Widdekind

In my understanding, the only difference between a "photon" and an "anti-photon", is a pi phase shift (so that they add in deconstructive interference). Photons are their own anti-particles. I would guess, that "anti-combustion" would look allot like regular combustion.

In Neutron Interferometers, using thermal neutrons,

Physicists Jim Al-Khalili (Quantum, pp. ~150) discusses David Bohm's "Pilot Guide Wave" (my words) HV interpretation of QM, as well as Richard Feynman's Sum-over-Histories approach. He produces colorful pictures for both, highlighting the the "spider-man-web-from-wrists spread" of various paths that the point particles actually "consider" taking (as it were). I will try to scan these images as soon as I can. The similarity of the pictures prompts me to ask, if the Bohmian HV approach might be particularly well-suited, to Feynman's SOH approach ?? Also, if Feynman's SOH approach considers "every possible path", between two points (spacetime events) A to B, "weighting" them by their path-integrated actions, could some of those "possible paths" trespass outside of the future lightcone of A ? To wit, could some paths possibly "warp", at "Mach 1000", from A, out to the Coma Cluster 100 Mpc away, and then "warp back" to Earth, at B ?? How could you calculate the action, for such superluminal paths ??

(the color is for emphasis) So, such is the standard means, of making anti-matter ?

Please ponder the Bonding & Anti-Bonding Molecular Orbitals of molecular Hydrogen, arising from the spatial overlap, of the individual atomic Hydrogen 1s orbitals: So, certainly simplistically, please ponder the following (thought) experiment: ionize isolated neutral molecular Hydrogen (4.5 eV) to create singly-ionized H2+ in B-MO bond length increases from [math]0.74 \rightarrow 1.06 \AA[/math] since Dissociation Energy of H2+ is 2.6 eV, excite the lone electron with ~5.2 eV photon, to boost it into the AB-MO Hydrogen nuclei now repel apart, each carrying half of the electron AB-MO wave function attract dissociating "H+1/2" fragments towards a charged cathode detector screen when "H+1/2" wave functions collide with the detector screen, quantum measurement will destroy the lone electron's atomic orbital super-position state, collapsing its wave function onto one, or the other, of the Hydrogen nuclei both the bare proton (H+) and neutral atom (H) will have reached the screen, in a time, corresponding to the accelerations, of particles, having half of a positive charge unit Wouldn't this demonstrate the "reality", of the "smeared out" electron wave function, assuming the delicate super-position state could be maintained until measurement ?

I presume, that the point of "inversion" in space, would correspond to the "zeros" of the Energy Operator (as modified for the presence of electro-static potentials), [math]\hat{E} \rightarrow i \hbar \partial_t - e V[/math], as per the SWE & KGWE. (This parallels the modification of the Momentum Operator, for the presence of electro-dynamic potentials, [math]\hat{p} \rightarrow - i \hbar \vec{\partial}_x - e \vec{A}[/math].) But, unless such a simple process, of 'slamming' an electron against a high-voltage cathode, involved the Weak Force, then otherwise I would guess, that (as per Bob_for_Short) the KGWE would evolve the incident high-energy electron wave, into a low-energy "stalled out" electron wave, plus an electron-positron pair (presumably appearing near the point of such "stalling out"). If so, the positron would be attracted to the cathode, where it could -- w/ suitably sophisticated "something something" -- be captured and "bottled up" in a "fuel tank" (as it were). The two electrons would "bounce back", towards the sending source. Is this a physically plausible sort of scenario ?

If I might make a "war-game" analogy, to (say) Squad Leader or some such, a (human) experimenter setting up some apparatus, is like "choosing the terrain" map-boards on which to play. And, those particular "map-boards" will, obviously, affect how the "battle" of the experiment unfolds. Using different "map-boards" will generate different dynamics. But, in this analogy, the "playing pieces" of the (actual) particles "move themselves" across the chosen terrain, according to the dynamical equations of motion (Schrodinger / Klein-Gordon Wave Equations; Type I Process) and measurement (Wave Function Collapse; Type II Process), which can (sometimes, seemingly) show non-local instantaneity.

Is Lepton number (always) a "good" quantum number ?

The Klein-Gordon wave equation is the logical, Relativistic, extrapolation of the Schrodinger wave equation, from applying the standard Quantum Mechanical operator ansatz [math]( \hat{E} \rightarrow i \hbar \partial_t, \; \hat{p} \rightarrow - i \hbar \partial_x)[/math], to Einstein's mass-energy relation, [math]E^2 = (m c^2)^2 + (c p)^2[/math]. The KGWE, like the Dirac Equation, predicts the presence of antimatter: The antimatter prediction may not have been realized at first (before Dirac): Now, a previous version of the Wikipedia article said, that probabilities, in the KGWE, dip below zero, near where the wave function encounters relativistic potentials (e.g., millions of volts for electrons, billions of volts for baryons). Apparently, the "tunneling wave" (my words), which extends out into the classically forbidden region, "inverts" and turns negative, representing the particle "traveling backwards in time", to wit, interpreted as antimatter moving forward in time. QUESTION: If this is so, couldn't you "slam" charged matter particles, against relativistic electrostatic potentials (millions of volts, or more), and "invert them" into antimatter ??? (Something along the lines of an "matter isospin flip" ???)

When very high energy projectiles collide with other nuclei, they sometimes apparently knock pions out... Some rather difficult experiments, involving beams of electrons & photons... can only be understood, if it is assumed, that there are pions within nuclei. Ray Mackintosh, Jim Al-Khalili. Nucleus: A Trip into the Heart of Matter, 56.

Wouldn't that (basically) be the Planck Mass ? [math]\frac{2 G m}{c^2} = R_S = \lambda_C = \frac{\hbar}{m c}[/math] [math]m = \sqrt{\frac{\hbar c}{2 G}} \approx 10^{-8} kg[/math] Could, then, we attribute some "substantial reality", to the Wave Functions, of quantum objects -- treating, for example, [math]\rho_m = | \Psi |^2 = < \Psi^{*} | \Psi >[/math] as, effectively, a "microscopic mass density", of said quantum object* ? What about "Bohmian" Hidden Variables interpretations of QM, which treat quantum objects as classical point particles (albeit "guided", non-locally & instantaneously, by "pilot waves") ? Could one exclude such classical conceptions, of quantum objects, on the grounds that, even if their pilot waves "smear out" our macroscopic knowledge of those point particles, the particles themselves would still actually & really be "micro-Black-Holes", which would radiate away in under a Planck Time ?? * Along like lines, one could call [math]\rho_p = < \Psi^{*} | c \hat{ p } | \Psi >[/math] the (relativistic) momentum density; and, [math]\rho_E = < \Psi^{*} | \hat{ E } | \Psi >[/math] the (relativistic) energy density (?).

(thanks for clarifying that concept.) Including radiation, could we say, that gravity reduces the entropy of the matter (whilst "shunting" (increased) entropy "out" into the radiation field, of photons) ?

Visualization of Wave Function collapse in Double-Slit experiment Electrons: Atoms: The atom's wave function is localized in space, as it leaves the gun, but spreads out as it travels towards the slits... As the wave function reaches the slits, it begins to squirt through both of them simultaneously. On the other side, the two pieces of the wave function form a super-position which has a very different looking probability distribution (due to interference between its two pieces). By the time the wave function reaches the screen, the distribution is such that the atom has a high probability of reaching certain places, and no probability of reaching others. Although the atom will only manifest itself in one place, statistically, many atoms, each with a similar probability distribution, will build up to form the pattern seen. Jim Al-Khalili. Quantum, pp. 86-87. Overlap Integrals underlie "collapse" phenomena The probability amplitude, for a "quantum jump" transition, from one current & actual state [math]\Psi[/math], to another potential & possible state [math]\Phi[/math], is equal to the "matrix element" or "Overlap Integral", of those two states: [math]a = \; < \Phi | \Psi > \; = \int d^3 x \; \Phi^{*} ( \vec{x} ) \Psi ( \vec{x} )[/math] The probability, then, of the transition, is equal to the modulus squared, of the probability amplitude, p = |a|2. Now, the wave functions, of electrons in double-slit experiments, frequently span some "several inches" across, as the fly forward towards the detector screen (Rosenblum & Kuttner. Quantum Enigma, pg. 75). Consider such relatively large, macroscopically scaled, distended & delocalized wave functions, "spread out over inches". And, consider the comparatively coarse-grained resolutions, of typical detector screens, which utilize phosphor particles (Herbert. Quantum Reality, pg. 58), typically [math]5-10 \mu m[/math] across. Then, for all practical purposes, of macroscopic experimental measurement, we can approximate the (nano-scopic) bound-state electron orbital [math]\Phi_{ \vec{ R } }[/math], in a (micro-scopic) phosphor particle, at position [math]\vec{ R }[/math] on the detector screen, and into which the electron's wave function [math]\Psi_e[/math] could conceivably "collapse" and be absorbed, as a Dirac Delta Function [math]\delta^3 ( \vec{R} )[/math], placed at the appropriate (coarse-grained) particular position on said screen. The probability amplitude, for the "quantum jump" transition, then, is: [math]a ( \vec{R} ) = \; < \Phi_{ \vec{ R } } | \Psi_e > \; = \int d^3 x \; \Phi_{ \vec{ R } } ( \vec{ x } ) \Psi_e ( \vec{x} )[/math] [math] \approx \int d^3 x \; \delta^3 ( \vec{R} ) \Psi_e ( \vec{x} )[/math] [math] = \Psi_e ( \vec{R} ) [/math] And, the (full-fledged) probability of the transition is: [math]p( \vec{ R } ) = | a ( \vec{ R } ) |^2 = | \Psi_e ( \vec{R} ) |^2[/math] Thus, the probabilities, of wave function "collapses", are simply the standard "Matrix Elements" or, equivalently, "Overlap Integrals", governing the "quantum jumps" between states available to the quantum system in question (here, electrons). This, then, highlights a crucial concept -- "quantum jumps", like the Localization of wave functions upon "collapse", involve an interaction, between the current state of a quantum system, and another state, with which said system comes into "contact", defined by the physical overlap of the current & actual wave function, with that of the potential & possible future wave function. The more the current & actual wave function overlaps that of the second state (usually bound, in a deep attractive potential well), and the more similar the present state is to the possible future state, the larger the Matrix Element or Overlap Integral will be, and the larger is the likelihood of the "quantum jump" transition, from the former, to the latter. In my understanding, since such Localizations absorb a formerly free electron, into a now-bound orbital, the electron, losing energy (c.f. Recombination in ionic plasmas), must emit a photon. Since a photon is a quantum of electromagnetic energy, photons can only be created whole, in integer numbers. Thus, a large & delocalized electron wave function, must "collapse" into a single bound-state orbital, in order to emit a single & whole photon. If the electron were to "collapse", into some super-position of many bound-state orbitals, scattered across the surface of the detector screen, then it would have to emit many "partial photons". But that is impossible, so the electron must "choose" one particular place to appear, and from where it emits its photon. In "Interactionless Isolation", quantum objects' Wave Functions always evolve deterministically (Type I process) Isolated quantum micro-systems are always "un-collapsed" Matter Waves: It is, as yet, a mystery (to Mankind), as to what specifically prompts "collapses" (Type II process) to interrupt, or punctuate, the deterministic evolution of the Wave Function (Type I process). As an attempt to connect these two types of evolution processes, please ponder, that, at all instants of time, the Overlap Integral, of a Wave Function, with itself, is always one. This fact flows from the very probability "normalization", of the Wave Function, [math]< \Psi^{*} | \Psi > \; = \int d^3 x \; \Psi^{*} ( x ) \Psi ( x ) \equiv 1[/math]. Thus, the Matrix Element, describing the probability of the "self-collapse" transition, [math]\Psi \rightarrow \Psi[/math], is always one. Now, in "interactionless isolation", there are no other possible quantum states to occupy, and no Matrix Elements & Overlap Integrals to compute. Hence, there is no possibility of transitions, to any other quantum state, than that currently occupied by the quantum object. In essence, there is nothing to "compete" with the self-collapse process [math]\Psi \rightarrow \Psi[/math], of probability one. However, upon interaction, the Wave Function of a quantum object begins to contact, and overlap, available alternative states [math]\phi_i[/math] (e.g., localized, bound-state orbitals, in a detector screen). Suddenly, for the first time (since the quantum object's last Wave Function collapse event), there are one, or more, Matrix Elements & Overlap Integrals, of positive-definite value, [math]0 < p ( \Psi \rightarrow \phi_i ) = | < \phi^{*}_i | \Psi > |^2[/math]. This produces the curious effect, that the total transition probability, of the quantum object, from its current state ([math]\Psi[/math]), to any of its available states ([math]\Psi, \phi_i \; \forall i[/math]), including itself, now exceeds one [math]\left( p(\Psi \rightarrow \Psi) \equiv 1, \; p(\Psi \rightarrow \phi_i) > 0 \; \forall i \right)[/math]. Perhaps, then, it is specifically to avoid these "impossible probabilities", that causes the collapse of Wave Functions, to "prune off" possibilities, and keep every quantum object's total transition probability "sufficiently close to" one (?)*. * One might imagine, that, say, as an electron "splats" against a phosphor-particle coated detector screen (in roughly 10-14s from "tip to tale"), its Wave Function starts to "tunnel into" the awaiting, available, bound-state orbitals, in those phosphor particles. Then, the Matrix Elements, for transitions to those states, start to increase, from zero (when the electron was in "interactionless isolation", far from the screen) to positive-definite values (as the electron "splats" onto the screen). But, soon, the electron's reflected wave will start to "bounce back", off of said screen. At that point (after ~10-14s), the electron's Wave Function will "ebb away", reducing the Overlap Integrals & Matrix Elements with those bound-state orbitals, in the phosphor particles. Thus, the probabilities of transitions, into those bound-state orbitals, will begin to fall, declining back down to zero. And so, were one to plot these transition probabilities, as functions of time, over the dozen or so femtoseconds of the "splatting" interaction, one would see a hill-shaped function (p: 0 --> peak --> 0). What, then, would actually "trigger the dice roll", to determine if the electron's Wave Function localizes into, or dis-localizes away from, those bound-state orbitals (the Type II process) ? Perhaps it is that "probability-peaking" process -- as interactions begin, and transition probabilities begin to climb from zero, the Wave Function is "put on notice", as the electron's "portfolio of investments" begin to accrue & accumulate increasing probabilities. But then, this "stand-by mode" is interrupted, when one of those "investments" (Transition Matrix Elements) "maxes out". When the electron "senses" that some "investment" has "panned out", and peaked in its probability, then the electron "rolls the dice", and either Localizes into, or Dis-localizes away from ("prunes off" the relevant "tunneling" transition wave), that bound-state orbital. (?) Or, as a second supposed scenario, imagine an unpolarized photon, incident upon a polarizer crystal. As the photon encounters the polarizer, its Wave Function "splits" or "breaks", into two pieces -- a transmitting wave, and an absorbing wave. The absorbing wave is a little like the aforementioned "tunneling wavelets", of electrons, impinging upon phosphor grains, on detector screens. In this case, the photon's absorbing wave begins to interact with the valence electrons, of the atoms, in the polarizer. The electrons' Wave Functions are "ruffled", into super-positions, of their original ground-states, and new excited states, into which they would "quantum jump", were they "chosen" or "selected", to be the electron who absorbed the whole incoming photon, and its commensurate quantum of energy. Now, before the photon enters the polarizer, and encounters all those valence electrons, there is no probability, of any electronic excitation transitions. Then, as the photon's transmitting wave is transiting through the polarizer, the interaction, of the photon's absorbing wave, with all those valence electrons, causes all of their excitation transition probabilities to increase. Finally, as the photon's transmitting wave exits the polarizer, its "other half" (the absorbing wave) starts to ebb away, as well. Thus, the electrons' excitation transition probabilities begin to decrease. As before, the transition probabilities rise & fall (p: 0 --> max --> 0). So, again, the "natural cut-off trigger point", would appear to be, this "peaking" of the transition probabilities. It seems as if, when the valence electrons "sense" the perturbation, from the incoming photon's absorbing wave, they "let it ride", allowing the "investment" (of energy potential, from the photon's absorbing wave) to "grow"; then, when that "investment" starts to "decline in value", they "sell" (as it were), and "dice are rolled", to see, on a valence-electron-case-by-valence-electron-case, whether said valence electron "grabs" the photon (in which case, the photon's wave function collapses, and vanishes, depositing its energy into the now excited, and slightly de-localized, electron); or, whether it "misses" or "whiffs". If all the valence electrons "miss the catch", then the photon's absorbing wave has been completely dis-localized, and vanishes. Such leaves only the transmitting wave in existence, and the photon flies off, appropriately polarized. Note that this picture predicts, that photons forced through longer & longer polarizers, would encounter more & more valence electrons, each of which would try to "grope" the photon's absorbing wave. Thus, fewer & fewer photons should "survive the gauntlet". But, this only says, that sufficiently long polarizers would transmit no light whatsoever, absorbing all photons, into their crystal structures, as thermal heat radiation (passing a laser through a polarizer heats it up).

I think you are objecting, to notions, of super-luminal influences, in QM: 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: Indeed, the CMB apparently already defines a preferred Frame of Reference:

Why does the entropy increase, on the emission of (thermal, heat) radiation ? Is it because the entropy, of a "larger" system, having more "particles", is larger ? Is entropy a measure of the "volume of state space" available to the system -- the number of microstates? In crude numbers, for classical particles, that would be proportional to the spatial volume of the system (~R3) times the momentum-space "volume" available to the system (~<p>3). Now, for classical particles, the square of the characteristic momentum, being the (characteristic) KE of the system, scales with the inverse spatial size of the system: <p>2/2m ~ kBT ~ GM2/R <p>2 ~ R-1 If so, then w.h.t. R3 x <p>3 ~ R3/2, so that as the system shrinks, its entropy decreases. What am I missing ?

B.R.Martin's Nuclear & Particle Physics (pp. 14,17,95) lists other pionic nuclear reactions: [elastic scattering] [math]\pi^{-} + \; p \; \rightarrow \; \pi^{-} + \; p[/math] [single pion charge exchange] [math]\pi^{-} + \; p \; \rightarrow \; n [/math] [charge exchange] [math]\pi^{-} + \; p \; \rightarrow \; n \; + \; \pi^{0}[/math] [inelastic scattering] [math]\pi^{-} + \; p \; \rightarrow \; \pi^{-} + \; p \; + \; \pi^{-} + \; \pi ^{+}[/math] Now, in nucleons, when gluon bonds break, "tearing" into a newly-formed quark-antiquark pair, down antiquarks appear up to 50% more often, than up antiquarks: Since the proton dissociation reaction ([math]p \rightarrow n + \pi^{+}[/math]) relies on the creation of a comparatively common down-antiquark, whereas the neutron dissociation reaction ([math]n \rightarrow p + \pi^{-}[/math]) relies upon the creation of a relatively rare up-antiquark, this could be construed as consistent, with the observation, that the former rate appears to be about 20% more frequent than the latter (p spend 30% of their time in the dissociated state, n spend only 25%) [see PPs]. Since (anti-)down quarks are actually more massive than (anti-)up quarks, perhaps the preference for production of the former flows from the fact, of the latter's larger electric charge -- which might make for more intense electro-magnetic attractions, between bond-broken [math]u \bar{u}[/math], vs. [math]d \bar{d}[/math], and which might make the former somewhat stronger, and less likely to unlink.

The Schwarzschild Radius of an electron is ~10-57 m, whist that of ("bare", "undressed") u & d quarks is about 10x larger. Never-the-less, a pure point particle has zero radius, and, so, R = 0 < RS. Wouldn't, therefore, demanding that all particles be treated as classical pure point particles imply, that all matter is made up of mini-Black-Holes ? And, such mini-Black-Holes would evaporate by Hawking Radiation in ~10-98s. How could matter, made up of mini-Black-Holes, which evaporate in less than a Planck Time, make up (stable) matter ? (Doesn't the "smeared out" distribution of matter, in the Wave Functions of quantum theory, 'neatly circumvent such issues? To compress an electron, down to its Schwarzschild Radius, would require giving it a relativistic energy, via the Heisenberg Uncertainty Principle, of 300 gigatons mass equivalent.)

Quantum Entanglement is "correlation" of particles' Wave Functions "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 ???

Material objects are, quantum mechanically, "Matter Waves", which Localize upon "measurement" interactions Quantum Mechanics relies, at root, upon the Schrodinger Wave Equation [or, relativistically, the Klein-Gordon Equation] to evolve the Wave Functions of material quantum objects. To wit, Quantum Mechanics treats material quantum objects as "matter waves", which can be Localized, or Delocalized, depending upon environmental conditions, to which they are subjected. The act of "measurement" induces the Localization of matter waves, called the "collapse of the Wave Function". But, besides being "compressed", the matter wave remains a matter wave -- which it must, to remain compatible, with the governing Wave Equation of Quantum Mechanics. Classical, "particle" behavior, of material objects, arises from Localization When matter waves are spatially confined, in deep attractive potential wells (e.g., electrons in atoms), they acquire a large statistical spread of momentum. This flows from the identification of momentum with the spatial gradient of matter waves [math]( \hat{p} \rightarrow -i \hbar \vec{\nabla} )[/math], and agrees with the Heisenberg Uncertainty Principal. It is this combination, of Localized confinement, and ensuing momentum spread, which gives matter waves their "outward pushing" ("hard elbows") and apparent solidity. Such "solid" states of matter, are the only ones, with which humans have common contact or conception. Estimation of "solid" matter Bulk Modulus from compression of Quantum objects From Heisenberg's Uncertainty Principle (3D), w.h.t. [math]\Delta p_i \approx \hbar / x_i[/math] [math]\Delta p^2 = \sum \Delta p_i^2 \approx \frac{3 \hbar^2}{\Delta x^2} \; \; \; (\Delta x_i \equiv \Delta x \; \forall i)[/math] [math]KE \approx \frac{\Delta p^2}{2 m} \approx \frac{3 \hbar^2}{2 m \Delta x^2}[/math] Now, Work = Force x Distance. And, as pressure is applied to our quantum object, confined in its 3D "box", it compresses slightly. The Work done, by the applied pressure force (on all 6 faces of the "box"), increases the Kinetic Energy of the quantum object: [math]W = F \times d = ( P \times \Delta x^2 \times 6 ) \times \delta x = \delta KE[/math] [math]\delta KE = \frac{\partial}{\partial ( \Delta x )} \frac{3 \hbar^2}{2 m \Delta x^2} \delta x = \frac{3 \hbar^2}{m \Delta x^3} \delta x[/math] [math]\therefore P = \frac{\hbar^2}{2 m \Delta x^5} = \frac{\hbar^2}{2 m \, V^{5/3}}[/math] And, so, w.h.t.: [math] K \equiv -V \frac{\partial P}{\partial V} = \frac{5 \hbar^2}{6 m \Delta x^5}[/math] For [math]m = m_e[/math] and [math]\Delta x = 1 \AA[/math], w.h.t. [math]K \approx 10^{12} Pa[/math], which is well within an order-of-magnitude, of measured Bulk Moduli, for standard substances (K = 100s GPa). As pressure is inexorably increased, the electronic Wave Functions become increasingly compressed. Ultimately, the "squishy" & "springy" electron waves can, under inordinate pressure, be compacted down to subatomic, and even nuclear, size scales (~1 fm). When the Wave Functions of electrons are made to so significantly overlap the Wave Functions of nuclear protons, the former can (apparently) be absorbed by the latter, forming Neutronium (which would probably "pop" back out, to standard sizes, were pressure reduced, perhaps obviating the possibility of producing Neutronium armor plating).

D.Griffiths (Intro. Elem. Part. (2nd ed.), pg. 131) describes Deuteron production: [math]p \; + \; p \rightarrow d \; + \; \pi^{+}[/math] [math]p \; + \; n \rightarrow d \; + \; \pi^{0}[/math] [math]n \; + \; n \rightarrow d \; + \; \pi^{-}[/math] I.S.Hughes (Elem. Part. (3rd. ed.), pg. 51) describes Pion production, from fleeting nucleon dissociations: [math]p \rightleftharpoons n \; + \; \pi^{+}[/math] [math] n \rightleftharpoons p \; + \; pi^{-}[/math] Are these processes related ? [math]p \; + \; p \rightarrow p \; + \; (n + \pi{+}) \rightarrow d \; + \; \pi^{+}[/math] [math]n \; + \; n \rightarrow n \; + \; (p + \pi^{-}) \rightarrow d \; + \; \pi^{-}[/math] It is almost as if, one nucleon runs into the "sticky slime molasses", of the Meson Cloud, enveloping the other nucleon; and, that the Meson Cloud acts like an "aero-brake" on an incoming nucleon, slowing it & sticking it, into a fused state, with the original nucleon -- even as that Meson Cloud, absorbing the impact energy, is "kicked out", & "blown free & clear", of the new nucleus. Would head-on collider beams improve the performance of fusion engines ??? (It also seems, that all of the "jittering, jiggling, & jostling" going on in the nucleus, ought possibly to perturb [the central cores of] any bound electron Wave Functions.) It seems as if you can conceive of three (3) different Ranges, of relevant forces, on nuclear sizes scales: Long-Range (Electro-Magnetism via Photons) -- extra-nuclear (r > 7 fm) Medium-Range (Strong Force via Pions) -- intra-nuclear, inter-nucleon (r > 1 fm) Short-Range (Color Force via Gluons) -- intra-nucleon (1 fm > r) If Deuterons are spin-parallel combinations of protons & neutrons (S=1), would it help, for fusion, to "spin prepare" the fusion fuels, to make them "spin-aligned", and, thereby, expedite the fusion process ???

Consider the classical cartoon, describing entropy -- a state where all the molecules of gas, confined to a box, happen to occupy only half of that available volume, is said to have "lower entropy", than a state where the same molecules are spread out more evenly. Now, make that "box" into the whole cosmos. And, make "half the box" into the gravitationally condensed over-densities, associated with Large Scale Structure. By "ordering" the matter, which was once spread out evenly across the cosmos (at the Big Bang), into Galaxies, Clusters, Super-Clusters, and the Cosmic Web, does gravity reduce the entropy of the cosmos ???

Thanks for the response. I found this reference, as well: If the uncertainty, in the transverse momentums, of the electrons in the beam, is [math]\Delta p_{\bot} = p \; sin( \alpha ) \approx p \; \alpha[/math] (Small Angle Approximation), where [math]p = \hbar / \lambda[/math], then, from, the Heisenberg Uncertainty Principal [math]\Delta x \approx \hbar / \Delta p[/math], w.h.t. [math]\Delta x_{\bot} \approx \lambda / \alpha[/math]. Then, for the wide-angle electron beam [math]\left( \alpha = 7 \times 10^{-3} \; rad \right)[/math], w.h.t. [math]\Delta x_{\bot} \approx 6 \; \AA[/math]; whereas, for the more focused electron beam [math]\left( \alpha = 2 \times 10^{-7} \; rad \right)[/math], w.h.t. [math]\Delta x_{\bot} \approx 20 \; \mu m[/math]. Perhaps these transverse spatial extents, of the electrons' wave packets, can be compared, to their longitudinal lengths [math]\left( \approx 1 \; \mu m \right)[/math] ? Note that, if p = m v (Classical limit), then [math]\beta \approx \lambda_C / \lambda[/math], where the Compton Wavelength of electrons is [math]\lambda_C = 0.0243 \AA[/math]. For 100 KeV electrons, with wavelengths [math]\lambda = 0.04 \AA[/math], w.h.t. [math]\beta = 0.6[/math]. Thus, if the electrons' wave packets are [math]1 \; \mu m[/math] long, then their emission must take about t = L / v = 6 x 10-15 s.

According to Wikipedia, when a Wave Function "fails" to interact with a detector device, then, not being absorbed or transmitted, the Wave Function reflects off the obstacle, producing a diffraction pattern. Such a situation is not [full] Wave Function collapse & Localization: EDIT: When part of a Wave Function impinges upon, and reflects off from, some surface, that incident Wave Packet evolves through three broad states of development: incoming wave packet: <p> towards surface, spatially varying phase collision: <p> = 0, spatially constant phase & doubled probability density (incoming & outgoing waves "pile up", effectively compressing to half the original longitudinal length, as they cross through themselves) outgoing wave packet: <p> away from surface, spatially varying phase Now, bound-state electron orbitals, if they're Hydrogen-like, are stationary states, with no net momentum (<p> = 0), and, hence, spatially constant phases. Logically, therefore, it would seem that the probability for absorption, of the electron, into the obstacle surface, would be maximum during that second, "turn around", phase. If the absorption occurs, the Wave Function collapses & localizes, into an available bound-state orbital. Otherwise, the reflection process continues normally, according to the Schrodinger Wave Equation, with no discontinuous "quantum jump". In theory, that reflected wave could then, later, still interact with other obstacles into which it impinges. For example, please ponder a beam splitting experiment, which "quantum splits" the Wave Functions of photons, along two separate pathways, where one such path is much longer than the other. Thus, each half wave packet reaches the nearer detector screen first. Each time that those half wave packets fail to interact, they reflect off, backwards, back towards the original source. Imagine that, during that collision & reflection process, another detector is inserted along the shorter path. If, before the other half wave packets, on the longer leg, reaches the far-off detector, the failed-to-interact reflected wave packet reaches the newly inserted detector, then the whole Wave Function could collapse, into said newly inserted detector, instead of that far-off detector, at the end of the longer leg (???).

If one can say, that "there are no free quarks"... then, can one say, that "there are no Delocalized quark Wave Functions" ? Does not the Color Force, carried by the "glue", constantly "compress" the Wave Functions, of quarks, "jacking up" their effective mass-energy, from 4-5 MeV, to 100 MeV (Mesons) or 300 MeV (Baryons) ? And, wouldn't this Color Confinement also imply, that the resulting composite-particle Hadron Wave Functions would also be highly localized (~1 fm) ?

I found this reference, which contradicts my suggestion: Does a "failed interaction", resulting in a "failed Localization", at one point in space, force that Wave Function to Localize elsewhere? Or, does a "failed Localization" here, merely mean, that the probability distribution is "expelled" from the "failure locality" (with the "missing probability" being "redistributed", back into the rest of the Wave Function, in a sort of Renormalization) ?