Talk:Measurement in quantum mechanics

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[edit] Discussion on paragraph

[edit] 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:

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)

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

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:

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

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)

[edit] 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), $\Delta x \cdot \Delta p \ge \hbar$ . If we can measure the exact value of momentum p than

Δ 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.

[edit] 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)

[edit] The paragraph itself

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.

[edit] The Quantum entanglement problem

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.

[edit] 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.

[edit] 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.

[edit] Example with only few assumptions (?)

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)

[edit] Is this a silly comment ?

Isn't ' ${\hat x}$ , where ${\hat x} = {-\hbar \over i}{\partial \over \partial p}$ ' in the definition of the distance operator equivilant to ${\hat x}$ , where ${\hat x} = {i \hbar }{\partial \over \partial p}$ - 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 ( $i \hbar$ ) 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)

[edit] Attention message

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)

[edit] Wavefunction collapse

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.

[edit] Case for merging?

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)

[edit] Wavefunction collapse, von Neumann scheme, and philosophical implications

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)

[edit] Afshar experiment

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)

[edit] changes

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)

[edit] Merge with "Measurement problem"?

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)

[edit] "mathematical" in section title

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)

[edit] subsection need rewrite

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)

[edit] Resolution

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)

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Talk:Measurement in quantum mechanics

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Contents

[edit] Discussion on paragraph

[edit] The Quantum entanglement problem

[edit] The uncertainty principle

[edit] What is a measurement? When the wave function collapse?

[edit] The paragraph itself

[edit] The Quantum entanglement problem

[edit] The uncertainty principle

[edit] What is a measurement? When the wave function collapse?

[edit] Example with only few assumptions (?)

[edit] Is this a silly comment ?

[edit] Attention message

[edit] Wavefunction collapse

[edit] Case for merging?

[edit] Wavefunction collapse, von Neumann scheme, and philosophical implications

[edit] Afshar experiment

[edit] changes

[edit] Merge with "Measurement problem"?

[edit] "mathematical" in section title

[edit] subsection need rewrite

[edit] Resolution

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