his textbook as one's only source of information about interpretations.

I do agree that all that makes his textbook an "advanced" one, as has been commented, and after seeing this discussion I would be much more hesitant about recommending it to someone with no prior background in QM, unless that person is also learning from other sources.

 

[PeterDonis]

is the book proper for someone that wants to learn QM from the complete unknown

Per my post #12 just now, I would be hesitant about this if it's going to be your only source, particularly about QM interpretations.

I do think there is a lot of useful information in Ballentine, and also good problems in the problem sections at the end of each chapter.

 

[atyy]

The term "Copenhagen" never appears in that section, so I don't see how Ballentine is claiming that experimental evidence contradicts "Copenhagen". I do think it is true that in that section, Ballentine is using the term "reduction" in an idiosyncratic way, driven by his preference for the ensemble interpretation. However, since this thread is not in the QM interpretation forum, I don't want to get into detail here about how Ballentine deals with QM interpretation/foundations issues vs. other sources--if there is enough interest for that type of discussion I can spin off a separate thread in the interpretations/foundations subforum.

Yes, that's true, one must read it in the context of his earlier review to associate his book with an attack on "Copenhagen". Nonetheless, Ballentine's book rejects the state reduction postulate "This postulate of reduction of the state vector creates a new problem that is peculiar to interpretation A: namely, how to account for the mechanism of this reduction process." (in the text just after Eq 9.9). In other words, his preferred interpretation "B" lacks a fundamental postulate of QM, and the experiment of Section 9.5 is meant to show that interpretations with state reduction as a fundamental postulate are wrong. This is an error, since there are interpretations with state reduction as a fundamental postulate that are consistent with all current experimental evidence, including the experiment in 9.5, and this includes the Copenhagen-style orthodox interpretation.

In case you think that is not a clear enough statement of the state reduction postulate, Ballentine states his rejection of the postulate again in section 12.2 and on p 584.

Also, within the orthodox interpretation, a pure state can be associated with an individual quantum system. However the orthodox interpretation, also contains the Born rule, which means that quantum mechanics only makes probabilistic predictions, and the notion of probability has most usually meant consideration of an ensemble. So in fact there is no difference between correct versions of what he calls Interpretations A and B.

I think this is too strong (as is the claim that Ballentine is "plain wrong on fundamentals"). To the extent Ballentine differs from other sources with regard to which interpretation of QM he prefers, and points out issues he sees with other interpretations, that is not "personal theory" unless every QM textbook that discusses interpretation at all is "personal theory". The fact is that there are multiple interpretations of QM, and with our current state of knowledge, none of them are "wrong" (or "right", for that matter). I think a better description of this aspect (and, as noted above, if more detailed discussion is desired I can spin off a separate thread in the appropriate subforum for that) would be that Ballentine's preferred interpretation is not shared by most other sources, and that one should not use his textbook as one's only source of information about interpretations.

Well, if a fundamental postulate - one that is in many standard texts - is omitted - then how can that not be a fundamental error? Furthermore, Ballentine omits it without introducing any new element, whereas Many Worlds and Bohmian Mechanics omit it but respectively introduce new elements of many worlds and hidden variables (one doesn't have to agree that they are without problems, but they at least agree that you can't drop a postulate without introducing another).

 

[PeterDonis]

Ballentine's book rejects the state reduction postulate

For "interpretation A", i.e., for interpretations that say the quantum state represents the real, actual state of individual quantum systems. But that is an interpretation-dependent claim. He does not reject "state reduction" when the basic math of QM (e.g., the 7 Basic Rules as we present them here at PF, in the Insights article that the SAs and Mentors created a while back) says to use it; for example, he does not reject using the projection postulate after what he calls a "filter measurement" (i.e., when Rule 7 in our Insights article says to use it).

his preferred interpretation "B" lacks a fundamental postulate of QM

I disagree. See above. It's true that he doesn't emphasize this aspect of interpretation B, but he clearly accepts it, as shown by his discussion of filter measurements elsewhere in the book.

within the orthodox interpretation, a pure state can be associated with an individual quantum system

Which is an interpretation-dependent claim. So you can't say Ballentine is "wrong" for not agreeing with it. All you can say is that your preferred interpretation is different from his.

the orthodox interpretation, also contains the Born rule, which means that quantum mechanics only makes probabilistic predictions, and the notion of probability has most usually meant consideration of an ensemble

This seems to contradict what you just said in the quote from you that I gave before this one. But arguing about what the "orthodox interpretation" says is out of scope for this thread (as I have already said, we can spin off a separate thread in the interpretations subforum if that kind of discussion is desired). The point for this thread is that I don't think you can use the term "wrong" to describe a difference of opinion about interpretations. I described what I think can be said in post #12.

if a fundamental postulate - one that is in many standard texts - is omitted

It isn't. See above. As I said, this is a difference of opinion about interpretation. I don't think it justifies the term "wrong".

 

[strangerep]

Hm, yes, I agree that [Ballentine's] argument at the end of the "watched pot" subsection of 12.2 is just handwaving: no math and no explanation, just a bare assertion.

The fallacy clearly results from the assertion that if an observation indicates no decay, then the state vector must be $|\Psi_u\rangle$. [...]

... which depends on the notion of reduction of the state vector which he criticized and rejected in sect 9.3. So,... certainly not a "bare assertion". More like "Line so-and-so in your proof is wrong. This invalidates the proof." No need for anything more.

Nevertheless he also references his 1990 paper "Limitations of the Projection Postulate"
(Found. Phys., vol 20, No. 11, 1990, p1329), which does contain more math. Unfortunately, it's behind a paywall.

 

[strangerep]

I searched for the book at amazon, which one of these is the first or the second? Amazon.com : ballentine quantum mechanics

The 2nd edition just has extra (important) material on quantum information, but the rest is essentially identical to the 1st edition.

As for the "criticisms" of Ballentine (which I reckon are BS), I have in the past challenged some of his critics to get their arguments published in a reputable peer-reviewed journal (and thus put this nonsense into a form more suitable for discussion on PF). They either ignore me, or abuse me for suggesting that this should even be necessary, which is a mark of crackpottery.

Mindful of Brandolini's Law (a.k.a. the BS asymmetry principle), I refuse to waste any more of my time on this, except to say that you will learn a lot by studying Ballentine.

 

[PeterDonis]

which depends on the notion of reduction of the state vector which he criticized and rejected in sect 9.3.

That actually wasn't what I was referring to as a "bare assertion". I was referring to this statement:

Here we see that it is disproven by the simple empirical fact that continuous observation does not prevent motion.

There is no argument at all about why continuous observation should prevent motion if the notion he is rejecting were true. Presumably there is more argument in the 1990 paper he references, which, as you note, is unfortunately behind a paywall.

 

[atyy]

For "interpretation A", i.e., for interpretations that say the quantum state represents the real, actual state of individual quantum systems. But that is an interpretation-dependent claim. He does not reject "state reduction" when the basic math of QM (e.g., the 7 Basic Rules as we present them here at PF, in the Insights article that the SAs and Mentors created a while back) says to use it; for example, he does not reject using the projection postulate after what he calls a "filter measurement" (i.e., when Rule 7 in our Insights article says to use it).

I disagree. See above. It's true that he doesn't emphasize this aspect of interpretation B, but he clearly accepts it, as shown by his discussion of filter measurements elsewhere in the book.

The more straightforward reading of his text is that he rejects the state reduction postulate in the basic math of QM (ie. that Ballentine's QM is not consistent with the 7 Basic Rules) and yet uses it without knowing that he is using it in the filter measurements, and that his text his therefore internally contradictory. At the very least, it is a matter of interpretation of Ballentine's text whether he rejects "state reduction" in the basic math of QM.

Ballentine states his postulates in Chapters 2 and 3 the book. Can the state reduction in the basic math of QM be derived from them? If they cannot, then overall it is not an accidental omission or slight under-emphasis, but a deliberate omission of part of the basic math of QM.

 

[PeterDonis]

Ballentine states his postulates in Chapters 2 and 3 the book. Can the state reduction in the basic math of QM be derived from them?

See the note in brackets in the Insights article (right after listing various other textbooks).

 

[atyy]

See the note in brackets in the Insights article (right after listing various other textbooks).

I did not notice that in the Insight before, perhaps it wasn't in the draft I read or I missed it.

Is Ballentine's derivation of state reduction as an effective rule correct? If in the basic math of QM, postulate 7 on state reduction is needed as a separate postulate, then it cannot be derived from the first 6 postulates. Perhaps it can be derived, but as far as I know, there is no consensus derivation of postulate 7 from the first 6 postulates alone. If Ballentine omits postulate 7, then he has to either (1) claim that postulate 7 can be derived from the first 6 postulates (which is not a mainstream claim, although no one has shown it cannot be done) or (2) add postulates that enable the derivation of postulate 7 as an effective rule (which Bohmian mechanics does by adding hidden variables and hidden variable dynamics).

The statement "[Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it in the form (9.21) as an effective rule.]" in the Insight is also not quite correct. Eq 9.21 is not a derivation of state reduction - that is in the spin recombination experiment in which there is no state reduction in both interpretations A and B. Ballentine introduces state reduction in Eq 9.28, and the question is whether he can justify Eq 9.28 from his postulates, or whether he has simply introduced that as a new postulate.

 

[vanhees71]

I think this belongs to the interpretation section of the forum. Whether or not collapse is needed or not is a purely interpretational issue. I don't think that one needs and I also think it's misleading to state a collapse postulate, leading to tons of unnecessary discussions about causality in the relativistic context.

In experiments no collapse occurs, one just measures the values of observables with more or less accuracy and statistical significance.

 

[atyy]

within the orthodox interpretation, a pure state can be associated with an individual quantum system.

Which is an interpretation-dependent claim. So you can't say Ballentine is "wrong" for not agreeing with it. All you can say is that your preferred interpretation is different from his.

Ballentine does not have to use the claim that a pure state can be associated with an individual quantum system. However, his error is that he claims that this leads to a form of state reduction that is not supported by experiment. In other words, his criticism of the interpretation is wrong.

 

[martinbn]

As it has been aknowledged here in the forums, there is more than one version of the CI. I personaly think that CI should be used for Bohr's view. And I don't think he ever talked about collapse/reduction, but I might be wrong.

As to the book, I cannot find anywhere in the text, where he reffers to Copenhagen. So bringing that up is inapproprate for this discussion.

 

[vanhees71]

Ballentine does not have to use the claim that a pure state can be associated with an individual quantum system. However, his error is that he claims that this leads to a form of state reduction that is not supported by experiment. In other words, his criticism of the interpretation is wrong.

What precisely is his version of state reduction (I have the book at hand, so for me it's sufficient to point to the section, where the statement is made). AFAIK in the minimal statistical interpretation there's no need for some "state reduction" or "collapse" assumption.

 

[vanhees71]

As it has been aknowledged here in the forums, there is more than one version of the CI. I personaly think that CI should be used for Bohr's view. And I don't think he ever talked about collapse/reduction, but I might be wrong.

As to the book, I cannot find anywhere in the text, where he reffers to Copenhagen. So bringing that up is inapproprate for this discussion.

It's hard to say, what Bohr really meant ;-)). I'm not sure whether or not he proposed a state reduction.

 

[PeterDonis]

I don't think that one needs and I also think it's misleading to state a collapse postulate

Rule 7 in the Insights article is not a "collapse" postulate. It's the projection postulate.

 

[vanhees71]

I thought we talk about Ballentine's book and I wanted to know, to which section @atyy is referring to. Rule 7 is indeed not a collapse postulate but is part of the minimal postulates you need to work with the formalism.

Collapse implies that there's something outside the dynamics as described by the formalism which leads to an instantaneous change of the state and which thus is incompatible with the fundamental assumptions of local relativistic field theory, but I don't think we need another discussion about this issue.

 

[PeterDonis]

The statement "[Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it in the form (9.21) as an effective rule.]" in the Insight is also not quite correct. Eq 9.21 is not a derivation of state reduction - that is in the spin recombination experiment in which there is no state reduction in both interpretations A and B. Ballentine introduces state reduction in Eq 9.28

Yes, I have the same equation numbering in my copy. It's possible that the numbering was changed between the first and second editions.

 

[PeterDonis]

I thought we talk about Ballentine's book and I wanted to know, to which section @atyy is referring to.

Ballentine talks about what he calls "reduction" of the state vector in several places, but I think the discussion that has generally been referred to in this thread is in Chapter 9.

 

[PeterDonis]

Collapse implies that there's something outside the dynamics as described by the formalism which leads to an instantaneous change of the state and which thus is incompatible with the fundamental assumptions of local relativistic field theory, but I don't think we need another discussion about this issue.

I agree we don't need another interpretational discussion on this issue. To the extent that Ballentine's discussion of "reduction" of the state vector talks about actual experimental results, however, it's on topic for this thread. His basic position appears to me to be that we have no experimental evidence that there is any actual physical process of "reduction", which, as just a statement about experimental evidence, seems to me to be correct. Whether that justifies his claims about the impact on interpretations of QM is a separate question that is out of scope for this thread.

 

[vanhees71]

Of course, there is no experimental evidence for "reduction". For this you'd have to define what's to be observed to claim that there's "reduction".

If we talk about Sect. 9.3 and 9.4 in Ballentine (my edition is that of 1998), where "reduction" is discussed, I don't see any error but a pretty clear argument against it.

 

[atyy]

As to the book, I cannot find anywhere in the text, where he reffers to Copenhagen. So bringing that up is inapproprate for this discussion.

It is appropriate as that is the term Ballentine uses in his 1970 review, and it is also used by Messiah (Vol 1, Chapter II, p48, footnote), to which Ballentine's book refers.

 

[strangerep]

There is no argument at all about why continuous observation should prevent motion if the notion he is rejecting were true.

? It's not obvious? Well,... ok,... suppose you measure the particle's position to be $x = x_0$. By the postulate, the state after the measurement must (supposedly) be the position eigenstate corresponding to eigenvalue $x=x_0$. Therefore, if another position measurement is made immediately thereafter, the measurement result must be $x_0$ again. Hence the silly conclusion that the particle cannot move.

Such nonsense is avoided by recognizing that the system's state post-measurement is not, in general, the eigenstate of the observable that was just measured -- as explained by Ballentine in much more detail in his sect 9.2 et seq.

 

[strangerep]

Yes, I have the same equation numbering in my copy. It's possible that the numbering was changed between the first and second editions.

Afaict, the only change between Ballentine's 1st and 2nd editions was the addition of ch21 at the end. Chapters 1-20 remain unchanged. Indeed, I checked a few pages at random throughout the book, and they appear identical -- which is probably a good thing.

 

[PeterDonis]

suppose you measure the particle's position to be $x = x_0$.

You can't. The mathematical operator that would do that is not physically realizable.

By the postulate, the state after the measurement must (supposedly) be the position eigenstate corresponding to eigenvalue $x=x_0$.

Which is impossible; the mathematical state in question is not physically realizable.

Possibly there is an argument that can be made that applies to operators that are actually physically realizable (perhaps such an argument is in the paywalled paper), but I don't agree that it's just "obvious", since the "obvious" argument you have stated is open to the obvious objections that I have just given.

the system's state post-measurement is not, in general, the eigenstate of the observable that was just measured -- as explained by Ballentine in much more detail in his sect 9.2 et seq.

The argument given there (which I agree with) appears to me to be addressing a different issue: the issue that, since the process of measurement (and indeed more generally any interaction, as Ballentine points out) entangles the measured system with the measuring device, after the measurement neither of those subsystems by itself has any definite state at all: only the joint system containing both measured system and measuring device does. So obviously the measured system can't be in an eigenstate of the measurement operator, since it isn't in any definite state at all.

Ballentine's solution to that problem is to adopt an ensemble interpretation. But that, by itself, doesn't address the "watched pot" issue, because the computation of the joint probability of successive measurement results being the same on a given state, given in section 12.2, is not interpretation-dependent; it's a straightforward application of the math of QM. So it should be valid under the ensemble interpretation. And the handwaving claim that the simple conclusion reached based on this simple computation must be false because "continuous observation does not prevent motion" doesn't help, because that claim is not a claim about ensembles, it's a claim about individual objects. But Ballentine's whole point is supposed to be that QM doesn't tell us about individual objects, it tells us about ensembles.

So no, I don't agree that his claim is just "obvious". Again, possibly these gaps are filled in in the paper that is unfortunately paywalled. But it seems clear to me that there are gaps.

 

[strangerep]

So obviously the measured system can't be in an eigenstate of the measurement operator, since it isn't in any definite state at all.

... which is the crucial take-home message.

The rest of what you said just shows that the usual naive mathematical machinery motivating Zeno's paradox is not valid for continuous processes when examined more carefully. More sophisticated setups involving POVMs and measurements-with-uncertainties must be carefully analyzed.

 

[atyy]

Ballentine does not have to use the claim that a pure state can be associated with an individual quantum system. However, his error is that he claims that this leads to a form of state reduction that is not supported by experiment. In other words, his criticism of the interpretation is wrong.

To add to this point, it is not only textbooks of quantum mechanics that use the sort of interpretation that Ballentine criticizes. Standard texts of quantum statistical physics also use the interpretation.

Reif Section 2.1: "Specifically, the system can be described by a wave function"
Kardar (not his book, but the lecture notes on which the book is based: "The (micro-) state of a quantum system is completely specified by a unit vector |Ψ〉, which belongs to an infinite dimensional Hilbert space."

So overall, I dislike Ballentine for criticizing textbook physics. If one reads it, one may get the opinion that Ballentine is the first person to have properly understood quantum mechanics, and learn false rejections of the orthodox interpretation, which is a statistical interpretation. Messiah explicitly identifies "Copenhagen" with the "Statistical Interpretation". What Ballentine has failed to grasp is that in the orthodox interpretation, the quantum state is not necessarily real, and just a way of calculating the probabilities of measurement outcomes. So if we can notionally assign a pure state to a single system, that is simply a way of calculating that yields correct predictions.

 

[PeterDonis]

The rest of what you said just shows that the usual naive mathematical machinery motivating Zeno's paradox is not valid for continuous processes when examined more carefully. More sophisticated setups involving POVMs and measurements-with-uncertainties must be carefully analyzed.

Does a more careful analysis show that there are some cases where "watched pot" experiments are predicted to work? Because such experiments have been done, and do work.

In other words, the issue is more complicated than just "well, this naive analysis makes it seem like a watched pot experiment should work, but that's obviously false because continuous observation does not prevent motion". Yes, it's true that continuous observation doesn't prevent motion, but it's also true that, at least under some conditions, you can make a watched quantum pot never boil. So there must be some cases where the "naive" analysis does give the right answer, which means that we need to understand what makes those cases different from a case like simple particle motion.

 

[atyy]

You can't. The mathematical operator that would do that is not physically realizable.

Minor point, but in some sense an exact position measurement is possible. For the measurement, we can use the Born rule. If we need to apply state reduction, the state of the system after the measurement is not a position eigenstate, but a normalizable state.
https://arxiv.org/abs/0706.3526 (Section 2.3.2)

 

[atyy]

Not my main complaint, but Ballentine's comment on renormalization and the quote from Dirac about infinities (section 19.4) is also out of date. Here a modern text would point the reader to notions of effective field theory.

Ballentine's argument is about zero point energy (section 19.4) is also suspect.
https://arxiv.org/abs/hep-th/0503158

But as I said, these are minor (you can find similar comments quite widely in older literature). My main complaint is that Ballentine omits the state reduction postulate. Certainly there are more general forms of state reduction than projection, but they all involve a change in state due to measurement that is different from unitary evolution by a Schroedinger equation, so would not escape his erroneous main objections to textbook physics in Chapter 9.

 

[Demystifier]

Rule 7 in the Insights article is not a "collapse" postulate. It's the projection postulate.

What's the difference? (Except that the word "projection" sounds more technical and hence lacks a mystic aura.)

 

[Demystifier]

Ballentine's argument is about zero point energy (section 19.4) is also suspect.
https://arxiv.org/abs/hep-th/0503158

That can be said for 90% texts (in both books and research papers) on Casimir effect.

 

[PeterDonis]

What's the difference?

"Collapse", as @vanhees71 was using the term in the post I was responding to in what you quoted from me, is an interpretation-dependent concept. "Projection" is just the basic mathematical operation described in Rule 7.

 

[strangerep]

Does a more careful analysis show that there are some cases where "watched pot" experiments are predicted to work? Because such experiments have been done, and do work.

I vaguely recall reading such claims, but it's a long time ago now, and I don't remember the references. You'll have to remind me...

I also vaguely recall rebuttals along the lines that the apparatus was explicitly contrived to regenerate the measured state, but I don't remember those references either. Maybe I can find time later next week.

[Also, check your PMs in the next few minutes...]

 

[atyy]

I vaguely recall reading such claims, but it's a long time ago now, and I don't remember the references. You'll have to remind me...

I also vaguely recall rebuttals along the lines that the apparatus was explicitly contrived to regenerate the measured state, but I don't remember those references either. Maybe I can find time later next week.

One doesn't have to use state reduction to produce a change in dynamics that causes a quantum Zeno effect, but it doesn't mean that the analysis involving state reduction is incorrect. One of the features of quantum mechanics is that what is considered a measurement outcome is observer dependent, so if there is no measured outcome, the process does not have to be modeled by a measurement. This is not unique to the quantum Zeno effect. It is also seen in the indirect measurement formalism, where the identity and timing of the outcome is dependent on the observer's assessment. It is also seen, for measurements restricted to a subsystem, in the consistency between the density matrix obtained with decoherence without any measurement, and that obtained when a measurement is performed and information about the outcomes is discarded.

https://arxiv.org/abs/quant-ph/9512012
Projection Postulate and Atomic Quantum Zeno Effect
Almut Beige, Gerhard C. Hegerfeldt

 

[martinbn]

@vanhees71 @Demystifier

Perhaps a new thread?

 

[Demystifier]

I thought we talk about Ballentine's book and I wanted to know, to which section @atyy is referring to. Rule 7 is indeed not a collapse postulate but is part of the minimal postulates you need to work with the formalism.

Where is the Rule 7 in the Ballentine's book? I cannot find it written down explicitly. (The Rule 7 is the same as the rule (ii) in my #49 above.)

 

[PeterDonis]

@vanhees71 @Demystifier

Perhaps a new thread?

Indeed. The new thread is here:

https://www.physicsforums.com/threads/difference-between-collapse-and-projection.998545/

Discussion of "collapse" vs. "projection" should take place in that thread.

 

[PeterDonis]

Where is the Rule 7 in the Ballentine's book? I cannot find it written down explicitly.

As discussed (and as noted in the 7 Rules Insights article), Ballentine does not include Rule 7 in his axioms. His equation 9.9 is more or less equivalent to Rule 7, but as the Insights article notes, Ballentine does not accept that equation as fundamental. He derives it as an effective rule in his equation 9.28.