It's my understanding that electrons are particles, and it's also my understanding that their location while orbiting an atom cannot be determined precisely and must be determined by statistics and probability, almost like electrons can be in multiple places at the same time. That made me think, hmm, could electrons exist more as smears instead of hard-edged particles? A smear can be in more than one place at a time. The only difference is a smear doesn't have a hard edge like a spherical particle would. It would sort of "blend" matter and space. I'm also wondering if perhaps smears would demonstrate wavelike properties that hard-edged particles can't. Is there any knowledge out there that states that matter particles either must have a hard edge or can't have a hard edge?
1$\begingroup$ Related: physics.stackexchange.com/q/24001/2451 , physics.stackexchange.com/q/41676/2451 , physics.stackexchange.com/q/119732/2451 and links therein. $\endgroup$– Qmechanic ♦Nov 30, 2014 at 21:44
$\begingroup$ It actually is smeared out, until it must interact (i.e. transmit a force)! $\endgroup$– dotancohenDec 1, 2014 at 11:22
$\begingroup$ Thank you for this wonderful question, by the way. The answers generated are terrific and really cover the topic from all angles. Welcome to Physics.SE! $\endgroup$– dotancohenDec 1, 2014 at 11:42
This is one of the key results of quantum field theory: particles are not single points, they are disturbances in quantum fields that are spread out over space. Typically the disturbance is not spread out very much, otherwise it looks more like what we know as a wave than a particle. The technical term for what you're calling a "smear" is a wavepacket.
In fact any object with non-zero physical extent is required to be deformable by relativity (but see below); otherwise pushing on it can transmit energy and information faster than light. And quantum mechanics rules at those scales so everything does have fuzzy edges. At the scale of the very small there are no sharply defined objects.
A consequence of this is that all fundamental particles are required to be dimensionless.
Alas, this introduces endless singularities into the math, so the plan is to find a theory which can combine relativity and quantum mechanics at the very small scale in a way to both preserves locality and prevents all those divergences. The best product to date is string theory.
I like the way you're thinking, but I'd like to try to open your mind a bit further.
A lot of our everyday concepts are a product of the net behaviour of billions of particles following the laws of physics. For example, consider a steel ball. It has a "hard edge". What does this mean? It means another ball can be anywhere outside of the hard edge, but cannot overlap. Why is this so? Because of billions upon billions of iron and carbon atoms in a stable arrangement which will not permit this (without enormous energy input). Now imagine two balls of cotton wool. They have "fuzzy edges" and so can overlap somewhat. Why? Because their particles rearrange themselves with only minor investment of energy.
If we assume (for purposes of argument) that electrons are fundamental particles that aren't made of smaller particles, then there is nothing to give them a "hard edge" or "fuzzy edge". They certainly have an effect on the particles around them, just as Jupiter has an effect on the Earth. But just because Jupiter is a "smear" (it is made of gas) and the Earth has a hard edge, doesn't mean electrons have any physical shape at all!
The lesson here is that concepts that seem to apply to all objects in our experience, like volume, density, and shape, are not necessarily "universal" and there is no reason to expect they apply to the very particles that give rise to them!
Electrons have a position, though it's not a traditional $(x,y,z)$ co-ordinate but a set of probability distributions. This is not the shape of the electron. It is its location. Without a shape, how can we visualise what is going on in the subatomic world?
Well, let me ask you this. Does your group of friends have a shape? You could write their names in a list, or in a circle, ordered in countless ways, but there would be nothing that lets you visualise the "physical reality", except maybe a map of all their current locations. Such a map would tell you very little about the system. Even photographs of all your friends would tell you very little about them. On the other hand, learning more about the characteristics of each of the people would give you a lot more understanding of the system and how it behaves.
Although it's nice to visualise things, sometimes the fact that we can't visualise them helps prevent us from trying to apply inappropriate intuitions to them.
(To expand your mind just that little bit more: I suspect that the reason we can't seem to find a unified theory of physics is that we're not willing to let go of concepts like position and time, which makes it impossible to imagine the "particles" from which space and time arise.)
A particle does not need to have a hard edge. It can for example, be a density function, which sort of fades to zero.
One might note that waves can intersect each other and come out as if the other wave was not there.
Particles with hard edges are more an artefact of our minds rather than 'what's really there', until we get some real evidence otherwise. Note however, that the scattering rules would still apply if the particle is nebulous, because instead of hitting a solid wall and bouncing off, one essentially rolls up some steep hill, and roll down at a different angle.
It's just that tennis balls are easier to understand, but they might not be the correct model at femtometre-scale physics.