Widdekind Posted August 12, 2010 Posted August 12, 2010 Quantum Entanglement is "correlation" of particles' Wave Functions Correlated pairs of particles... These particles interact, and then separate; thereafter, measurements made on one particle can be used, via the correlations, to generate predictions about the other. These predictions have probability one... It is possible to prepare pairs of particles, such as an electron-positron pair, whose total spin in any direction is zero. If the pair then separates, theory suggests that if for instance an Sz experiment is carried out on each system, then the results will always be opposite in sign: if the result of measuring Sz on the electron is +, then on the positron it will be -, and vice versa. The same holds true for all directions in space (that is for Sz, Sy, and so on), provided that both experiments measure the same component of spin. R.I.G.Hughes. The Structure & Interpretation of Quantum Mechanics, pg. 159-160. In their 1935 Physical Review article, Einstein Podolsky, & Rosen considered a system consisting of two particles, that are permitted to interact with each other, and then separate. During this interaction, the particles share momentum, angular motion, and position. As they separate, they share their total momentum, their angular motion, and a common point of origin... The Schrodinger [Wave] Equation can be used , to describe the possible states of these two objects, as a result of their original interaction.... It describes reality as a collection of potential realities, a collection of pictures of the world, that includes all the things we expect could be found in a measurement. If we call the two particles mentioned above A & B, then quantum mechanics will say there is a series of pictures -- mathematical functions -- that describe the possible conditions of A and of B. Let us say, that these pictures for A are named PA(#1), PA(#2), and so forth. For B they are similarly PB(#1), PB(#2), ... Now, the Schrodinger Equation can combine all of these pictures together, into a single description of the system, that is represented by [math]\Psi[/math]. The quantity looks like this: [math]\Psi[/math] = (first picture of A) multiplied by (first picture of B) + (second picture of A) multiplied by (second picture of B) + ... That is, [math]\Psi[/math] = PA (#1) x PB (#1) + PA (#2) x PB (#2) + PA (#3) x PB (#3) + ... This description says, that if particle A is given by picture #1, then particle B is given by its picture #1. If particle A is given by picture #2, then particle B is given by its picture [#2] that shows how it moves, and so on. This is because the two particles share all of their dynamical properties during their interaction. This means that one we measure the position of particle A, we can use this equation to find out the position of particle B exactly, without ever disturbing particle B. And, if we decide to measure exactly the momentum of particle A, then we will know the momentum of particle B exactly, without ever having disturbed particle B. E.H.Walker. The Physics of Consciousness, pp. 91-92. [Einstein's] 1935 paper, which became known as the EPR paper... describes a thought experiment, in which two quantum systems interact. We can imagine one particle, with a momentum of, say, ten arbitrary units, decaying into two particles, p1 & p2. The two particles then speed off, in opposite directions. Since the system's total momentum was known before the [decay], then, from the law of Conservation of Momentum, the total momentum of the pair is also known after the [decay], for it must equal the same ten units. What is not known, is how that momentum is shared between the two particles... The two particles are said to entangled, in what has become known, as an EPR pair. The particles do not have an independent reality, but are described by a single Wave Function, containing within it, all of the possible states of both particles (for example: [math]\Psi[/math] = { p1 w/ momentum of 1 unit & p2 w/ momentum of 9 units } + { p1 w/ momentum of 2 units & p2 w/ momentum of 8 units } + { p1 w/ momentum of 3 units & p2 w/ momentum of 8 units } + ... Since the total momentum of the entangled pair is known, it is possible to measure the momentum of one, and thereby predict 'with certainty, w/o disturbing the system', the momentum of the other particle... Consider again our example of two EPR-entangled particles, with a total momentum of ten units. Suppose a scientist (Alice) performs a measurement on the first, and finds it has a momentum of seven units. Classical conservation laws are now all she needs, to predict that the second particle must be left with a momentum of three units. She could ask her colleague (Bob) to perform the measurement of the second particle (which might be one mile away by now), and perhaps inform her of his result -- by phone. A dutiful scientist, Alice would not rely on a single observation but would perform a series of measurements on many EPR-entangled particle pairs. In each case, she would measure the momentum of the first particle, simply subtracting its momentum from ten, to predict the result of Bob's measurements... This result would not distinguish between the classical, and the quantum, version of events. Both theories make the same prediction, about the expected correlations, between Bob & Alice's measurements: they should always add up to ten. The theories differ only in their model for the state of both particles prior to measurement. Classical theory predicts that the momentum of the particles was fixed at the moment the original single particle decayed into two particles. Quantum theory insists, that the [definite] property of momentum for either entangled particle, did not exist, until one was measured. J.McFadden. Quantum Evolution, pp. 197-199. "Entanglement" as "cards in shared card-sleeves" ??? To protect & preserve expensive cards, some people buy "card sleeves" -- laminated plastic pockets, into which cards can be inserted. Now, please ponder two card players, each with a "hand" of (five) cards. Player One (P1) has Ace (1),2,3,4,5; Player Two (P2) has 6,7,8,9,10. Let these cards represent (Linear) Momentum Eigenstates, with the face value of the card representing the amount of (Linear) Momentum in some spatial direction. Each "hand" of (five) cards, then, represents the state of one of a pair of particles (p1 for P1, p2 for P2), whose Wave Functions are super-positions of (five) Momentum Eigenstates. Note that such "spreads" of super-positions of momentums correspond to particles (partially) Localized in space. Now, when these particles meet, and interact -- presumably as p2, with more momentum, overtakes, and passes through, p1 -- they "share their total momentum", and become "correlated". Then, as their Wave Functions separate, their momentums are (apparently) "correlated" & "linked" -- presumably in such a way, as to conserve total momentum (???). Thus, we might imagine, that the "hands of cards", representing the Wave Functions of p1 & p2, become "correlated", "cross-connected", or "linked", along the lines of: p1 : p2 1 : 10 2 : 9 3 : 8 4 : 7 5 : 6 To represent this "inter-leaving linkage" between the Wave Functions of the particles, imagine putting each of those five pairs of cards, back-to-back, into five card-sleeves. Then, if one player "plays" one (or more) of his cards -- representing a "collapse of the Wave Function" upon measurement, to some single (or set of) state(s) -- his partner is "compelled" to play the correspondingly correlated cards, stuck in the same shared card-sleeves. So, imagine that a momentum measurement is performed upon p1, and P1 decides to "play" his Ace. Since that Ace is "stuck in the same shared card-sleeve", as p2's 10, P2 is essentially "compelled" to "play" her 10. Likewise, if P1 decided to "play" his 2 (representing the "collapse of p1's Wave Function" into the momentum eigenstate having "2 units" of momentum), then P2 would basically be "compelled" to "play" her 9 (representing the "collapse of p2's Wave Function" into the momentum eigenstate having "9 units" of momentum). And so on, for other decisions by P1; and, similarly, were p2 to be "measured" first, so that P2 was forced, first, to "play" a card, from the above "cross-linked hand" ("correlated" Wave Function for both particles). Now, imagine, instead, that a position measurement is performed upon p1. Such would cause the "collapse" of p1's Wave Function, to a particular point somewhere in space, which could be represented as P1 playing his "whole hand" (a distributed "spread" of momentum eigenstates amount to a Localized position eigenstate). Then, by playing his "whole hand" of cards, P1 would effectively "compel" P2 to "play" her "whole hand" of cards, stuck in the same "correlated" card-sleeves. Thus, a position measurement performed upon one of a pair of "correlated" or "entangled" particles, which causes its Wave Function to "collapse" to a particular point, instantaneously causes the "collapse", of the Wave Function of its "entangled twin", to some other particular point. Since Schrodinger's Wave Equation is deterministic, it is possible to precisely predict the particular place, in space, of the "induced" or "sympathetic" Wave Function Localization, of the other particle, from the spatial location, of the Localization, of the "entangled twin" particle. Time-Evolutions of separated particles -- "draws & discards" ??? The Schrodinger Wave Equation evolves Wave Functions, forward in time. This can be represented, mathematically, by the Time Evolution Operator T(t) (Walker, ibid., pg. 347): [math]\Psi (t) = \hat{T} (t) \Psi (0)[/math] In particular, this also applies, to the component eigenstate Wave Functions, whose superposition comprises the particle's Wave Function, so that: [math]\Psi (t) = \hat{T} (t) \Psi (0) = \hat{T} (t) \left( a \psi_1 (0) + b \psi_2 (0) + c \psi_3 (0) + \dots \right)[/math] [math] = a (\hat{T} (t) \psi_1 (0)) + b (\hat{T} (t) \psi_2 (0)) + c (\hat{T} (t) \psi_3 (0)) [/math] [math] = a \psi_1 (t) + b \psi_2 (t) + c \psi_3 (t) + \dots[/math] This can be modeled, by imagining, that our two "hands of cards" (representing the super-position state Wave Functions of our two particles) "evolve" over several rounds of game-play, as each player draws new cards, and replaces & discards old cards: p1(t) : p2(t) Jack 1 : 10 ...... 2 : 9 Queen King 3 : 8 . 8 6 4 : 7 ...... 5 : 6 7 8 9 Thus, although both particles' Wave Functions evolve, according to the Schrodinger Wave Equation (independently, and from locally applied forces & potential fields), after their interaction & separation, their Wave Functions remain "correlated" & "entangled" (here represented by their "joint hand of cards", linked together, in their shared card-sleeves), until a "measurement" causes the collapse of one of the particles' Wave Function -- which collapse instantaneously triggers a "sympathetic" collapse, in its entangled twin. Such collapse "randomizes" the phases of those Wave Functions, thereby destroying their "entanglement" (N.Herbert. Quantum Reality, pp. 155,168,194). In our "card game analogy", whatever card(s) were played, for both players, upon the "correlated collapses", of the particles' "linked" Wave Functions, become their "new hands" (new, post-measurement, Wave Function states); these states are fully independent, so the card(s) are removed from the shared card-sleeves, and returned to their original players ("link broken"); and, all other cards are discarded. Is this an accurate analogy, for quantum entanglement ???
swansont Posted August 12, 2010 Posted August 12, 2010 Locally applied potentials and fields will tend to destroy the entanglement. The collapse of the wave function is for the system. Characterizing it as a sympathetic collapse is one of the things that gets you in trouble with relativity.
Widdekind Posted August 12, 2010 Author Posted August 12, 2010 I think you are objecting, to notions, of super-luminal influences, in QM: On the other hand... the phenomenon of the reduction of the quantum state [Wave Function Collapse], when a measurement is made... must be seen as implying a violation of [Einsteinian] causality, if we conceive of... the quantum state to be physically real, and to be 'reduced' (in general) by an 'ordinary' measurement... The need to consider a reduction of the quantum state, when a measurement is carried out, has always been seen as a rather unsatisfactory aspect of the conception discussed here, according to which Wave Functions of quantum systems are deemed to be elements of reality... Bernard D'Espagnat, J. C. Whitehouse. Reality and the Physicist, pg. 166. Schrodinger speculated, that an objects waviness [Wave Function] was the smeared out object itself. Where, for example, the electron fog is densest, the material of the electron is most concentrated. The electron itself would thus be smeared over the extent of its waviness... Though an object's waviness may be spread over an extremely wide region, when one looks at a particular spot, one immediately finds either a whole object there, or no object in that spot. For example, an alpha particle emitted from a nucleus might have waviness extending over kilometers [3 km / c ~ .01 ms ???]. But, as soon as a Geiger counter clicks, one can find a whole alpha right there inside the counter. Or, consider the waviness of a single electron having just passed through both slits in an interference experiment. It will be in several clumps, separated perhaps by inches, with each clump headed toward an allowed region on the screen But, an instant later, a flash is seen at a single spot on the scintillation screen, and the whole electron can be found there. The electron's previously extended waviness is suddenly concentrated at that single spot, at which the electron can be observed. If, on the other hand, the electron were observed while in transit to the screen, it would be found wholly at some single spot, in one of the several clumps of waviness. If an actual physical object were smeared over the extent of its waviness, as Schrodinger initially thought, its remote parts would have to instantaneously coalesce to the place where the whole object was found. Physical matter would have to move a speeds greater than that of light. Rosenblum & Kuttner. Quantum Enigma, pp. 74-75. What happens, if I quote the famous physicist John Bell ? According to the late physicist John Bell, Quantum Non-Locality & Instantaneity (FTL) is not incompatible with Relativity Theory -- only Einstein's particular "brand" of Relativity: the resolution is something like going back to Relativity, before Einstein, when people like Lorentz & Poincare thought that there was an aether -- a preferred reference frame -- but that our measuring instruments were distorted, by motion [through the aether], in such a way, that we could not detect [our] motion through the aether. Now, in that way, you can imagine, that there is a preferred Frame of Reference, and, in this preferred Frame of Reference, things do go faster-than-light. But, then in other Frames of Reference, when they seem to go not only faster-than-light, but backwards in time, that is an optical illusion... What is not sufficiently emphasized in textbooks, in my opinion, is that the pre-Einstein position, of Lorentz & Poincare, Larmor & Fitzgerald, was perfectly coherent, and is not inconsistent with Relativity Theory. The idea, that there is an aether, and these Fitzgerald contractions & Larmor dilations occur, and that as a result, the instruments do not detect motion through the aether -- that is a perfectly coherent point of view... There are lots of problems, which are solved more easily, by imagining the existence of an aether... In these EPR experiments, there is the suggestion, that behind the scenes, something is going faster-than-light. Now, if all Lorentz frames are equivalent [Einstein's aether-less Relativity Theory], that also means that things can [truly] go backward in time. John Bell, interviewed in: Davies & Brown. The Ghost in the Atom, pp. 48-49. Indeed, the CMB apparently already defines a preferred Frame of Reference: Relativity holds differentially. It does not necessarily hold in the integral form. Thus, whereas in flat Minkowski space, motion is entirely relative, in our Big Bang universe, described by the Robertson-Walker-Friedmann (RWF) metric, motion is absolute ! One can measure the motion of the Earth relative to the Big Bang background microwave radiation [CMB], but this motion cannot be arbitrarily transformed away -- as would have to the the case for the motion to be relative -- without imposing on the overall space a rotation that would be easily discernible as a distortion of the RWF metric. In addition, one cannot arbitrarily transform the time coordinate prior to the Big Bang event. Differentially, space, time, and translational motion are relative; in the whole-world integral form, they are not. E.H.Walker. The Physics of Consciousness, pg. 354.
Widdekind Posted August 24, 2010 Author Posted August 24, 2010 (edited) Whenever a wave function undergoes quantum splitting, such as the reflected & transmitted waves refracting from a potential barrier, "The two parts are, in practice, joined, b/c the wave function is never quite zero, just very small between them" [E.Squires. The Mystery of the Quantum World, pg. 29]. And, in quantum entanglements, "Something seems to be linking the two quons [quantum objects] instantly (faster-than-light)" [N.Herbert. Quantum Reality, pg. 170]. Indeed, once two particles have had any interaction, they do somehow remain linked as parts of the same indivisible system. Separated particles seem to be as connected as two ends of the same rod... the powerful rod-like connection has to be there, because Bell's sums say it is [J.Klaff. Bluffer's Guide to the Quantum Universe, pg. 33]. Now: The essence of a local interaction is direct contact... Body A affects Body B locally when it either touches B, or touches something that touches B... On the other hand, the essence of non-locality is unmediated action-at-a-distance. A non-local interaction jumps from body A to body B without touching anything in between... A non-local interaction is, in short, unmediated [Herbert, ibid., pp. 211-213]. So, when two quantum objects interact, inter-mingle, & entangle, then, even when they separate, their wave functions will remain "physically" linked, by a drawn-out & tenuous "tail" stretching between the two main "lumps" of probability. Could this "physical tendril", of mingled & entangled probability, mediate the instantaneous correlations, observed in EPR experiments ? If so, current interpretations of EPR correlations, as both "non-local" and "instantaneous", could be reduced, in "quantum weirdness", to "local" but "instantaneous", mediated through the "tendril" of entangled wave function, spanning from one "main lump" of particle probability, to that of its entangled twin. Possible rationale for quantum instantaneity Mainstream physicists are rather reticent to accept the "reality" of the wave functions of quantum objects: The fact that observing a quon "here" instantly changes the wave function "there" (where "there" may be billions of miles away) is another good argument for the fictitious nature of the proxy wave. If the wave function were real, it would have to change its shape drastically, over large distances, at faster-than-light speeds [Herbert, ibid., pp. 170-1]. However, such instantaneous "internal communication", through the wave function, could account for the elementary nature of fundamental quantum objects: There are serious logical problems with the quantum theory, when it applied to the electron, or other point-like particles. One of the important terms, in the mathematics of QED, is the "self-energy" of a charged particle, such as an electron, which has an electrical potential energy assumed to be given by V = e2/r. The self energy of a charged particle depends on the radius r according to 1/r. Thus, if the particle size is shrunk down to a point r --> 0, the self energy goes to infinity. Besides being impossible, the equation becomes useless. This is a problem. To avoid the infinity dilemma, one is tempted to abandon the idea of a point particle. But relativity will not allow this, as seen from the following argument. If a particle is elementary, it must react as a unit. However, if it has a finite size, and an electro-magnetic signal should arrive at one side, the other side must simultaneously know of the arrival of the signal, in order to react as a unit. But this implies that the signal travels with infinite speed, which is prohibited by relativity. The only way out, is to have a point particle. (Or, no particle at all, if you could find a way to represent mass & charge w/o it) [M.Wolff. Exploring the Physics of the Unknown Universe, pg. 132]. Such a "realistic" interpretation of the wave function was initially favored by Schrodinger himself: Schrodinger [initially] speculated, that an object's waviness was the smeared out object itself. Where, for example, the electron fog is densest, the material of the electron is most concentrated. The electron itself would, thus, be smeared over the extent of its waviness... If an actual physical object were smeared over the extent of its waviness, as Schrodinger initially thought, its remote parts would have to instantaneously coalesce to the place where the whole object was found [upon Measurement & Localization]. Physical matter would have to move at speeds greater than that of light [Rosenblum & Kuttner. Quantum Enigma, pp. 74-75]. Thus, quantum instantaneity could account, for the elementary nature, of spatially extended (De-Localized) quantum objects, in SR -- sidestepping issues of infinities in QED. This suggestion, of "quantum local instantaneity", avoids un-mediated non-locality, and (renormalizeable) infinities, in quantum physics, admitting only the "quantum weirdness" of instantaneous, FTL, influences, which are kept completely confined within the wave functions of singular quantum systems*. Such could be called "damage control for quantum weirdness". * This seemingly suggests, in turn, that, upon "collapse", wave functions "reel in", "slurping" the rest of the wave function in, from across the cosmos. This "drawing in", or "in-folding", could be compared to origami -- or, the Transformers, like Optimus Prime, which when "un-folded" was a large battle-station, but which could "fold up" into the shape of a smaller truck trailer. One could also consider swimming squid, in the seas (of Earth), which "flare open" to feed, but which rapidly "draw in" to squirt away. Edited August 26, 2010 by Widdekind
Widdekind Posted September 1, 2010 Author Posted September 1, 2010 (edited) Typically, interacting Wave Functions do not "collapse", but become "entangled", entering a single state for all of the entangled particles. "Measurement" destroys this singular entangled state, causes all of the entangled Wave Functions to collapse, and, thereby, puts all of the particles into individually separate states ("product states"): Virtually every time two particles interact and aren't measured, they become entangled... When two particles interact, there is no Wave Function Collapse. But, when a particle interacts with a 'measuring device', the measurement has a definite outcome: the particle's Wave Function "collapses" into a pure state, of whatever the device measures. Marc Lange. An Introduction to the Philosophy of Physics, pp. 259,297. Would wave function collapse coincide with "free" ("externally oriented") vertices, in Feynman Diagrams, of the interactions, associated with (electron) Localization ?? Such "free" vertices would beget full-fledged photons ("micro-signal (s)" generation), which would leave the locale of the original interaction, and thereby (potentially) inform the rest of the universe, of the "measurement" outcome of that interaction (representing the "registration" of the phenomena, as demanded by Bohr & Wheeler). Edited September 1, 2010 by Widdekind
Widdekind Posted September 10, 2010 Author Posted September 10, 2010 (edited) Whenever Wave Functions collide with potential barriers, they bifurcate, into Reflected & Transmitted waves. Now, electrically charged particles create potential barriers. Thus, the contact collision, of (say) two electrons' wave packets, would amount to a "double barrier" Reflection & Transmission event, which could account for the ensuing Entanglement. Is this what happens*? * Entanglement, of two different types of particle (e.g. electron + photon) would result in scattered wave packets which would overlap, accounting for the phase entanglement, but which would propagate with different (group) velocities. Edited September 10, 2010 by Widdekind
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