In classical mechanics, positions and momenta (and energies and times) are independent variables, but in quantum mechanics, they are dependent variables and interact in wavefunctions. In classical mechanics, when two or more particles interact, system properties sum particle properties. In quantum mechanics, when two or more particles interact, system properties multiply and sum particle properties, and particle wavefunctions combine constructively and/or destructively to make a system wavefunction {entanglement}|. If two (indistinguishable) particles entangle, they both travel together on all possible state paths available to them, and they interfere with each other's independent-particle wavefunctions along each path. For example, two particles created simultaneously form one system with one wavefunction.
Entanglement does not put particles into unchanging states (that observers measure later). Neither do particle states continually change state as they move through space-time (not like independent neutrinos, which change properties as they travel). Therefore, observation method, time, and space position and orientation do not determine observed particle state. In quantum mechanics, particles have probabilities, depending on particles and system, of taking all possible space-time and particle-interaction paths, and measurement finds that the particle has randomly gone into one of the possible particle states.
system wavefunction
When two particle wavefunctions add, system-wavefunction frequency is the beat frequency of the two particle-wavefunction frequencies, and is lower than those frequencies. System wave packet has smaller spatial extension than particle wave packets, and has higher amplitude (more energy) at beat-frequency wavelengths. Quantum-mechanical particle and system wavefunctions have non-zero fundamental frequency and its harmonic frequencies and have non-zero amplitudes over all space and time. Systems spread out over space and time.
system wavefunction decoherence
After entanglement, system wavefunction lasts until outside disturbances, such as measurement, particle collision or absorption, and electromagnetic, gravitational, or nuclear force field, interact with one or more particles. At that definite time and position, system wavefunction separates into independent particle wavefunctions (decoherence). Whole system wavefunction ends simultaneously over whole extent.
measurement
By uncertainty principle, experimenters can precisely measure either particle energy or particle time (or momentum or position) but not both. After two entangled particles separate, separate instruments can measure each particle's energy (or momentum) precisely and simultaneously and then communicate to determine the exact difference.
measurement: speculation
Perhaps, unobserved particles and systems are two-dimensional (but still in three-dimensional space). Observation then puts particles and systems into three dimensions. People observe only three-dimensional space. For example, observers see that gloves are right-handed or left-handed. Perhaps, unobserved quantum-mechanical-size gloves actually have no thickness and so have only two dimensions, so unobserved right-handed and left-handed gloves are the same, because they can rotate in three-dimensional space to superimpose and be congruent. Perhaps, unobserved clockwise and counter-clockwise particle spins are two-dimensional and so are equivalent. (Note that a two-dimensional glove appears right-handed or left-handed depending on whether the observation point is above or below the glove.)
Perhaps, unobserved particles and systems randomly, continually, and instantaneously turn inside out (and outside in), in three-dimensional space. Observation stops the process. For example, turning a right-handed glove inside out makes a left-handed glove, and vice versa. Perhaps, unobserved quantum-mechanical-size gloves continually and instantaneously turn inside out in three-dimensional space and so are equally right-handed and left-handed. Perhaps, unobserved clockwise and counter-clockwise particle spins continually interchange. (Note that a glove appears right-handed or left-handed depending on when the process stops.)
Perhaps, unobserved quantum-mechanical-size particle and system states are indeterminate and follow quantum-mechanical rules because space-time is not conventional four-dimensional space-time. Observation requires conventional three-dimensional space, and randomly makes definite three-dimensional particle and system states, with probabilities. Perhaps, time is not real-number time, but complex-number or hypercomplex-number time. Real-number times are separate, but imaginary-number times are not. Perhaps, space is not real-number space, but complex-number or hypercomplex-number space. Real-number distances are separate, but imaginary-number distances are not.
Observations measure real-number part of complex-number variables. Perhaps, wavefunction imaginary-number part continues after observation.
Perhaps, Necker cubes illustrate the effects of observation. Observer angle to Necker cube determines whether observer sees right-facing or left-facing Necker cube. Effects may be linear with angle or depend on cosine of angle.
interacting electrons and spin
If a process creates two electrons, momentum sum is the same before and after creation, by momentum conservation, and electrons move away from each other at same velocity along a straight line. Angular-momentum sum is the same before and after creation, by angular-momentum conservation. (If two separate electrons entangle, momentum sum and angular-momentum sum are the same before and after interaction.)
By quantum mechanics, measured spin is always +1/2 or -1/2. Because the electrons are in a system, one cannot know which has +1/2 spin and which -1/2 spin. Both electrons share a system wavefunction that superposes the state (wavefunction) in which first electron has spin +1/2 unit and second has spin -1/2 unit and the state (wavefunction) in which first electron has spin -1/2 unit and second has spin +1/2 unit, with zero total angular momentum in any direction. Two wavefunctions can superpose constructively (add) or destructively (subtract). Because two electrons are distinguishable, the two wavefunctions add, so system wavefunction is anti-commutative.
One possibility is that one particle has positive 1/2 unit spin along z-axis (motion line), and other particle has negative 1/2 unit spin along z-axis. See Figure 1.
After two particles interact and move apart, separate spin detectors can measure around any axis for first particle and around any axis for second particle, simultaneously or in succession. For example, the axes can be z-axis (motion line), x-axis, and y-axis. See Figure 2. Measuring spin around an axis fixes one electron's spin at +1/2 (or -1/2) and fixes the other electron's spin around an axis at -1/2 (or +1/2), to conserve angular momentum.
spin: possible axis and spin combinations
By quantum mechanics, left electron has spin +1/2 half the time and spin -1/2 half the time, around any axis, say z-axis. Around same z-axis, right electron always has opposite spin: left=z+ right=z- or left=z- right=z+. Around x-axis, right electron has opposite spin (while y-axis has same spin), same spin (while y-axis has opposite spin), opposite spin (while y-axis has opposite spin), or same spin (while y-axis has same spin): x-y+z-, x+y-z-, x-y-z-, x+y+z-; x-y+z+, x+y-z+, x-y-z+, x+y+z+. For right-electron z-axis compared to left-electron z-axis, spins are opposite all of the time: z+z-, z+z-, z+z-. z+z-. For right-electron x-axis or y-axis compared to left-electron z-axis, spins are same 1/2 of time and opposite 1/2 of time: z+x-, z+y+; z+x+, z+y-; z+x-, z+y-; z+x+, z+y+. See Figure 3. Because quantum mechanics has random probabilities, left and right electrons have same spin half the time and opposite spin half the time.
However, quantum mechanics with non-randomness (due to local real hidden factors) makes a different prediction. Non-random hidden factors correlate right and left spins, to conserve angular momentum. If left=x+y+z+, right=x-y-z-. If left=x-y+z+, right=x+y-z-. If left=x+y-z+, right=x-y+z-. If left=x-y-z+, right=x+y+z-. If left=x-y-z-, right=x+y+z+. If left=x+y-z-, right=x-y+z+. If left=x-y+z-, right=x+y-z+. If left=x+y+z-, right=x-y-z+. See Figure 4. For right-electron z-axis compared to left-electron z-axis, spins are opposite all the time. For right-electron x-axis or y-axis compared to left-electron z-axis, spins are same 4/9 of time and opposite 5/9 of time, higher than the 1/2 level for quantum mechanics. Local hidden variable theories correlate events through hidden variable(s), making probabilities non-random. Quantum mechanics has no more-fundamental factors and introduces uncertainties, and so is random. Therefore, correlated outcomes in classical theories have different probabilities than in quantum mechanics. Experiments show that outcomes are random, so there are no local hidden factors and/or no real hidden factors.
if infinite light speed
Perhaps, entanglement over large distances and times has no non-locality problems if light speed is infinite, as in Newton's gravitational theory. Assume that relativity is true but with light speed infinite. Time is zero for light, and speed is always infinite for all observers, so all objects are always in contact. However, light speed is finite.
action at distance
Wavefunctions do not represent physical forces or energy exchanges, so space and time do not matter. If system wavefunction does not decohere, system particles and fields remain connected, even over long duration and far distances. Experiments that measure energy and time differences, or momentum and position differences, show that particles remained entangled over far distances and long times, and that wavefunction collapse immediately affects all system particles and fields, no matter how distant (action at a distance). Seemingly, new information about one particle travels instantly to second particle. See Particle Interference, Scientific American 269(August): 52-60 [1993].
However, information about collapse only travels at light speed, preserving special relativity theory that physical effects faster than light speed are not possible. Observers must wait for light to travel to them before they become aware of information changes. All physical laws require local interaction through field-carrying particle exchanges, which result in space curvatures. All physical communication happens when particles are in contact and interact, so there is no actual action at a distance.
teleportation
After particle entanglement, particle wavefunctions have specific relations. By manipulating particle properties at interaction and at wavefunction collapse, experimenters can transfer particle properties from one particle to another particle, even far away, though the particles have no physical connection at collapse time.
Bombs can have photon or light pressure triggers. Bombs explode if trigger does not jam, but jamming happens often. How can testers find at least one working bomb without exploding it {Elitzur-Vaidman bomb-testing problem} [1993] (Avshalom C. Elitzur and Lev Vaidman)? Using photon entanglement can find good bomb without triggering it.
Particles can seemingly move from one place to another without ever being between the two places {teleportation}|. Teleportation requires that both locations share a particle pair {EPR pair}. Particles are identical, with entangled properties. For example, if one photon splits into two photons, new photons can be same-state superpositions. If instrument observes one particle's state later, it then knows other particle's state. If EPR pair exists, putting one pair member into one state can result in property disappearance at one location and other-pair-member property appearance at another location.
5-Physics-Quantum Mechanics-Waves
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Date Modified: 2022.0225