Talk:Measurement in quantum mechanics/Archive 1

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Archive 1

The Quantum entanglement problem

One should also notice that the collapse proccess is happening instantly, and thus violating the locality principle that the no physical interaction can exceed the speed of light. It is easily derived, if we consider the following gedanken experiment dealing with entangled pair of particles:

Suppose we have a machine that produces conjugate particles, in which measurable property P of particle A corresponds with the value of P in particle B (Experimental physicists can actually build such a system, using conservation laws).

Experimental physicists can measure whether (and to which accuracy) some particular setup, in some particular trial, was closed.

Based on such measurements in past trials they may have some expectations about what would be found; but that does not spare to actually measure (again), in the following trial(s).

When I talked about particle A and particle B I didn't ment for seperate trials which are carried out one after another, but for a trial which A and B are corresponded because of conservation law. Such trials have been conducted, using the conservation of spin. In this case, A and B are two particle which exist simultanisly, each travel to different direction, but they were generated together.

In other words, if we know the value P in particle A we also know the value of P in particle B.

No: there may be expectations (from previous trials), but knowing (in this following trial) requires having measured (in this following trial).

No. Knowing. When A is measured B can only accept a single value which not violates the conservation law.

No. If "another value of B (and B's detector)" is found, then this value is merely unexpected. Thereby may be found, for instance, that the system as a whole didn't remain closed. This can happen; and it does happen frequently.

But according to the conservation law, the probability to measure value of B different than val(B) = TOTAL - val(A) is zero. The thought experiment that I described is based on the assumption that the system is close and there is conservation of relevant measurables.

Therein lies the problem, AFAIU: You/we can presume to know (the outcome "on the other side", in advance) only with the caveat of this assumption. IOW: we don't really know, until the measurement is obtained explicitly.

In discussing "= The Quantum entanglement problem =" such assumptions should at least be made explicit; and in particular, how it ought to be decided whether they were "valid" (and any particular was "valid" in this sense) to begin with. Frank W ~@) R 16:50, 5 Apr 2004 (UTC)

Indeed, that's the solution to the paradox that ensure us information can't be transmitted faster than the speed of light. But I dealt with a little bit different subject - that the collapse "interaction" travels faster than the speed of light. MathKnight 22:16, 5 Apr 2004 (UTC)

Then how could this supposed subject be dealt with at all ?

Consider (and please expand on) my "Example with only few assumptions (?)" below, for instance. Frank W ~@) R 15:22, 6 Apr 2004 (UTC)

Now, we shall let the particles get very far from each other before we conduct a measurement (for instance, we will wait until they are one light-second apart). Now, we measure P in particle A. At the same moment we also know P of particle B, although particle B is a light-second away from us, i.e. we have managed to get information on particle B faster than the speed of light. More fundamentally, when we measure particle A, its wave function collpases, but in the same moment (instantly) the wave particle of B also collapses. According to locality principle, however, it should take at least a second for particle B to notice that particle A's wave function has changed. In other words, the influence (interaction) of particle A over particle B traveled faster than the speed of light, which is in contradiction to Einstein's the theory of relativity.

(Side note: this is the known problem of Quantum entanglement, a developement of this paradox is the EPR paradox)

One of the solutions to this paradox, or contradiction, is based on the actualism philosophy of Immanuel Kant, George Berkeley, David Hume, Henri Poincaré and Niels Bohr. According to it, the collapsing of the wave function is not a physical process or interaction, but rather a logical process which is an outcome of the observer's mind that interprets the events according to his apriori conventions.

See also: Wavefunction collapse, actualism, Copenhagen interpretation, Quantum entanglement, EPR paradox.

Sorry, the notion that the wavefunction collapse happens instantly is bosh. See quantum decoherence. Also, there are no nonlocal effects in quantum mechanics; relativistic quantum field theory takes locality as an axiom. Dave Kielpinski 00:16, 19 December 2005 (UTC)

it does happen instantly (according to, say, G. Brassard). Mct mht 04:16, 15 June 2006 (UTC)

According to the decoherence modellists collapse takes a finite time (the decoherence time!). BTW I noticed that

1. The trace of a Hermitian operator, is real. It follows that the average, or expectation value, of outcomes which may be obtained from performing measurement on an ensemble would also be real.

was lost on the grounds that it was considered too obvious. Is that really wise? Obvious to who? To us, yes, but to the lay audience? I doubt it. --Michael C. Price ^talk 00:47, 25 June 2006 (UTC)

i removed it. if you wanna put it back, go ahead. it needs a little modification though. that statement as it is doesn't accomplish much and is not quite accurate, IMHO. for vectorial pure states, a better way to convey the message to someone not familiar with the stuff would be a remark like, say, "a matrix M is Hermitian iff <x, Mx> is real for all x. this is precisely the expectation value of M assuming the system is prepared in state x. as we would expect, the expectation value of an observable is real, as each measurement outcome is real. similarly, for a system prepared in mixed state ρ, the expectation value of an observable M is tr(ρM), which must also be real, as a consequence of the spectral theorem." it is not completely honest to say trace of a Hermitian matrix being real leads to the avg. value being real. Mct mht 07:15, 25 June 2006 (UTC)

Reit is not completely honest to say trace of a Hermitian matrix being real leads to the avg. value being real. I don't see why not:

${\displaystyle \operatorname {Tr} (\rho M)=\operatorname {Tr} (\rho ^{1/2}M\rho ^{1/2})}$

The latter is selfadjoint, obviously.--CSTAR 15:07, 25 June 2006 (UTC)

ok, bad wording on my part. i just meant it needs a little more explanation, like the one you got. (perhaps add "by cyclic property of the trace" as per Michael C. Price's comment) Mct mht 19:43, 25 June 2006 (UTC)

The uncertainty principle

According to the uncertainty principle, one cannot measure simultaneously the definite (exact) momentum and place of particle.

"place" ?? (Translation) distance !

I think "place" or "position" of the particle is better term to use to clear the issue, I know that "place" is merely a distance from the origin of pre-set frame of reference, but it is important to note when we measured x we want to measure where the particle is and not how much did it move.

Decisive is that results of measurements (in application of the same, reproducible measurement operator) are commensurate to each other; as are values of distance.

Please explain more what do you mean.

Values of distance can be compared to each other, real-number ratios between distance values be evaluated; as is required of measured results, and as applicable to eigenvalues of Hermitean operators.

In contrast: what could be made of "positions" ?? Frank W ~@) R 15:22, 6 Apr 2004 (UTC)

Simply define position as a distance from a fixed reference point. I used the term position for more intuitive reason, to point out the meaning that the particle is localized around x. MathKnight 16:15, 6 Apr 2004 (UTC)

Well -- if this is the definition (of "what and how is to be measured") then I prefer wording by which the definition is most explicit and transparent ...

Other nomenclature might have to be accompanied by a

Disclaimer: Referring to <...> instead as <...> is considered more intuitive by some; however, it is not gueranteed that their intuition is as far reaching as Measurement in quantum mechanics presumes to be applicable. Frank W ~@) R 15:14, 7 Apr 2004 (UTC)

I accept this offer. MathKnight 20:53, 7 Apr 2004 (UTC)

Splendid! Let's hope that any articles that are or still would be linked are no less scrupulous ... Frank W ~@) R 02:33, 8 Apr 2004 (UTC)

A big question arises now: "Is it just that we can't measure definite momentum and place, or is it there aren't definite momentum and place?" In other words, are definite place or momentum exist only when measured?

This question is still under debate.

Hardly: the Momentum operator is defined as generator of translation. What else ??

Momentum is defined as operator, but is it defined as value? Can we associate a particle a value which is its definite (exact) momentum? It is clear we can never measure such a value, but does it exists even if we can't measure it?

Not clear at all. What would prevent us from evaluating ∂/∂x, case by case, at least in principle ?? Regards, Frank W ~@) R 15:08, 5 Apr 2004 (UTC).

Please explain more what do you mean.

You explain, please! Why could we "never measure a momentum value" ? Frank W ~@) R 15:22, 6 Apr 2004 (UTC)

Because of the uncertainty principle (UCP), ${\displaystyle \Delta x\cdot \Delta p\geq \hbar }$. If we can measure the exact value of momentum p than ${\displaystyle \Delta p=0}$, contradicting the UCP (we talk about a particle which localized in some way or another. MathKnight 16:15, 6 Apr 2004 (UTC)

Interesting and important argument -- I just updated the uncertainty principle to indicate that and why it's also a wrong argument.

In our specific example: the derivative of a function (in one particular point) can not be evaluated if the function is given/defined only in this point. Frank W ~@) R 15:14, 7 Apr 2004 (UTC)

See: uncertainty principle, actualism.

What is a measurement? When the wave function collapse?

The Schrödinger's cat example shows how quantum uncertainty and superposition of states can be expanded to macroscopic bodies.

The article as it stands is not limited to "microscopic bodies"; nor, IMHO, should it draw such distinctions.

The bizzare quantum effects are mostly seen directly only at the sub-atomic levels. It is much easier to you to think about superposition of places for an electron or interference of neutrons or electrons rather on uncertainty in a position of a tree and intereference of a chair.

Even today, there isn't a decisive definition or method to determine in which conditions exactly the wave function collapsement takes place.

Finally, concerning

"Notice that treating d/dt as operator is far more complicated"

How is ∂/∂t more complicated than ∂/∂x,

The Hilbert space in which we work is defined on a square-integrable functions according to x. We have an inner product in x\p representation but we don't have such to time. (Maybe there is, but it is not just clear as the x one). Anyway, when I said "Hamiltonian" I ment the H = P^2/(2m) + V(x) operator.

not even to mention "m" and "V(r)" ??

For right now, m is a scalar and "V(r)" or "V(x)" (potential which not dependent on time) can be treated as operator.

And why the apparent restriction to "Sqrt( (c p)^2 + (m c^2)^2 ) =(approx)= (m c^2) + 1/2 p^2/m" ??

I didn't understand. As far as I recall, it is the relative energy. Please explain more.

The good old "non-relativistic limit":

"Sqrt( (c p)^2 + (m c^2)^2 ) =(approx)= (m c^2) + 1/2 p^2/m" ??

"Sqrt( (c p)^2 + (m c^2)^2 ) = m c^2 Sqrt( 1 + (p/m c)^2 )".

If "(p/m c)^2 << 1" (i.e. in a "non-relativistic limit"):

"m c^2 Sqrt( 1 + (p/m c)^2 ) =(approx)= m c^2 (1 + 1/2 (p/m c)^2 - 1/8 (p/m c)^4 + ...) =(approx)= m c^2 + 1/2 p^2/m,

where only "1/2 p^2/m" may be "dynamically relevant".

The article might (and presently does) simply mention both: the general Hamiltonian/energy operator, and the non-relativistic approximation. Frank W ~@) R 15:22, 6 Apr 2004 (UTC)

O.K. I just didn't heard yet on a relative Hamiltonian opertor in QM.

Best regards, Frank W ~@) R 04:50, 5 Apr 2004 (UTC).

My notes are in odd-spacing ident. Best regards. MathKnight 16:36, 5 Apr 2004 (UTC)

Again, the statement "Even today, there isn't a decisive definition or method to determine in which conditions exactly the wave function collapsement takes place." is incorrect in the light of decoherence theory. Dave Kielpinski 00:18, 19 December 2005 (UTC)

Below, the revised paragraph for inline editing (discussion should be done above in order not to break continueity of the paragraph). A revised version of it would appear in the article. You are free to add clarifactions to this text or more accurate side-notes.

The Quantum entanglement problem

One should also notice that the collapse proccess is happening instantly, and thus violating the locality principle that the no physical interaction can exceed the speed of light. It is easily derived, if we consider the following gedanken experiment dealing with entangled pair of particles:

Suppose we have a machine that produces conjugate particles, in which measurable property P of particle A corresponds with the value of P in particle B (Experimental physicists can actually build such a system, using conservation laws). In other words, if we know (have measured) the value P in particle A we also know the value of P in particle B. Now, we shall let the particles get very far from each other before we conduct a measurement (for instance, we will wait until they are one light-second apart). Now, we measure P in particle A. At the same moment we also know P of particle B, although particle B is a light-second away from us, i.e. we have managed to get information on particle B faster than the speed of light. More fundamentally, when we measure particle A, its wave function collpases, but in the same moment (instantly) the wave particle of B also collapses. According to locality principle, however, it should take at least a second for particle B to notice that particle A's wave function has changed. In other words, the influence (interaction) of particle A over particle B traveled faster than the speed of light, which is in contradiction to Einstein's the theory of relativity.

(Side note: this is the known problem of Quantum entanglement, a developement of this paradox is the EPR paradox)

One of the solutions to this paradox, or contradiction, is based on the actualism philosophy of Immanuel Kant, George Berkeley, David Hume, Henri Poincaré and Niels Bohr. According to it, the collapsing of the wave function is not a physical process or interaction, but rather a logical process which is an outcome of the observer's mind that interprets the events according to his apriori conventions. Making the collapsement a logical process and not physical, along with the work of of physicts which proved the Quantum entanglement cannot transmit information - the contradiction with the locality principle of Albert Einstein vanishes and the paradox is solved.

See also: Wavefunction collapse, actualism, Copenhagen interpretation, Quantum entanglement, EPR paradox.

The uncertainty principle

According to the uncertainty principle, one cannot measure simultaneously the definite (exact) momentum and place (position) of particle. A big question arises now: "Is it just that we can't measure definite momentum and place, or is it there aren't definite momentum and place?" In other words, are definite place or momentum exist only when measured?

This question is still under debate.

See: uncertainty principle, actualism.

Disclaimer: Referring to place instead as distance is considered more intuitive by some; however, it is not gueranteed that their intuition is as far reaching as Measurement in quantum mechanics presumes to be applicable. Place is measured by a distance from a fixed reference point. When we say that particle's place is x we mean that the particle is localized around x.

What is a measurement? When the wave function collapse?

The Schrödinger's cat example shows how quantum uncertainty and superposition of states can be expanded to macroscopic bodies.

Even today, there isn't a decisive definition or method to determine in which conditions exactly the wave function collapsement takes place.

---

Suppose that a signal source had been observed in a large number of distinguishale trials by two detectors who each detected one signal at a time in one of two distinct own channels or outcomes: A detecting and counting a signal either as (A↑) or (A↓), and B detecting and counting a signal either as (B «), or (B »).

Suppose further, that the observations by A and B had been perfectly correlated:

• in each trial in which A had found (A↑), B had found (B «),
• in each trial in which A had found (A↓), B had found (B »), and vice versa
• in each trial in which B had found (B «), A had found (A↑), and
• in each trial in which B had found (B »), A had found (A↓).

Finally, suppose that in each trial A had observed the signal before B did (the signal source had always been closer to A than to B).

Therefore ... ??

The question is rather

whether (?)

the value of A determines the value of B. In the thought experiment I described - it is known physically that val(A) + val(B) = TOTAL and thus by measuring val(A) we can calculate val(B).

That's satisfied in my example.

I explicitly assumed such a relations between A and B (unlike the example you presented).

Fair enough -- in my example, that's not just an unsubstantiated assumption, but explicitly given and known.

In Quantum Mechanics, measuring A is making its wave function collapse, and it is clear that if we measure val(A), than val(B) = TOTAL - val(A). So in some sense, B's wave function also collapsed, and I further dare to say - simultaneously with the collapse of A's wave function. So, seemingly, there is a violation of the locality principle. MathKnight 16:15, 6 Apr 2004 (UTC)

To focus on one particularly glaring question (as far as such "collapse" is relevant at all):

By your description, apparently "B's wave function" (and/or "the wave function of what's on its way to B" ?) may have "collapsed" before B's observation of the signal.

May "A's wave function" (and/or "the wave function of what's on its way to A" ?) not similarly have "collapsed" before A's observation of the signal ?

In particular: may such "collapses" not have occured (already) on emission of the signal (together, and therefore indeed simultaneous) ?

Best regards, Frank W ~@) R 02:33, 8 Apr 2004 (UTC).

Interesting idea. If the collapse takes place in the emision, locality is not violated. In this case I think we should not see A and B as different particle with different wavefunction, but as a system with a single wavefunction with eigenstates ( val(A) , TOTAL - val(A) ). Therefore, measurement makes all the system collapse and not just the measured particle.

BTW, we have found a mechanism that makes the wavefunction collpase without actually makes a measurement (if we do accept that collapse can take place only at emision).

It is really interesting discussion here. :-) MathKnight 13:20, 11 Apr 2004 (UTC)

Isn't '${\displaystyle {\hat {x}}}$, where ${\displaystyle {\hat {x}}={-\hbar \over i}{\partial \over \partial p}}$' in the definition of the distance operator equivilant to ${\displaystyle {\hat {x}}}$, where ${\displaystyle {\hat {x}}={i\hbar }{\partial \over \partial p}}$ - NeilTarrant 21:41, Aug 30, 2004 (UTC)

Yes. This is because of the mathematical identity: i*i = -1 and thefore i = -1/i. MathKnight 22:12, 30 Aug 2004 (UTC)

In my opinion I think the second form (${\displaystyle i\hbar }$) is a more logical representation, however I can see arguments for keeping the page in the first form as every operator is defined in the same form... Any comments? --NeilTarrant 07:40, Aug 31, 2004 (UTC)

---

See also Delayed-choice experiments

Who ever put him, care to explain why? MathKnight 16:37, 10 Sep 2004 (UTC)

(William M. Connolley 16:56, 10 Sep 2004 (UTC)) I listed this on cleanup a long time ago. Someone else has moved it from there to pages in need of attention.

As to why... lets see: Firstly, looking at the history I see I've hacked quite a bit out already, which answers some of my original complaints...

• "Eigenstates and projection" assumes eigenstates are countable not continuous
• The delta function isn't a function. This affects some later summations, which aren't
• And (being somewhat harsh) in general the rest of the discussion in that section is unhelpful and confusing (possibly confused too). Essentially there are a lot of unhelpful Big Maths equations where a few sentences of English would do. I used to know this stuff but have fallen off recently: I may be being unfair.

Thanks for the input. I will try to work it out a little bit, I hope others in the field will contribute as well. MathKnight 18:14, 10 Sep 2004 (UTC)

The following part was added by User:Phys, and was disputed by User:William M. Connolley:

However, in the ensuing decades, after physicists came to terms with quantum entanglement, decoherence etc., the apparent "collapse" turned out to be a phenemological consequence of entanglement coupled with decoherence and is not only consistent with a deterministic Schroedinger's equation but is a consequence of it! Not only that, the insight was that it's was not the process of measurement (which can't even be defined precisely!) which drives the phenemological collapse but a decohering entanglement with the environment. See the relative state interpretation and decoherence for more details. A later reanalysis of the Bohr-Einstein debates in light of our current knowledge of decoherence and entanglement revealed that Bohr's analysis, which were at the heart of the foundations of the Copenhagen interpretation were flawed.

(William M. Connolley 12:17, 26 Sep 2004 (UTC)) Indeed it was. BTW, note that RSI redirects to Many-worlds interpretation.

I saw this article listed on Wikipedia:pages needing attention, and boy, does it. I've tidied up some of the grammar for now; I hope my tidying of some of the clumsier phrasings hasn't introduced some subtle error beyond my understanding. I've left the example alone. (Someone who gets MWI should probably expand the "Reject it as a physical process ..." bullet point.) I was going to plough on and try a full rewrite, but it occurs to me that wavefunction collapse and measurement problem both cover similar ground (albeit in the latter's case from a horribly pro-Many Worlds Interpretation POV). What I suggest is this:

• Merge an NPOVed version of the content from measurement problem into this article.
• Merge the general discussion of how measurement works under Copenhagen currently in wavefunction collapse into this article; simultaneously, move the detailed mathematical discussion from here to wavefunction collapse and include internal links in both directions (ie wavefunction collapse, which is after all a more technical term, becomes the place to go for mathematical formalism; this article describes things in non-mathematical terms and also tries to explain the philosophical aspects, from all angles -- I think an ideal version of this article should give various different viewpoints' answers to at least some of those questions).

I would probably just get on and do this if I wasn't worried that I'm biased towards wavefunction collapse, substantial parts of which are still my turgid prose from a couple of years ago. Bth 18:57, 14 Oct 2004 (UTC)

The "wavefunction collapse" section needed serious work. It employed obsolete measurement ideas that predate the development of decoherence theory, and contained many incorrect statements like ``The collapse process has no trace or corresponding mathematical description in the mathematical formulation of quantum mechanics" and claims that the wavefunction collapses "instantly". I cleaned it up in light of decoherence theory.

The section on the von Neumann scheme contained some tendentious remarks at the end, which I have rewritten. References to the Bohm interpretation should include serious disclaimers since only a small minority of physicists accept it.

The "philosophical implications" section was also written without reference to recent work. The questions "Does a measurement depend on the existence of a self-aware observer?" and "What interactions are strong enough to constitute a measurement?" are addressed satisfactorily by decoherence theory. However, the answers to these questions are nontrivial and so I have included them in this section.

The Copenhagen interpretation seems to mean a lot of different things to different people, so I've weakened the statement "According to the Copenhagen interpretation, the answer is an unqualified "yes"." Based on this, I'm guessing the Copenhagen interpretation article needs work. Dave Kielpinski 00:08, 19 December 2005 (UTC)

An anonymous editor has removed my warning to readers on the Afshar experiment external link. No reason was given. I am willing to discuss this point, but for now I have replaced the warning (somewhat toning it down.) Dave Kielpinski 01:29, 19 December 2005 (UTC)

I started editing this page; feel free to revert me if you can justify. I remove this, it's incomprehensible. "The goal of a particular measurement of a particular system, in any experimental trial, is to obtain a characterization of the system in mutual agreement between all members of this system, and therefore by a particular method which is reproducible by all members of the system, at least in principle." If you know what it means, please rewrite. The rest is attempted better exposition. GangofOne 13:52, 30 December 2005 (UTC)

Should this be merged with Measurement problem? Also, Measurement (quantum mechanics) redirects to Measurement problem and not here. --Apoc2400 12:04, 22 March 2006 (UTC)

Yes, definitely merge. --Michael C. Price ^talk 00:53, 25 June 2006 (UTC)

i changed "mathematical formalism of measurement" to "formalism of...". that has been reverted. a small point but article should be honest. a mathematical and rigorous description of the von Neumann measurement scheme would include the PVM associated to an observable, etc. The short paragraph on the case of "continuous spectrum" is kinda funny. Of course it can be rectified using rigged Hilbert spaces but that's not mentioned either. Mct mht 22:46, 14 June 2006 (UTC)

That reversion was vandalism by Hyrun. Reverted. --Michael C Price 23:41, 14 June 2006 (UTC)

the subsection giving the e.g. re particle in a box, i'm sry to say, needs to go or a complete rewrite. clean up tag will be added. Mct mht 04:41, 26 June 2006 (UTC)

The whole thing needs a rewrite.--CSTAR 19:30, 26 June 2006 (UTC)

the wikilink to resolution needs to point to a more exact page. [[Resolution (logic)] makes sense to me but I could be wrong. STHayden ^{[ Talk ]} 02:35, 22 August 2006 (UTC)

The comment(s) below were originally left at Talk:Measurement in quantum mechanics/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Comment(s)

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I rated this "top" importance, but I don't know much about what the standards are. Others are welcome to downgrade. --Steve 23:58, 30 November 2007 (UTC) Steve (talk) will all due respect I don't know if you noticed that the importance is set on "high" as for Wikipedia:WikiProject_Physics and not on "Top" as you stated. I have set it for "High" too as for what Wikipedia:WikiProject_Philosophy should be concerned. As well I don't know much about what the standards are but like you I am open to debate. LOL.   M aurice   Carbonaro  19:03, 17 February 2013 (UTC)

Last edited at 19:03, 17 February 2013 (UTC). Substituted at 15:22, 1 May 2016 (UTC)

This talk page hasn't received any comments since May 11, 2007. It's been since June 26 2006 since an actual conversation took place. I've archived the old talk page to start fresh on cleaning up this article. I'll put the appropriate tags on the main page. Here's hoping we can all work together to make this article better. JFlav 02:33, 22 September 2007 (UTC)

This topic is in need of attention from an expert on the subject. The section or sections that need attention may be noted in a message below.

This article needs to be revised or rewritten. It is heavy on derivations and examples and light on any explanation of concepts. In fact, it reads as if it was copied out of someone's quantum mechanics notes. JFlav 02:52, 22 September 2007 (UTC)

you wanna volunteer? :-) Mct mht 03:13, 22 September 2007 (UTC)

Yeah, I'll do my best. But I don't know how much I qualify as an "expert." JFlav 16:34, 22 September 2007 (UTC)

You'll notice I did a thorough cleanup on this article recently. Does it address the problems? In what ways could it be improved? --Steve 00:00, 1 December 2007 (UTC)

The example can't be salvaged. Wavefunctions can't collapse to single position eigenkets without violating the uncertainty relation. This can be seen as follows: if a measurement is made which collapses the wavefunction to the position ket ${\displaystyle |S\rangle }$, we can project this into position space as ${\displaystyle \langle x|S\rangle =\delta (x-S)}$, and the position standard deviation for this wavefunction would be zero. The assumption that the function has collapsed to this nonphysical state leads to an error in the derived probability of measuring the nth energy value immediately after the position measurement. Suppose, for example, that a particle is measured at L/2 with such accuracy that it is left in the state ${\displaystyle |{\tfrac {L}{2}}\rangle }$. Then: ${\displaystyle \Pr(E_{n})=|\langle \psi _{n}|{\tfrac {L}{2}}\rangle |^{2}={\frac {2}{L}}~{\rm {sin}}^{2}\left({\frac {n\pi {\tfrac {L}{2}}}{L}}\right)={\frac {2}{L}}~{\rm {sin}}^{2}\left({\frac {n\pi }{2}}\right)={\frac {2}{L}}}$ for every odd n. This is, of course, impossible as these probabilities must add to 1. Auspex1729 (talk) 17:56, 2 August 2009 (UTC)

First, since (non-relativistic) quantum mechanics imposes no limitation on the accuracy with which a parameter can be measured there is no problem with collapsing onto eigenkets.

Second, the formula:${\displaystyle \Pr(E_{n})=|\langle \psi _{n}|{\tfrac {L}{2}}\rangle |^{2}}$ is incomplete. You have assumed a discrete orthonormal basis. More generally it should be:

${\displaystyle \Pr(E_{n})={\frac {|\langle \psi _{n}|{\frac {L}{2}}\rangle |^{2}}{|\langle \psi _{n}|\psi _{n}\rangle ||\langle {\tfrac {L}{2}}|{\tfrac {L}{2}}\rangle |}}}$

Since ${\displaystyle |\langle {\tfrac {L}{2}}|{\tfrac {L}{2}}\rangle |=\infty }$ each individual probability vanishes, but they still sum to one.--Michael C. Price ^talk 19:21, 3 August 2009 (UTC)

You have assumed a discrete orthonormal basis. Because the infinite square well has a discrete orthonormal basis! You've assumed, to the contrary, that they aren't (fine, it's your life) and thrown some orthonormalization terms into the probability. You should get the same thing as someone who started with an orthonormal basis, but you don't. You don't because you're using unbelievably sloppy math here, and drawing incorrect conclusions as a result. Then you threw some redundant notation into the position kets in the example... why? And doesn't your conclusion that "each individual ${\displaystyle \Pr(E_{n})}$ vanishes" disturb you, especially in an example which was supposed to demonstrate the probability of collapse to the energy eigenstates after a position measurement? The position has a continuous spectrum, and a position measurement will cause the wavefunction to collapse onto a continuous superposition of position states about the measured value. If you don't believe it because you're getting unphysical results from this example, if you don't believe it because delta functions such as ${\displaystyle \langle x|S\rangle =\delta (x-S)}$ aren't normalizable, then at least sit down and use some uncertainty estimates or the DeBroglie relation to see the energy requirements of making a measurement which would justify writing the state of the system as a single position ket (Spoiler: They're infinite). If that doesn't make you give this up, the error is explained in Cohen-Tannoudji Vol. 1 on p. 278, again on Griffiths p. 106 in a footnote, and again in Sakurai.Auspex1729 (talk) 18:42, 4 August 2009 (UTC)

Obviously the energy eigenbasis is orthonormal and discrete, but the position basis is not. I should have thought that it was obvious which eigenbasis I was referring to. I suggest you digest that and then perhaps calm down. And, no, the fact that each individual probability vanishes does not disturb me; since the position measurement was infinitely precise the associated disturbance has been infinite (as a consequence of the very uncertainty relations which you maintain have been violated by this example -- I don't think so....). I was being polite in my first response, but I will be blunter now; your statement: Wavefunctions can't collapse to single position eigenkets without violating the uncertainty relation. reveals that you really don't have a clue about the physics. --Michael C. Price ^talk 18:58, 4 August 2009 (UTC)

The formalism of the problem was based on the energy basis. The only thing you could obviously have been referring to was the set of energy states. ...reveals that you really don't have a clue about the physics. Based on the little gems you've peppered this thread with, I don't doubt for a second that you believe there's a ${\displaystyle \sigma _{p}}$ satisfying ${\displaystyle 0\cdot \sigma _{p}\geq {\frac {\hbar }{2}}}$. Maybe you should review a text on elementary quantum mechanics before you cram this article full of your egregiously wrong original research.Auspex1729 (talk) 19:40, 4 August 2009 (UTC)

${\displaystyle \sigma _{p}}$ can be infinite, and is in this case. But Auspex1729 has a bit of a point...if we're going to put in an example, it might as well be one that's at least somewhat physically plausible. (No energy expectation values being infinity, etc.) It's enough already to confront some poor reader with their first example of doing a QM measurement calculation...it's too much if, at the same time, we confront them with infinities and delta-functions and all the other stuff that we're used to but is hard to swallow the first time you see it. Can we come up with a better/simpler example? Maybe measuring the spin of a spin-1/2 particle along the x-axis then y-axis then x-axis? Or something else? Do we even really want or need an example? This is sorta moving into textbook territory out of wikipedia territory if we're not careful. :-) --Steve (talk) 20:02, 4 August 2009 (UTC)

Examples are not out of place on wikipedia, especially when we have a lot of abstract concepts floating around. The example is only illustrating the disturbance induced by the measurement of one parameter on another non-commutting parameter. The actual values of the probabilities involved aren't that important, so this is all a bit of a red herring. The example illustrates what it is intended to illustrate quite well.

The trouble with an example involving spin is that this isn't something a non-physics reader can really relate to. But of course, no reason why we can't have another example as well.

BTW there are neither delta functions nor infinities in the example (only on this talk page), so that is another red herring we don't have to worry about. Please don't delete this example, based on some superious irrelevant objections -- at least until we have some equally/more acessible examples to replace it with. --Michael C. Price ^talk 21:12, 4 August 2009 (UTC)

The only thing spurious here is your handwaving upthread. I have cited two standard textbooks on this, the footnote in Griffiths p. 106 and the example in Cohen-Tannoudji. Incidentally, I was wrong when I said the example couldn't be salvaged. Cohen-Tannoudji does write the system as prepared in a state represented by an integral over a small interval of position eigenkets (as I suggested it should be upthread), and obtains solutions and probabilities which, you know, actually work.Auspex1729 (talk) 05:05, 5 August 2009 (UTC)

Salvage it then, if you feel so strongly. However, as I said, the explanation currently does not mention infinities, so why are we bothered? Why complicate a simple example?--Michael C. Price ^talk 06:04, 5 August 2009 (UTC)

It seems that the edit by IP address 86.73.72.40 : reference ^[1] is not relevant to the encyclopedic nature of wikipedia. It may even be self promotion. The IP address 86.73.72.40 has gone about adding this reference and information rtelated to this reference at several Physics pages. Kanwarpreet Grewal 09:31, 2 April 2010 (UTC) —Preceding unsigned comment added by Kp grewal (talk • contribs)

In the section on observables it mentions the properties of observable operators. Amongst these it says that the operators are Hermitian, and they have real eigenvalues. But ALL Hermitian opertaors have real eigenvalues, so it might be better to put these in the same point. Currently it reads like having real eigenvalues is specific to operators corresponding to observables. —Preceding unsigned comment added by 82.6.96.22 (talk) 21:44, 24 May 2011 (UTC)

I'll be adding a sentence here and there, or inserting a few words, to hopefully improve comprehension. I'll start with the lead which is not very informative. I'm somewhat familiar with the discussion on this topic among experts, but wish to make things more accessible to non-experts.

regards, Jonathan JonathanD (talk) 03:57, 12 June 2011 (UTC)

I've reverted the changes, since the measurement problem is not largely the result of Quantum indeterminacy - the cause is a matter of debate that relates to whatever interpretation of quantum mechanics one holds. Jonathan, I would advise you not continue making large additions/changes to the lead of articles on this subject (or any subject for that matter). The lead should only mention material that is discussed in more detail within the body of the main article. -- cheers, Michael C. Price ^talk 06:19, 12 June 2011 (UTC)

AFAIK the best defintion of a measurement is the interaction of a quantum system with a classical one (as mentioned in the Copenhagen interpretation article). Of course this is unsatisfactory as the observer can be described by quantum mechanics too: the observer is nothing more or less than a (hugely complicated quantum system). Therefore a huge problem is why "measurement" causes wavefunction collapse, when the interaction of two quantum systems doesn't. Perhaps this could be addressed in the article? — Preceding unsigned comment added by 86.26.13.2 (talk) 00:21, 31 October 2012 (UTC)

Did you not see that there's a section Measurement in quantum mechanics#What physical interaction constitutes a measurement? Or are you saying that you find that section to be inadequate to answer those questions? --Steve (talk) 13:25, 31 October 2012 (UTC)

This and the following sections need major overhaul. The author(s)did not have sufficient understanding to engage in this. (1) The double-slit experiment is primarily about wave-like interference, and sometimes a major player in discussions involving wave vs. particle like behavior (especially of photons). The issue of collapse applies to all measurements. (2) The language in description of von Neumann's measurement scheme needs a bit more clarity. ".. the operator that needs to be measured" needs better language {that has been fixed.} (3) In von Neumann's analysis of measurement, you show 2 different equations (A -> B) where a pure state becomes an entangled state. There is no clue how that happens, either physically or mathematically. (4) There is a system upon which a measurement operator represents a measurement, then there is the quantum representation of a classical device that produces a classical measurement such as a pointer reading—and that needs a different name. Etc. Most likely some symbols (e.g., M1 & M2 or CMD [classical measurement device], whatever) will need to be used in order to keep things clear. Whoever is doing this, please get someone who is knowledgeable to proof-read what you are writing. von Neumann's analysis of measurement is a serious challenge; it would be really worthwhile to have a real expert contribute to this!

And, from later on:

[This is confusing |psi> is the state prior to interaction with the original measuring device and with von Neumann's apparatus. The next equation shows the transition from the original unmeasured state to an entangled state involving both pieces of apparatus. It is not clear how that happens or how one can, by a unitary transformation, go from an original pure state to an entangled state. Without that, this is all smoke & mirrors.]

I'm not sure exactly what this last comment means in this context, given that the initial state ${\displaystyle |\psi \rangle |\phi \rangle }$ is clearly an entangled state (belonging to the direct product of the Hilbert spaces of the system and of the apparatus). It doesn't seem to me that the unitary operator needs to be explicitly constructed for this argument. --Anagogist (talk) 15:20, 3 June 2013 (UTC)

Since the article uses the phrase "pure state", I suggest it be defined. The article says " A pure state is represented by a state vector in the Hilbert space." Is that a definition? Is each vector in the Hilbert space considered to be a "pure state". Tashiro (talk) 22:06, 2 February 2015 (UTC)

Pure states are defined and explained in the Quantum state article. I added a link to the notion "pure state" as well, to clarify. Xaggi (talk) 10:44, 3 February 2015 (UTC)

The article says there is a connection between measurement as a mathematical operator and ordinary measurements in a laboratory. Ordinary measurements in a laboratory are connected in some way to physical theories that predict them. The predictive aspect clear from article is that if we happen to observe outcome X in the laboratory then we can model this as the action of an operator and use the eigenvectors associated with the eigenvalue X to predict the state of the system after the measurement. (Do the eigenvalues all have the same physical dimensions? , e.g. kg m / sec or some other physical unit, It would be useful to state explicitly how physical units enter the picture.)

The method of predicting the probability of outcome X is (I think) only gradually explained in the body of the article. It involves the representing the system being measured as a sum of eigenvectors of the measurement operator. That representation can be used to compute the probability of a given outcome. That should be explained to some degree in the qualitative introduction. At least it should be said that a known state of the physical system and a given measurement operator let us compute the probability distribution for the experimental outcomes. Since the mathematical explanation assumes the state of the system before the measurement is a "given", the qualitative introduction should explain how this assumption relates to setting up a laboratory experiment. (I think the standard jargon is making a "preparation".)Tashiro (talk) 22:06, 2 February 2015 (UTC)

I do not think this reference adds any particular value to the article. Compared to the systematic review article cited just prior, it is far less detailed, being superficial without being more clear. Self-published expositions of decoherence are a dime a dozen. We should not be elevating works on technical and/or philosophical topics that have yet to go through the wringer of peer review. StarryGrandma recently removed it from Quantum decoherence for much the same reasons. (Both the additions were made from Michigan IP addresses, use almost identical phrasing in the edit summaries, and misuse the {{cite journal}} template in the same way, so we have to careful about citation spamming here.) XOR'easter (talk) 01:27, 7 January 2023 (UTC)

I have to say that I find this rationale for putting it back to be unsatisfying. "Well-known" is not substantiated — it's one uncited preprint among the dozens posted on quant-ph every day — and would not suffice to establish reliability even if it were true. XOR'easter (talk) 14:26, 7 January 2023 (UTC)

David spector, I am proposing removing the addition of Stephen Hsu's paper from the article. It was not used a reference for writing the material, just added later to an already sourced sentence. We frequently get editors trying to add recent preprints to Wikipedia articles without adding content. It is not proper to try to publicize unpublished articles this way, or most newly published papers for that matter. Can you explain how a paper uploaded to arXiv on Dec 5, 2022 is well known? And I am sure many university professors have lecture notes that explain this to their modern physics classes. In the early days of requiring articles to have references (there were many problems with science articles before sources in Wikipedia were required), links to professor's lecture notes were often used. Now we require reliably published sources. StarryGrandma (talk) 18:48, 7 January 2023 (UTC)

Dear Starry, it was "well known" to me, since I had already read it recently, and was impressed by its clarity (WP should be understandable by any intelligent adult, not just specialists in their field). But if it was added to the article by the authors, then this is in violation of at least two WP policies: no conflict of interest and no posting of own work, and possibly of Reliable Source. I'd like the reference to remain in the article because of its clarity, but I will stand aside if there is significant doubt about whether the circumstances of its addition met WP requirements. Thank you for presenting the issue here. David Spector (talk) 20:20, 7 January 2023 (UTC)

I'm torn. On the one hand, this is clearly a case of citation spamming, but on the other hand the preprint is a nice and pedagogical exposition. In fact this is the first time I have ever seen self-promotion that wasn't of crackpot gibberish. But at the end of the day I think Wikipedia would benefit most from a consistent ban on self-promoting unpublished pre-prints. Tercer (talk) 10:24, 8 January 2023 (UTC)

I agree on both points. David Spector (talk) 12:26, 8 January 2023 (UTC)

If we're lucky, it'll be formally published in some place like Am. J. Phys. (who have done articles on the topic before), and this whole discussion will be largely moot. XOR'easter (talk) 23:24, 9 January 2023 (UTC)