There are an infinite number of possible waveforms; pretty much any waveform in the electron field is a valid one. What we see as a single electron is simply a "spike" waveform localised in a particular position. There's no reason you couldn't have a more smeared-out waveform that was "an electron somewhere in the universe", though entanglement comes into play at some point.
When we have particular constraints (e.g. known energy) that constrains the space of possible waveforms. E.g. when we talk about there being an electron in an orbital around an atomic nucleus, what we actually mean is there's a waveform. of a particular shape around the nucleus.
Are you asking why we only ever see waveforms corresponding to whole numbers of electrons? That's the "quantum" part of quantum mechanics; certain values are quantized (e.g. electric charged). I don't have a good intuition for why that's so though, except to observe that the time evolution of a system preserves this quantization, so there's no way to ever go from having one electron to having half an electron (for example).
At some point "why" becomes impossible - everything just becomes a set of relationships between things that we can define and predict. The only real answer to "why", from a scientific point of view, is ultimately "because that's how it is"
Most good science comes of asking "why" - if you took "because that's how it is" as the answer to "why did the apple fall" science wouldn't have got as far as it has. Whenever a theory has some seemingly arbitrary property it's worth asking "why"; sometimes the answer is "we don't know yet", but that doesn't mean it's not worth asking the question.
Couldn't agree more - I wasn't thinking about how that might be interpreted when I wrote it.....
Absolutely we ask "why" - and look for explanations.
I meant nothing with regards to being defeatist and not looking at things - only that, as far as I can see, even though we'll keep going deeper and deeper and discovering more and more, we'll never get to a final answer (other than perhaps getting to a point where we can't research further without blowing up the universe? I read too much sci-fi.
A final answer would be boring..... WHY is a fantastic question - it's just not something pure science can answer with finality, only layers until we get to an unknown.
When we have particular constraints (e.g. known energy) that constrains the space of possible waveforms. E.g. when we talk about there being an electron in an orbital around an atomic nucleus, what we actually mean is there's a waveform. of a particular shape around the nucleus.
Are you asking why we only ever see waveforms corresponding to whole numbers of electrons? That's the "quantum" part of quantum mechanics; certain values are quantized (e.g. electric charged). I don't have a good intuition for why that's so though, except to observe that the time evolution of a system preserves this quantization, so there's no way to ever go from having one electron to having half an electron (for example).