Talk:Born rule

Untitled

There's no English translation of Born's paper available online? —Preceding unsigned comment added by Teutanic (talk • contribs) 12:55, 15 October 2007 (UTC)

I don't know whether this is worth a disambiguation, but there is in fact a "Born Rule" in crystal mechanics as well, normally called the "Cauchy-Born rule" to avoid confusion. Not sure how widel used this term is, and anyway, that article is tiny, but I'm working on it! Thudso (talk) 00:04, 10 December 2009 (UTC)

I took the liberty of adding disambiguation note to the top of page. That should be sufficient 72.221.122.168 (talk) 16:48, 14 June 2012 (UTC)

Collapse of Wave Function

Was the collapse of the wave function following measurement part of Born's theory? In textbooks I see this listed as part of Born's rule but this article doesn't say one way or the other. Dstahlke (talk) 00:15, 15 October 2010 (UTC)

Born doesn't mention it, but it is an implication with a lot of hindsight. I think von Neumann introduced the idea. --Michael C. Price ^talk 12:41, 15 October 2010 (UTC)

Roughly speaking . . .

I came here looking for confirmation of this idea: The Born Rule, roughly speaking, says that wave intensity at a location predicts the probability of finding the particle in question at that location. IF this is true, would someone please add such a statement? I heard it in a popularization featuring Benjamin Schumacher. Thanks -- 69.138.209.185 (talk) 19:35, 12 May 2013 (UTC)

Examples?

This is just a suggestion: Because applying the Born rule is so fundamental to making experimental predictions in quantum mechanics, I feel that the article would be helped by including an explicit example or two-- perhaps of the quantum harmonic oscillator and/or of the hydrogen atom, or even just a particle in a box? The terms used in the article are very abstract and confusing to non-experts, rendering the article impenetrable to most people. But even just one simple example of applying the Born rule could make it much easier for a layperson to discern the meaning of what's being described. (Some familiarity with complex functions and integral calculus would have to be assumed, I suppose, but nothing beyond that.)50.174.178.168 (talk) 21:40, 26 September 2014 (UTC)

Better Font?

Can something be done to make the mathematical symbols in this article more legible? Currently they render so faintly that it's impossible to decipher most of them.

This is, BTW, a problem common to many Wikipedia articles which use graphical representations for these symbols, rather than fixed font characters. — Preceding unsigned comment added by 74.95.43.249 (talk) 19:02, 20 October 2014 (UTC)

Language

The language in this article seems unnecessarily abstruse and overcomplicated. Especially in introductory quantum mechanics, the Born interpretation is often explained without reference to an arbitary operator or probability-projection subsets thereof of it; the fact that a wavefunction is defined (whether real or complex) and its modulus squared over an infinitesimally small point in space gives the probability is sufficiently robust to then introduce the concept of integrating over all the region through which the wf is defined in order to sum up all the probabilities (which should equal one due to the normalization requirement). --50.67.247.221 (talk) 22:33, 1 February 2015 (UTC)

Opening para

Should this "the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point" say amplitude rather than magnitude?

(I am not an active editor. Please go ahead and change if it makes sense to.

--Sarabseth (talk) 04:32, 3 March 2019 (UTC)

Why are there no experimental proofs listed

I would have thought the article should list experiments proving or at least giving evidence that the calculation works. Lowell Boggs (talk) 21:39, 5 January 2020 (UTC)

Basically, it's a step in every calculation in applied quantum mechanics, so every experiment ever done that supports quantum mechanics is evidence that Born's rule is the right way to go. There's not much point listing every physics experiment since 1900. XOR'easter (talk) 21:46, 5 January 2020 (UTC)

Wave Function vs "State Function"

It is historically correct, according to Born's original article on the interpretation of the wavefunction, and continues to be reiterated in Q.M. texts, that the inner product of Psi and its complex conjugate specifies the probability of determining a given eigenfunction (associated with a real eigenvalue), AT MEASUREMENT. Does anyone disagree with that?

My point: no matter that the wavefunction continues to too often be denoted as the "state function" of some quantum system at present time, t, that is not actually what Born told us, is it? Precisely, a present state function, as, for instance, in statistical mechanics, specifies the characteristic properties of the object at present time, t. (Think of the position and momentum of the molecule of an ideal gas.) Not so, for a quantum system according to Born. The characteristic properties of that quantum system, its eigenvalues, will be determined by a quantum measurement, which may, and often does, alter (disturb) the system being measured. Thus, whatever the state of the system just prior to measurement, it will not be that same state after a measurement is performed. Comments?

So, I would like to edit the Born Rule page to emphasize the distinction between wavefunction and "state function". Schrodinger's cat is an especially trenchant example. Shall I do that? D bar x (talk) 03:02, 25 December 2020 (UTC)

Wavefunctions are quantum states. They aren't probability distributions over properties (like Liouville densities in classical statistical physics are). There's no need to make the subject more confusing than it is by inventing a distinction that isn't there. XOR'easter (talk) 03:37, 25 December 2020 (UTC)

I agree with XOR'easter. The term "state function" doesn't appear in the article and I don't understand why we need to address that here. Dan Gluck (talk) 17:01, 25 December 2020 (UTC)

The aim is to eliminate confusion, not perpetuate it. The Born Rule isn't the only interpretation of the quantum wavefunction. There also is the Bohm-deBroglie explication, and Kastner's Transactional Interpretation, for instance. But, it's the Born Rule that's invoked to sustain such irrational suppositions as a real, tangible cat that is simultaneously both alive and dead. A physical impossibility, as, I expect, Schrodinger intended to tell us.

If, by state, we accept the usual meaning - the present characteristics of a system - then, according to Born, the wavefunction is NOT the state of the cat, or any other system. It is this ambiguous interpretation of a wavefunction as a 'state', exampled by Xor'easter here, which causes the confusion. Born tells us, unequivocally, that the wavefunction specifies the likely state (not a probability distribution) after a future measurement, so the cat, for example, is not in a state of both alive and dead. But it will be either one or the other after the next measurement. That is rational, not paradoxical.

This page could be very succinctly, and lucidly, improved to eliminate confusion caused by the ambiguous interpretation of the quantum wavefunction as a state, contrary to Born's specification. Above, it's suggested we provide an example of application of the Born Rule. Almost any quantum phenomenon would suffice, so long as we make abundantly clear the essential distinction Born stipulated: that the wavefunction predicts the state of the system only after the next measurement occurs. I suggest Schrodinger's cat as the perfect, trenchant example. 2601:8C1:C180:6D80:84D9:7B64:F419:4FC2 (talk) 16:43, 26 December 2020 (UTC)

But that is not the "usual meaning" of the term "state". Insisting that "state" mean "the present characteristics" is, implicitly, arguing for hidden variables. XOR'easter (talk) 17:22, 26 December 2020 (UTC)

Following a holiday break, careful reading of applicable literature, and consideration, I wish to reassert my argument that exorcism of the apparent Schrodinger’s cat paradox follows from a literal interpretation of Born’s rule, rather than the usual, mistaken interpretation. I’ve added that explanation to the interpretation section of the page. I trust my addition will not be removed, if that’s your inclination, without talking to me first.

It is the case that classical physics does specify the present characteristic properties of the system. Because the quantum wave function is usually called its “state”, this has led to the misunderstanding that the cat, described by its superposed wave function, can be construed as some paradoxical “state” of simultaneously alive and dead. Since the wave function is not the classical state of the system, there are no cats alive and dead at the same time.D bar x (talk) 06:00, 14 January 2021 (UTC)

This is still wrong. It's closer to an interpretation that has in fact been advanced and has a substantial following, but as written it's still Original Research (and terminologically confusing). To say that Schrödinger's cat is unproblematic for reasons along those lines requires taking a particular stand about what wave functions are first. If you say, as Asher Peres would have, that a quantum state is information, then it makes no sense to say that it's information "about the state".

Really, people have been arguing The Right Way To Think About Quantum Mechanics for a long time, offering many answers with little agreement among them. Pushing a paragraph into a Wikipedia article isn't going to solve anything. XOR'easter (talk) 16:11, 12 January 2021 (UTC)

Lets be clear. This is the page on Born's rule, which, obviously, does take a stand about what wave functions are. They predict the probability of characteristic eigenfunctions and eigenvalues of the quantum system after the measurement is performed; that understanding surely does have a substantial following among physicists. The Born rule explanation is not Peres' interpretation, nor that of Bohm-deBroglie, nor the transactional interpretation, nor any of the other, dozen or so, competing, contradicting interpretations. It's the Born rule we're talking about.

I simply argue that Born's rule, understood literally, just as it was published decades ago, not some original research, resolves the cat "paradox". It does solve that conundrum. I've carefully tried to make my argument lucid and comprehensible. If you find my terminology confusing, point directly to that.

I remain convinced that the addition to interpretation of Born's rule I've suggested does remove ambiguity about how that law should be understood. So, I protest its removal.D bar x (talk) 06:00, 14 January 2021 (UTC)

Schrödinger published the cat paradox years after Born proposed the probabilistic interpretation of wave functions, in order to illustrate what he found unsatisfying about Bohr–Heisenberg–Born. If Born's rule, understood literally dispensed with Schrödinger's cat, then there would never have been a Schrödinger's cat. The complaint that people have always had with Bohr and Heisenberg, and that the cat experiment was supposed to illustrate, is that "measurement" is an ambiguous notion. And if "measurement" is ill-defined, then talking about the probability of characteristic eigenfunctions ... after the measurement is performed doesn't resolve anything. XOR'easter (talk) 16:06, 14 January 2021 (UTC)

Once more, for emphasis: we are talking about the Born rule here, not one, or another, of the many, contradictory, confounding, adamantly-held, preferred proposals for quantum measurement. Quantum measurement is not an ‘ambiguous notion’ according to Born. What do we read on the Wiki, Born rule page? It says that “the measured result will be one of the eigenvalues” of the operator corresponding to that observable. That’s just what I’ve written previously. And, such eigenvalues are characteristic of the measured system, of course.

Schrodinger did not propose the cat example to illustrate that the Born rule is unsatisfying because it leaves measurement ambiguous (Schrodinger, Proceedings of the American Philosophical Society, 124, pp 323-38; see also, Wheeler and Zurek, "Quantum Theory and Measurement" I-11, section 5). Clearly, it does not do that, in spite of XOR’easter’s contrary assertion. Instead, he uses the cat to demonstrate the absurdity of interpreting the wave function as a classical state depicting well-defined properties. We ought not naively accept the wave function, with “the living and dead cat...smeared out in equal parts” as “representing reality,” he says (Wheeler and Zurek, p. 157). What Born said of measurement, Schrodinger repeats. He tells us that this “macroscopic indeterminacy”... “can then be resolved by direct observation.” (Wheeler and Zurek, p. 157)

So, where do we find this cat ‘paradox’ claiming that quantum mechanics implies a cat both dead and alive simultaneously, if not in the original Schrodinger article? That interpretation is nearly ubiquitous among contemporary quantum physicists. As one example, Zurek may now be the most renowned of quantum theorists. His decoherence theory of quantum measurement is almost canonical today. Consider his seminal article explaining that theory (Zurek, Phys. Today, 1991, pp. 36 - 44). He writes that “...at the root of our unease with quantum mechanics is the clash between the principle of superposition...and the everyday classical reality in which the principle appears to be violated.” (p. 36) He supplies an illustration of a superposed cat to further this misunderstanding. Note carefully that superposition describes the wave function, which, according to Born, provides the probability of measuring characteristic eigenvalues. Everyday classical reality is our tangible, material world, including singular cats. There is no conflict between a wave function superposition and our perception of definitive, autonomous (not superposed) physical objects, unless one supposes that the wave function actually specifies the classical state of some physical object. Contrary to what Schrodinger told us, Zurek claims here that the wave function is such a state in conflict with our normal perceptions. Without the careful, critical consideration Schrodinger advises, one might believe the quantum state (a wave function) is a classical state of the real world.

Further, Zurek writes: “Why do I, the observer, perceive only one of the outcomes?” of a quantum superposition. (Phys. Today, p. 37) Because, of course, the wave function does not describe superposed characteristics of some physical reality. It is not a classical state function. Rather, it prescribes the probability for the definitive outcome of a future measurement. Many other physicists now perpetuate Zurek’s misunderstanding.  

My approach is to selectively analyze just one particular aspect of quantum theory - Born’s rule - to resolve a single misapprehension. Such a limited investigation may be the most efficacious method. I believe the objections of XOR’easter are neither realistic, nor scientifically persuasive, and I formally protest his removal of my addition to the interpretation section of this page.D bar x (talk) 22:39, 18 January 2021 (UTC)

"Emerging Technology from the arXiv" is a blog by a pop-science magazine that blurbs for non-peer-reviewed preprints, not a reliable source. If you want to argue that Zurek is fundamentally wrong, that's fine, but Wikipedia is not the place to advance your own take as definitive. Nor is this Talk page intended to be a forum for debating the interpretation of quantum mechanics. You're welcome to write a paper arguing that a close reading of Born supports a psi-epistemic or psi-doxastic interpretation of wave functions in the Spekkens–Brukner–Fuchs region of the spectrum, and to try and get that paper published in Foundations of Physics or Stud. Hist. Phil. Sci. B. But even then, you'll be one advocate among many. As N. David Mermin once said, new interpretations of quantum mechanics appear every year, and none ever disappear. XOR'easter (talk) 21:39, 18 January 2021 (UTC)

In case anybody is confused: the first sentence of my previous comment refers to a line that has since been deleted. XOR'easter (talk) 20:11, 21 January 2021 (UTC)

I’ve continually made a very simple argument here. Born’s rule, literally interpreted, just as he explained it, exorcises the Schrodinger cat paradox. I’m not proposing my own theory of quantum measurement, or wave function interpretation, as XOR’easter continues to claim. Instead, I’ve plainly described Born’s own interpretation in the section of this page on interpretation. This is precisely the right place to do that, I believe.

Nor am I trying to rebut Zurek’s decoherence theory. XOR’easter has mistakenly claimed that Schrodinger suggested the cat as evidence that quantum theory implies a cat simultaneously alive and dead. Schrodinger did not say that. (I quoted him.) As I’ve written here, Schrodinger told us that the quantum wave function is not a realistic, classical state function describing a paradoxical existence of the cat. That would be absurd, of course. So, I’ve cited Zurek as one influential contemporary physicist, among many, who does assert that the paradoxical cat state is implied by quantum mechanics. I’ve cited Zurek unequivocally saying so. There is a legion of others who perpetuate the same misunderstanding.

I believe the misunderstanding within much of the present quantum theory community results from a lack of critical consideration of the term “state”. In spite of what XOR’easter said, the present characteristics of a physical object are, indeed, the classical state of a system, often applied in our physics textbooks, especially thermodynamics, and must not be confused with the wide-spread use of “quantum state” to label a wave function. That’s also what Schrodinger told us when he introduced his cat example.

I don’t believe this dialog with XOR’easter is making progress, and will request a second opinion. D bar x (talk) 19:38, 28 January 2021 (UTC)

You are making a proposal for what wavefunctions are — for the relation between the mathematics and physical reality. That's what an interpretation of quantum mechanics does. Proposing an explanation for what wavefunctions are while saying that it is not an interpretation is like claiming to speak without an accent.

I did not say that you were trying to rebut Zurek's decoherence theory. I wrote, If you want to argue that Zurek is fundamentally wrong, that's fine, but Wikipedia is not the place to advance your own take as definitive. In your reply, you call Zurek's assertions a misunderstanding. So, in your view, Zurek is wrong about something — not about how to do technical calculations in decoherence theory, but about the meaning of wavefunctions.

You began this conversation by saying, It is historically correct, according to Born's original article on the interpretation of the wavefunction, and continues to be reiterated in Q.M. texts, that the inner product of Psi and its complex conjugate specifies the probability of determining a given eigenfunction (associated with a real eigenvalue), AT MEASUREMENT. Later, you wrote, This is the page on Born's rule, which, obviously, does take a stand about what wave functions are. They predict the probability of characteristic eigenfunctions and eigenvalues of the quantum system after the measurement is performed. In order for a statement like this to be meaningful, one has to say something about what "measurement" means. Hence the question that has been asked in one form or another since Einstein and von Neumann at least: what counts as a "measurement"? In the thought-experiment with the poor cat, does the "measurement" occur at the ionization of a gas atom in the Geiger counter? When the counter sends out an electrical signal? When the vial of poison breaks? When the poison enters the cat's lungs? If "measurement" is an "act of irreversible amplification", how irreversible does it have to be? Latter-day Bohrians like Günther Ludwig and Léon Rosenfeld would probably offer an answer in thermodynamic terms [1]. QBists like Mermin and Fuchs would perhaps say that "measurement" is a conceptual primitive of the theory and all the arguments about von Neumann chains are beside the point. The trouble is, first, unless one says something about what "measurement" means then invoking it to explain what wavefunctions are is an empty statement. Second, none of these schools of thought enjoy consensus within the scientific community, so Wikipedia cannot advance any of their answers as definitive. That goes equally well for Born's view, or Schrödinger's, or anything that we claim to be the position of Born or Schrödinger. XOR'easter (talk) 21:20, 28 January 2021 (UTC)

Response to third opinion request:

Unfortunately, since more than 2 editors are involved in this dispute, your Third Opinion request has been declined. This is obviously a highly technical dispute. It would be best to post your concerns to WP:PHYSICS, WP:WPMATH or to some other related Wiki Project. Dr. Swag Lord (talk) 07:06, 6 February 2021 (UTC)

Which is the true Born rule?

This article effectively claims:

• The true Born rule involves projection operators.
• The commonly-taught Born rule is only one special case of the true rule (and differs in being inapplicable to non-degenerate observables).
• The Born rule can be derived from POVM (but not vice versa).

Are there any reliable sources to support these claims? Cesiumfrog (talk) 13:22, 28 January 2023 (UTC)

The Born Rule

What is a PVM? Len loker (talk) 00:25, 28 March 2024 (UTC)

Projection-valued measure. Cheers, Jähmefyysikko (talk) 05:41, 28 March 2024 (UTC)

Probability density

@XOR'easter, I noticed you’ve reverted my edit. First of all, I based the edit on a reference I cited, so I’m not sure why I should “avoid” the phrasing. If I’d removed the “when measured”, I could see why you’d have an issue with that, but didn’t.

Also, the phrasing regarding the probability density is the same as in the “details” subsection. So could you please explain what the problem is? Roffaduft (talk) 18:59, 13 October 2024 (UTC)

I tried to clean up the introduction, but I was not satisfied with the result, so I made a further modification. I found the phrasing slightly unsatisfying, by way of having a potentially misleading connotation, and tweaked it to try and avoid that. We should not imply that in quantum mechanics, the value of a particle position measurement exists prior to that measurement. The phrasing "the probability density for the position of a particle, when measured" is both a little awkward, with that "when measured" interjected, and possibly misleading. Or, to say it another way, the "when measured" qualifier might have originally been inserted to avoid the misinterpretation, but it does so in a rather unclear and ineffective way. Hall's book is fine on the math but not particularly scrupulous about connotations, which is to be expected for a book called Quantum Theory for Mathematicians that expressly disavows all questions of "interpretation" (pp. 15–16) and says nothing at all about Bell's theorem, Gleason's theorem, etc. See the discussion on pp. 15–16 of Peres' Quantum Theory: Concepts and Methods, for example. XOR'easter (talk) 21:40, 13 October 2024 (UTC)

Thank you for your elaborate reply and subsequent edit of the "Details" subsection. At least it's consistent now. I'll have a look at the suggested literature but, as I'm first and foremost a mathematician, if the ratio text vs. equations is too skewed I can't give any guarantees ;)

Kind regards, Roffaduft (talk) 03:58, 14 October 2024 (UTC)