Double-slit experiment: electrons 'everywhen'?

Question

Non-scientist here trying to get my head around the double slit experiment.

If whilst exhibiting wave-like behaviour electrons are potentially in more than one place at once, and if time is a dimension just as the spatial dimensions are dimensions (correct me if I'm wrong), then why whilst exhibiting wave-like behaviour would electrons not also be potentially in more than one time at once? And if they can be, wouldn't that explain the irrelevance of intervals when it comes to the appearance of the interference pattern? They're not only everywhere but everywhen, so to speak. The act of observation would then not only force the electron to be in a particular place but in a particular time. Or am I so far off that I need to go back to school? (Only did science up to GCSE level many years ago.) ;)

Perhaps a clearer way to ask this is: why is uncertainty limited to spatial dimensions, and generally not considered in the time dimension?

Answer 1

In quantum mechanics, time is not a dimension, it's a universal parameter (involved in the evolution of states/operators depending upon your so-called picture). Also in quantum mechanics, the electron is a particle described by a wave function. The wave function can go through both slits but what does that mean for a particle? That question is the point of the experiment.

Of course, time is not a parameter. Quantum mechanics is a (mostly) non-relativistic approximation. Incorporating relativity leads to quantum field theory, where there are no particles. Quantum fields permeate space-time, and an electron is quanta of that field.

Since computations are so difficult, we usually think in approximations. There is an initial state: incoming free particle electron, a final state: outgoing free particle electron detected at the screen. The electron takes all possible paths between the two (Feynman's path integral formulation), and we can calculate the scattering matrix describing it.

In that picture, there really is no mystery: beam impinges on target and scatters to a detector.

Answer 2

Electrons are not everywhere at once. All information about the electron is encoded in a function $\psi (x, y, z, t) $. And there is no hidden information about the electron other than this (at least not in the standard interpretation).

$|\psi (x, y, z, t) |^2$ is the probability density of observing the electron at the point $(x, y, z) $ at time $t$. Note that:

$$\int |\psi (x, y, z, t) |^2 dxdydz=1$$

for a fixed time $t$. So if you simultaneously conducted position measurements at all space locations, the probability of you finding the particle would be 1. The information about a particle is uncertain in space, but not uncertain in time.

It is not true that:

$$\int |\psi (x, y, z, t) |^2 dxdydzdt=1$$

If your hypothesis was right, the above equation would have been true. Position is a probabilistic observable in Quantum Mechanics. Time is only a parameter.

if time is a dimension just as the spatial dimensions are dimensions

Time and space should be treated equally in a relativistic theory. The theory I described above is a non-relativistic theory.

In the Relativistic theory, called Quantum Field Theory, both space and time become parameters instead of observables. They appear as parameters of Operator Fields which are mathematical objects defined at every point of spacetime.