ray3400 said:
Ok, but isn't it the case that when the electrons are being detected they act like particles, and instead of making a pattern like the one you posted, they make a pattern of two strips?
What aspect of the detection device in this experiment caused the electrons to act differently?
The confusion comes only from the fact that some people, obviously and unfortunately many of which write popular-science articles/books, that the world is not behaving as they think from everyday experience with macroscopic systems, which appear to behave as described by classical physics. This is, because for macroscopic objects we need to describe only quite rough macroscopic observables and neglect (average over) the many microscopic degrees of freedom they consist of.
The intuitive worldview from this experience breaks usually down when dealing with small systems like electrons. You cannot describe them as classical particles nor as classical fields. Rather, from the most fundamental point of view we have today about them, they are certain states of a quantized Dirac field.
Under "non-relativistic" circumstances, i.e., when the electron runs with speeds much less than the speed of light, you can describe it by a wave function ##\psi(t,\vec{x})## (where I neglect the spin degree of freedom, which is not important for the explanation of the double-slit experiment). This wave function obeys the Schrödinger equation, which describes the time evolution of the wave function.
Though there are many people, who cannot accept this view, and I'm sure there'll be tons of following answers to your question denying it, yet the only consistent interpretation of this wave function (it's called the minimal statistical interpretation, since there's a plethora of other interpretations all trying to somehow deny the very fact of generic indeterminism in our description of nature, which is due to some philosophical prejudices but not founded on any observations): The function ##P(t,\vec{x})=|\psi(t,\vec{x})|^2## describes the probability distribution to find the electron at a place ##\vec{x}## when looked there at time ##t##.
The specific form of the wave function is determined by the "preparation of the electron" at some initial time ##t=0##. Then its time evolution is given by the Schrödinger equation. So given the preparation you can determine the probability distribution of its location at any time. You can also calculate the probability distributions for any other observable, like momenta, by certain mathematical manipulations (in this case going from the position representation to the momentum representation it tells you to take the Fourier transform of the wave function).
This interpretation implies now that observables of a particle are usually not determined but have only certain probabilities do be found when you measure them, given by the wave function.
Now take the double-slit experiment. Now say you want to figure out through which slit each electron comes you shoot at the double-slit, but it should not be determined beforehand, that each electron goes through one of the specific slits (this possibility is not very surprising, yielding a priori the same thing as you expect from classical physics).
As an experimental physicist you now have to ask, how to realize this case, i.e., you have to think how you have to prepare each electron to describe this situation. Now the above described formalism tells you that if you prepare an electron to have a very well defined position initially, this implies that the wave function is narrowly peaked around this position. Now the momentum wave function is given by the Fourier transform of this wave function that's narrowly peaked in position space, which implies that it is a pretty broad distribution in momentum space. This is quantified by the famous position-momentum uncertainty relation: If the standard deviation (which quantifies the uncertainty, i.e., the width of the wave function around the expectation/average value) of the position-vector-##x## component is ##\Delta x##, then there's a an uncertainty momentum-vector-##x## component ##\Delta p_x## which is so large that ##\Delta x \Delta p_x \geq \hbar/2##. So if you initially localize the electron very well, you get a quite uncertain momentum, and the Schrödinger equation for a free electron accordingly tells you that the electrons position-wave function gets broader and broader with time.
In order to avoid the case that your electron's position wave function is still so narrow and the experimentalist aimed at on slit well enough (i.e., he's chosen the momentum, though broad, to be pointing into the corresponding direction) that with certainty it goes through one slit, you simply have to put your slit far away enough from the electron source, so that the wave function gets so broad that its peak covers well both slits. You can arrange this such that it goes with equal probabilities through one or the other of the two slits, if it goes through at all, i.e., many electrons will simply be absorbed by the material making up the double slit, but some will go through, and you cannot with certainty say through which of the two slit any individual electron will have come.
However, at the position of the slits, each electron only goes as a whole through one of the slits. This implies that those electrons that went through the slits at all, at the slit it's with certainty localized with an accuracy given by the width of this slit, and that implies that directly at the slits each single electron is uniquely at the one or the other slit.
Closely behind the slits the wave function thus now shows two narrow peaks corresponding to the pobabilities to go through one or the other slit. The two peaks are well separated, i.e., their width (given by the width of each slit) is much smaller than their distance (given by the distance of the centers of the slit).
Behind the slits the electron's wave function again goes on according to the Schrödinger equation, i.e., it broadens again, but not too far from the slit, i.e., after not too long times, the wave function will still show two well-distinct bumps. Although these bumps got a bit broader, they are still not too much overlapping, so that for each electron the probability distribution is such that with quite high certainty you can say through which slit they came.
This means that, if you want to know (with high certainty) through which of the two slits each electron came you simply have to put your electron dectector (e.g., a CCD cam like the one in your cell phone) close enough to the double slit. Sending through a lot of such prepared electrons in this setup you'll just see two well-disitinct bumps according to the probabilities given by the wave function at the position of the CCD cam. Each electron being in one of the bumps very likely came through the corresponding slit. Thus, here you obviously have which-way information with high certainty.
On the other hand, if the wave function develops further from the slits the two bumps get broader and broader and then overlap more or less. The longer you wait and the farther the electron goes away from the slits the less certain you can determine through which slit this electron came. I.e., if you put the CCD cam far enough away from the slits, there's no possibility anymore to know from the position the electron hits the cam, through which slit it has come.
Now the Schrödinger equation is mathematically a wave equation, and this implies that the partial waves of electrons coming through the one or the other slit simply add like ##\psi=\psi_1+\psi_2##. Now the probability is given by ##|\psi|^2=|\psi_1|^2+ |\psi_2|^2 + \psi_1^* \psi_2 +\psi_1 \psi_2^*##. The last to terms are called the "interference terms". In the situation where the CCD cam is placed far away from the two slits ##\psi_1## and ##\psi_2## have a large overlap and thus the interference term is significant, and depending on the specific position at the CCD cam the contributions of the partial wave may add (constructive interference) or cancel each other out (destructive interference) or something in between, and that's precisely the interference pattern you expect from wave phenomena, i.e., in this situation where you don't know which way the electron took the probability distribution shows a significant interference pattern.
Of course you can argue with the interference piece also in the case discussed above, when the CCD cam is close enough to the slits to still resolve which-way information with high certainty: Then the partial waves do not significantly overlap, corresponding to the still well separated two bumps, and thus ##\psi_1 \psi_2^*+\psi_1^* \psi_2 \simeq 0## everywhere at the CDD screen. Thus you get just well distinct bumps on the CCD screen but no (significant) interference.
Note that still, each electron makes one single dot on the CCD screen, i.e., a single electron cannot appear as some smeared distribution at the screen. In this sense it has always particle features when detected, and the above given probabilistic "interpretation of the wave function" is at least non-contradictory with these observational facts and also not contradicting any other fundamental laws of physics (like causality and all that). I know of no other interpretation that is consistent in this sense (except Bohmian mechanics, which however is consistent only in the here discussed non-relativistic limit, and also doesn't lead to other phenomenological implications of the quantum-theoretical formalism than the minimal interpretation).
