Are you talking about quantum tunnelling?

@Charlie No. I was lookin for what The Photon said.

The probability of finding an electron in any particular place is zero.

@RyanThorngren Do you mean if you take a sample of any specific space, the probability of finding an electron is zero in it? If so, why?

Keep in mind electrons aren't particles, they're quanta. A quantum leap doesn't mean that an electron at [x0, y0, z0] suddenly jumps to [x1, y1, z1]. Quantum leap isn't a leap in position, it's a leap in energy levels. Think about it more as a standing wave on a guitar string - when you press on the string, the "note" suddenly "leaps" to a different energy level; but it still doesn't really have a position (of course, on a guitar string, the "leap" still involves emitting "junk" sound waves; an electron does not produce intermediate photons in its leap).

Possible duplicates: physics.stackexchange.com/q/135520/2451 , physics.stackexchange.com/q/262918/2451 and links therein.

@RyanThorngren probably means the probability at one specific point, as there are infinitely many other points to be at. However any given volume will have a finite probability. The $\Psi \left( r \right)$ of an $e^-$ in an atom measures the probability of the $e^-$ being in a thin shell of radius $r$ centered at the nucleus

The probability of the electron being in any particular place is zero but this has nothing to do with physics and everything to do with continuous probability distributions. It's just like saying that the probably that any two randomly chosen real numbers are equal is zero.

I think the problem is that you are conflating two different things. Energy levels are about a bound electron (e.g. in an atom); the electron there is this fuzzy object (a "cloud") over a certain region. The different energy levels are more or less in that same region (in the "same place" loosely speaking, but the shape of the cloud varies), it's just that a bound electron can't have any energy - i.e., there's no continuous transition. That's the "leap". The cloud is really a probability distribution, that, in this case, "fizzles out" asymptotically outside of the atom.

The probability of dropping a pencil and having it land at a specific angle is also zero, but I bet if you do it it will land at one angle

@Shep the probability is zero, or the probability approaches zero / can be considered zero?

@Blueriver Well is zero, because it is infinitely close to 0... So all three clauses of what you said are true - the probability approaches zero, can be considered zero, and is zero.

Seems important to stress the distinction between current models and reality. I mean, this question's focused on a seemingly absurd prediction that's never been validated (or even supported) by any experiment, ever -- so while it's worth noting what our models predict, it'd seem a tad silly to say that it's reliable.

(Just to note it, I think that most folks wonder this when seeing the $1\text{s}$ atomic orbital diagram which, naively, would seem to predict that a hydrogen-atom's electron could be a particle a trillion light-years away.)

@Nat That is a possibility right? Just a small one because the electron cloud predicts it would be much much closer to the atom.

@TheGodlyBeast: Modern physics isn't really evidenced to accurately predict quantum effects over trillions of light-years, so such predictions ought to be understood as speculative (or in this case, extremely speculative).

@llama I'm not convinced that all angles on the real spectrum are possible for a pencil to drop in. The contrary to this is that space is infinitely divisible, but the existence of the planck length seems to dispute that. Therefore, no the angle your pencil drops in did not have probability 0.

@Cruncher the Planck length is an artifact of limits of current theories of physics, not the universe itself. There isn't any good evidence for space itself being quantized at this point, and since it appears to not be in every experiment that we can currently run, I'm going to call it a pretty good assumption for now

the Planck length is an artifact of limits of current theories of physics, not the universe itself This is incompatible with it appears to not be in every experiment that we can currently run. If it appears to not be the case in every experiment we run, then it would contradict existing theories, and thus the theories would be discarded/refined.

@llama Consider the other implications if a pencil could in fact be in any angle on the real spectrum. That would mean that you could, in theory, encode an infinite amount of data into a pencil.