The interference pattern that builds up in single-electron double slit experiments looks random at early times. Please Register or Log in to view the hidden image! Suppose you have several identical setups, each with independent electrons and double slits, and you collect the early patterns from each one, at about the time they look like image b. Lets make the number of different two-slit 'detectors' a thousand. If you combine them (take the union of the sets of points of the collection), what would you expect to see? If you assume the experiments can be simultaneous or alternatively each is sequential--ordered over time--how would the two runs compare?
I would expect the combined pattern to be essentially indistinguishable to that for single setup run till the total hit count is the same. After all, in any given setup each hit is presumed independent of any before or after. Why would it matter? You have some theory about spooky temporal entanglements or whatever?
Agreed. The pattern of dots builds up over time to yield the pattern of the classical interference fringes. It should not matter whether this is comprised of a single long accumulation or a sum of several shorter accumulations.
I agree also. Is there any consensus on the explanation for the interference fringe pattern from single electrons over time? Is it caused by the wave-particle nature of the particles, allowing the particle to go through one or the other of the slits, and the wave associated with the particle to go through both slits?
I also agree that it shouldn't matter how the pattern is made, a thousand single electrons at the same time over the same pair of slits in a single beam produces an interference pattern and so does a thousand electrons one at a time in the same apparatus. Using a thousand double slits and a thousand different electron sources so you get one dot per apparatus, then combining the images should produce an interference pattern too. It shouldn't matter if you shuffle the images around either, the only requirement seems to be that you need enough dots so a pattern "emerges", and this isn't seen in image a or b. Ok, we're in agreement. What about the reason it doesn't matter how the images are collected or combined, all that matters is the number of dots being close to some threshold (I understand this is a few hundred). So what's the explanation?
No. That is from Bohm's "guiding wave" version of quantum mechanics, and there is no evidence that "guiding wave" even exists.* The more standard POV is that all possible paths are used and waves can interfere when all arrive at some specific spot on the screen. Then the waves cease to exist and the particle does at one well defined location. This is true of each photon or electron - it only interferes with itself, so it can be the only one or one of many non interacting ones. * There is a serious conceptual error or at least conflict inherent in Bohm's QM in that we believe all electrons are identical yet each is guided ONLY by its associated "guiding wave." That implies the electrons are not identical, but must have some difference which are like unique "names," for example Bill & Tom. Bill is guided only by wave Wb and Tom is guided Wt. If they are as believed IDENTICAL they are not distinguishable. Thus Wb could guide either an different than observed results would occur especially in cases where statistics is of import.
No. I think you have jumped to a conclusion about what I asked. Bohm pilot wave theory is not a wave-particle theory, as near as I can tell. See text quote from Wiki link on Pilot Wave Theory: https://en.m.wikipedia.org/wiki/Pilot_wave Pilot wave "In theoretical physics, the pilot wave theory was the first known example of a hidden variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, remains a non-mainstream attempt to interpret quantum mechanics as a deterministic theory, avoiding troublesome notions such as wave-particle duality, instantaneous wave function collapse and the paradox of Schrödinger's cat." I bolded the phrase that indicates that pilot wave theory is not associated with wave particle duality.
Look at what state each electron is in, before and after passing through the two slits. As usual, the system is tuned to the particle wavelength, the de Broglie wavelength. After passing the slits, each electron's wavefunction is in a superposition of states, but what states? Obviously the "appearance" of interference is statistical somehow. What does it have to do with the uncertainty principle? Come on, I know there's a physicist out there. What can we tell from the images in the OP? 1) The statistics are independent of the time and the locality of individual scattering events. 2) If the per-particle statistics (images) are brought together in a sufficiently large number you get a measurement of interference fringes. Otherwise you have a collection of images that look totally random. 3) Projecting the images backwards from e to a (backwards in time) shows that most of the dots lie in a bright region, so each electron must have a greater probability of being in a bright rather than a dark region. The randomness isn't really what it looks like.
Why is "everything linear"? Suppose you have a pattern-recognition algorithm that can scan the images; a horizontal scan line moves up the image and counts how many dots it intersects at a time. Once the interference pattern emerges, there should be the same average number on each horizontal line through the image. You can also project vertical lines from the centres of each maximum in pattern e "backwards" through all the patterns, and count the dots near each line. Linearity. And time-slices of the image from the same or many double-slits can be added together because the images can. Time is linear too.
This blog entry may help answer just the first line above: http://motls.blogspot.com.au/2012/08/why-quantum-mechanics-has-to-be-complex.html Not sure what your issue is with the rest there.
Maybe you don't want to consider wave-particle duality when you ask for explanations? We are talking about how the wave interference pattern emerges from individual particles over time. To understand that, I think you have to consider what a particle is, and how it moves. In your view, does a particle have internal composition, i.e. are they composed of wave energy with in and out flowing wave components is in standing waves, or are they a dense point in space, or something else? Is particle motion discrect as in incremental steps, i.e. discontinuous, or is particle motion continuous? The various ways that you can answer these questions would seem to have to give you various explanations for the emergence of the wave pattern. Have you tried to think through the experiment using the various ways those questions can be answered?
Look, the double slits prepare each electron. The coin analogy doesn't seem that useful, but again, each electron arrives at the double slits in a mixed state and leaves in a prepared state--the double slits measure position and so the uncertainty principle applies. I've seen an analysis that first defines a position operator--the dirac delta--over the y direction from a single slit when the other is closed. That is, the analysis considers a single horizontal line through the pattern and the y deflection angle. Then with both slits there are two operators and two deltas. That's Fourier transforms, right? The momentum picture is the transform of the dirac delta one. They use the Born rule as an axiom, type of thing. In their conclusion: " . . . It is interesting that for particles scattered from a double slit, the probability amplitude that gives rise to the interference is due to a superposition of delta functions." It says that interference appears when two position operators introduce a relative phase, with one operator this relative phase can't be seen. I'm aware there are several approaches to a mathematical definition of what I'm going to dub, the probability phase space. A distribution of complex probability amplitudes, call it what you will. What about the symmetries of qubits? Why are the quaternions useful? . . . ?
K It's amazing the only thing you know about is the bullshit you've made up. For a free particle moving over the natural path, freefall, it's easy to predict the geodesic and everything associated with the path using GR. Other than that it takes force to accelerate the particle off the natural path. Does internal composition mean are particles made of something? The answer is yes. The general term for what stuff is comprised of is matter. It's best to model inertial and accelerated motion as continious until there's a reason to do otherwise. The various ways you can answer these questions are in the literature. Try reading some of it. Right now you're just trying to obfuscate any discussion with your made up bullshit.
I was thinking about the algorithmic approach; if you can scan the images horizontally so you eventually get lines with the same average number of dots on them, why can't you generate a set of horizontal lines with the same distribution, and construct an image? Suppose you cheat and use an actual average horizontal distribution of dots, but alter the positions of each dot slightly (along the line) as you scan up the image you construct, maybe use some small distance generator function that looks random or somesuch. So it kind of devolves to: can an algorithm be constructed that reliably reproduces the average distribution of dots, does it need to run on a quantum computer or can a classical one handle it? Another thought extension: all particles and even fairly large molecules exhibit this interference effect--the hallmark of quantum behaviour and the "waviness" of particles--so you could equally combine early images from single particle diffractions of any kind, and an interference pattern will be there; you could combine electron, photon, buckyball, etc scattering events. Bring them all into the same, ahem, frame of reference. Why is it about measurement? The position of a single particle is initially measured by a single slit, this is the position operator (everything except time is an operator). Subsequently the position of a single particle is measured simultaneously by two operators. Measurement is preparation--now the single particle is in a superposition of states, in a position-momentum picture. Each particle therefore can be said to "be" in two places at once. The superposition of delta functions is aka what Penrose says about the difference between a singly-peaked delta (one position operator), and a doubly-peaked delta. But, I'll let you read it for yourself . . .
Here's some help with the qubit basis: Note that a qubit is a modern term that just means a particle like a photon or electron, it can also be an atom or a molecule in the case of double slit diffraction. We have the wavefunction, I don't know that this is an apt kind of name, because it's just a mathematical object, a 2-vector with complex coefficients, that's usually notated: \( \Phi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \), so \( \Phi = \alpha \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 0 \\ 1 \end{pmatrix} \). This is \( |\phi\rangle = \alpha|0\rangle + \beta|1\rangle \) in Dirac notation.
So if you understand complex numbers, and how to represent vectors in the unit circle, that's all the domain is here. But these objects describe this domain too: Please Register or Log in to view the hidden image! the lower graph is the time domain transform of the actual signal, the animation demonstrates how each time domain component is a finite impulse; I think the idea is to look at the positive height of each harmonic. So, you can compose any waveform in the frequency (spatial) domain by adding impulse functions together in the time domain. Wavefunctions can be considered as objects with a similar structure, a wavepacket.
Not my understanding. How can it be said one particular slit initially 'measures' the particle? Both slits jointly interact with particle to produce superposition - but that afaik is formally a preparation not a measurement. Measurement usually comes when we have a single dot on the screen, at which time 'wave function collapse' occurs. If which-way detection occurs at the slits, that is an initial measurement, interference is destroyed and dots correspond to passage through one slit at a time. Although..... An interesting article from a few years ago: http://arstechnica.com/science/2012...rticle-duality-in-the-double-slit-experiment/.
How can it be said that a single slit doesn't measure the particle? Measurement and preparation are equivalent. Since with one slit the particle passes through it, the slit must measure the particle's position.
We are still discussing a double slit situation? Or has it somehow gone to a single slit scenario? Are they? It's not so, measurement (not 'weak' measurement) results in a definite eigenstate? One cannot prepare a system in a superposition of states? Again, is this now discussing a single slit situation somehow? Even then, knowing a particle must pass through the slit is not in my book a measurement of position, if that's what you are getting at. Having no background in QM, I will defer and refer to a lengthy discussion on the distinctions plural according to: http://www.phys.tue.nl/ktn/Wim/fop994.pdf Beyond that, maybe you should clarify just what exactly is the aim here. You want to create or at least justify a hidden-variable interpretation perhaps? Anyway, there are others here with much better backgrounds in QM that may risk getting involved to possibly help.
This from a 2007 paper: . One can, and one does, or rather the apparatus does. This is what two slits do: they prepare each particle in a superposition of position-states. In QM experiments, measurement and preparation are seen to be equivalent--the system prepares states by measuring them, the experimenter measures a classical interference pattern, or more exactly the screen does. Where is the distinction? In our minds.
