Classical: it’s like a miniature solar system, with protons and neutrons glued together in the nucleus, and electrons orbiting them.

Quantum mechanics: it’s wrong to think of the electrons as “orbiting” the nucleus, or indeed, as being anywhere at any given moment. An electron’s position is really described by a probability distribution, which allows you to calculate the probability of finding it at any point at any given time, within the limits of the Uncertainty Principle. An electron is essentially a smeared-out cloud of probability.

I thought the QM defintion was pretty much accepted, and the classical one considered hopelessly naive and outdated - but I chanced upon a comment in a Scientific American from a couple years back where they said something like, “the new lasers are capable of attosecond pulses; for comparison, it takes an electron about 150 attoseconds to orbit the nucleus of a hydrogen atom.” That sounds very “classical” to me.

It’s quantum mechanics, no question. The “classical” model has not been regarded as realistic for nearly a century. I wouldn’t take the offhand use of the word “orbit” by a science writer for Scientific American to have any particular significance.

Even though the electron can’t be viewed as “orbiting” the electron in any sense, you can still associate a time scale with the wave function using expectation values. Essentially, the expectation value of some property is what you’d expect to get if you measured that property of the electron a bunch of times and averaged all of the measurements. So you can figure out the expectation values of the momentum and distance from the nucleus for an electron in the ground state of an atom. Then you can pretend that the electron is really a classical particle orbiting the nucleus at that radius with that momentum, and figure out how long it would take to do so; the result is indeed about 150 attoseconds.

Physicists use this kind of heuristic “semi-classical” calculation all the time when dealing with quantum mechanics, but we always keep in mind that much more complicated things are really going on underneath the easily visualized classical picture. So either a physicist wrote the article and forgot to (or decided not to) mention that this was an oversimplification; or a science journalist was told this factoid by a physicist they were interviewing and didn’t bother to ask for explanation.

It’s worth pointing out that you common-sense, “billiard ball” picture of the electron leads you astray when trying to picture what’s going on. The s = 0 orbital has no angular momentum, and the wavefunction is actually highest at the center. It’s pretty hard to square that with any classical pictutre of an orbit, where the angular moment can’t be zero, and the probability of being at the center of mass of the system would also be zero.

The p orbitals aren’t any better. The “dumbell-shaped” orbitals aligned with the three axes look as if the electron must be zooming around from one lobe to the other. But if that were really happening the highest probability would be in the center. But it isn’t =-- the wavefunction is zero at the center. Classuically, with an angular momentumk of one unit, the electron shouldn’t be able to go through the center, anyway.