After some remarks on my comment to Dr Kaku's (a popular figure from bigthink.com) overpopularization of physics one more comment on the wave-properties of an electron seems appropriate:
There is a famous experiment illustrating the consequences of the wave-description for the probability to find an electron at a given point in space.
If an electron-source is placed in front of a metal-sheet with two narrow slits, a recording-device behind this screen would detect a distribution of incoming electrons that is identical to an interference-pattern of waves passing through the double slit (as would be seen when shining coherent, monochromatic light (a laser produces this kind of light) on the two slits).
This interference-pattern shows up no matter how low the rate of electrons coming through. Even if only one single electron per hour would pass through the slits, there would, finally, be a distribution of detected electrons given by the well-known interference-pattern.
This experiment clearly demonstrates the wave-characteristics of the electron - or more precisely, it demonstrates that the probability to find an electron at a given point in space and time is given by a wave-function. It is this probability-distribution that passes the double-slit and, quite logically, results in an interference-pattern on the other side - giving the resulting probability-distribution for the position of an electron behind the double-slit. So the electron *is* no wave. It's position is given by probabilities *that are described* by a wave-function.
Compare it to casting a dice - twice: the probability for any number on the dice is equal. So if you plan a game (an experiment) where you win depending on a specific combination of the two outcomes, the probability of possible results will be a combination of all probabilities.
If you had some kind of dice with a huge number of faces (say: 1000) and the probabilities for these numbers to show up would be 'modulated' by a sine-function (a simple wave-like distribution), then the combination of casting the dice twice would be the interference-pattern of the two wave-functions that describe the probabilities for the numbers showing up...
So the probabilities of all events and the combination of these probabilities describe the possible outcomes of the game. The probabilities for all number-combinations in (number-) space are given by a wave-function.
The same is true for the electrons in the two-slit experiment. The possible position of the electron after passing the screen with the two slits is given by the probability-distribution that results from the wave-like probability distribution of the electron passing through the double-slit screen - either the left slit, or the right slit.
This does *not* mean (as is so often implied) that the electron is passing thtough both slits simultaneously. Just as it does not mean that your dice show all numbers simultaneously. It does not mean either that the electron is at every point in space at the same time. As your winning numbers are not all numbers on the dice.
No, the probability to find the electron at a given point in space and time is described by a wavefunction. The probability to have your winning pair of numbers showing up in the experiment with the dice is described by a probability-distribution, that we just constructed to be a wavefunction.
If the path of the electron is determined before travelling through one of the slits then, of course, the probability-distribution for the position of the electron is completely different: the position of the electron is known (within principal limits), the probability to find the electron at a different point is zero.
If you cast the dice and record the number, the result of this 'experiment' is known, the probability to get another number is zero.
If the electron continues it's trip through the double-slit experiment there will be no resulting interference-pattern behind the screen. This is not surprising at all, because the probability-distribution for finding an electron at a given point is no longer a monochromatic wave showing equal probabilities for either path through any of the slits. The wave-function is a strongly localized package.
No surprise here: also the result of casting your dice twice will no longer be described by a combination of all probabilities of the first cast and the second - since the first outcome was 'measured' and is therefore determined.
Nobody would dare to say the dice have wave and matter duality - the wave describes the probabilities for a given point in (number-) space.
The same for electrons, photons, anyones...
They dont have wave and matter duality. The probability-distribution is described by a wave-function. The electron in a box is not at every point at the same time - the probability for being at any point might be equal at any time. The probability for any number on a dice is equal at any time - until you check, you measure. Then the probability for exactly one result is one - for the others it is zero. Schrödinger's cat is not dead and alive. The probability might be equal - until you check. (Schrödinger himself, however, is most probably dead)
let's try to explore your explanation of the double slit experiment. You assert "It is this probability-distribution that passes the double-slit and, quite logically...". Is this really that logical?
Why is it a probability function, that passes the slit? Why not a particle, why not a wave? I do not (well, only partially) pose the question what an electron actually *is*. I merely ask how exactly a thing may move in terms of probability functions?
If we think about Newtonian motion, say a thrown stone, we could ask a similiar question: How exactly do things move in parables? A parabola is a mathematical object, like a wave-function, too. Isn't it?
I'd say, there is a fundamental difference. We can interpret a parabola. It somehow describes where to find the stone - or its center of mass - at a given time point.
We are, however, unable to interpret quantum mechanical probabilities. If I measure the position of an electron once, what does this psi-square probability refer to? The point is btw not, whether or not we deal with abstract entities (or descriptions, for that matter). The point is rather, if we have a interpretational framework, into which these entities can be integrated. The dice-analogy does not help much either, because dice-probabilities are known a priory from many castings. If we had a new dice with an unpredictable behaviour (e.g. its mass is distributed irregularly) and were allowed to cast it only two times: what probabilities would we use and how could we justify our choice?
My understanding goes as follows. If we adopt a frequentist view, we must postulate that quantum mechanical objects are the manifestation of underlying processes, which can be seen as a (fast) repititions of dice-castings. These processes may be thought of as if they constitute the QM-object, much like spinning water constitutes (by its spinning motion) a vortex. Instead of the spinning water we'd have a dice-cast-generator.
If, on the other hand, we adopt a Bayesian view, we end up with the idea, that a physical system (be it as small as a QM-particle) has some sort of "knowledge" about the world. I think Leibniz purported something like this.
To me, both consequences are hard to accept. The first idea does neither explain much (we are referred to other, more basic "dice generators") nor does it really come to a happy end (what causes the "dice-generators" to work?). The second idea borders at the edge of esoterics and does not really explain what "knowledge" means.
Another side of the question "how things move in terms of probability functions" is the following. A probability other than 1 or 0 can only assigned to processes which have a propensity to happen or not to happen. In other words it just refers to the fact, that something is changing/moving. So. There we have a circular thought. A thing, which moves in terms of probability functions, is nothing more than a thing that moves. But why does it exhibit all that strange double slit behaviour then?
But the central point of my simplification must not be lost: the mathematical descriptions of observable processes or events in nature are *models* of those processes or events. They are not the events or processes themselves. In this sense a probability-distribution can be written down without any dices being cast or statstical processes actually running in the background. The distribution is a model that, if done correctly, may be able to describe some observation - and give predictions of states, processes, events to be observed in the future. But the model exists also without any real states, processes or events.
The equal distribution of probabilities for the six numbers in a lottery-draw does exist even without the lottery, or also when there is only one lottery-draw ever.
My admittedly oversimplified picture aimed at demystifying the 'collapse of the wavefunctions' - a term used to describe the fact that the probability to find a quantum-mechanical partical at one given point at one certain time is fixed and either zero or one, once the particles' position is measured - as compared to the distribution of probabilities (given by a wavefunction) at any instant before measurement.
The probability for any combination of six numbers (out of a set of 49) to win the lottery is equal - until the numbers are actually drawn. After that the probability is zero for almost any combination, except one - for which it is 1.
A wavefunction would be a 'real thing' - not just a concept... we will see.