From reading the article, I think they mean “determined” in the sense of controlled. They changed the potential distribution in the electrodes, and the single-electron wavefunction changed in a controlled way. The “geometry” is presumably the spatial probability distribution, including the spin.
Yes it does,from the abstract of the original article: "We show that in-plane magnetic-field-assisted spectroscopy allows extraction of the in-plane orientation and full 3D size parameters of the quantum mechanical orbitals of a single electron GaAs lateral quantum dot with subnanometer precision."
Well, if they're confined they're not free. I guess they mean the spatial probability density (modulus squared of the orbital wavefunction in an atom, you might skip the orbital label if the potential is not central, but it's the same thing) of the electron in these artificial atoms. The title is misleading because the "geometry" of the electron is point-like (to all effects in this case).
Free depends on the context. These electrons are in a potential well, they're in a bound state. Actual atoms are just an example of that. My point was that conceptually both situations are alike, as they are bound states you get discrete spectra (roughly speaking), if they were free particles you wouldn't. (Maybe those levels happen to be very close, but that's a different story. In principle the artificial atom name makes sense and talking about orbitals is not a bad analogy.)
The spherical harmonics represent the analytic solution to the Schrodinger equation for a hydrogen in a vacuum. Any other system (e.g. Hydrogen with two electrons, or a neutral Helium atom) does not have any analytical solution (its a many body problem).
In general, determining the wave function for a collection of electrons is a very hard problem. Depending on the accuracy needed, different commonly used methods range from O(n^4) to O(n^7) where n is the number of electrons.
Platonic? It isn't exactly the spherical harmonics, because of the radial term, relativistic corrections and stray electromagnetic fields that are always around.
That's what makes it Platonic -- there is no perfect sphere, triangle, etc... But material substances approximate spheres, etc as a platonic form.
Also "platonic", because their school of thought held that the properties of matter were based on the geometries of their component atoms, which they assumed to be the simplest forms possible (platonic regular forms). They were wrong on the specifics (atoms aren't made of tetrahedra or cubes), but the basic atomic theory (atomic geometry determines functional properties) remains an essential insight.
This is a fascinating notion to me, and while generally I get things I have no idea what foundation I would need to understand this. What exactly is going on?
The scale of this experiment is pretty big in quantum terms, so a semi-classical explanation is not inaccurate. The electron is behaving like water filling a bowl. The height of the bowl floor represents how much potential energy the electron needs to reach that point in space.
Suppose that the bowl is invisible, and detailed knowledge of its shape is of scientific interest for one reason or another. Here, it's interesting because they want to fine-tune their ability to spatially and electromagnetically control individual electrons so that they can explore "spintronics".
One way to measure the shape of the bowl is to measure the shape of the electron "fluid" filling the bowl. They've developed some technique for doing that. I haven't read the details.
That only gives you information about the bottommost part of the bowl, where the fluid lies. By applying voltages to the electrodes, the authors can raise the potential energy needed for the electron to hang out at one end of the bowl relative to the other. This effectively tips the bowl, and the electron "fluid" re-distributes itself, allowing them to measure the shape of the bowl at locations other than the bottom.
That's all I get from the article. You'd have to read the linked original paper to learn more. A traditional undergraduate series in classical mechanics, electromagnetics, waves, and quantum mechanics is very helpful, but cutting-edge research is never communicated in the same terms that are used in undergraduate teaching. There are going to be some jargon barriers no matter what. Happy reading!
Quantum mechanics mostly but with impacts in quantum computing. If you come from the computing side and want to tangentially back into quantum mechanics from the quantum computing side, I would highly recommend Andy Matuschak and Michael Nielsen's highly approachable intro to quantum computing which goes into surprising depth - https://quantum.country/qcvc
Once you feel like you have a good grasp, just dive into the deep end (why not?) with a pictorial survey of the geometry of quantum states - https://arxiv.org/pdf/1901.06688.pdf
Then see where the gaps are and decide if it's worth digging in deeper to understand this better :-)
Undergrad level quantum mechanics is enough to have a basic idea of what is going on. The electron position is calculated as a time- and space- dependent probability density that can also be time-independent if certain conditions are met. This probability density is proportional to the magnitude squared of the wavefunction. The wavefunction is a solution to a partial differential equation like Schrodinger's equation given a specified Hamiltonian and boundary conditions. In this experiment, the potential energy portion of the Hamiltonian (total energy) is controlled via externally applied electric fields. Designing the quantum dot to control the spatial pattern of these electric fields is the secret sauce of the experiment that is not so easily understood at an undergrad level, but as a first pass you can consider canonical problems like the quantum harmonic oscillator. It's been a while, so others should feel free to correct me if I said anything inaccurate.
As other commenters have noted, the title is mildly annoying. Electrons are structureless, one of the most beautiful facts about nature. Euclidean points actually exist all over. And have spin. :-)
Well, 'Euclidean points' (can) have precise coordinates, whereas elementary patricles don't. (Besides, thinking of particles as points is at odds with their being excitations in the respective field.)
They’re theorized to be fundamental particles. Nothing smaller exists within them. There’s no nuclear decay of or fusion which produces electrons. Similar to how photons are quanta of energy.
Edit: clarified that electrons don’t decay. Of course things decay into electrons plus other things.
Er ... there are lots of nuclear decays which produce beta particles electrons. Neutrons have a short half-life when free of a nucleus, and decay into a proton, electron and an electron anti-neutrino (to conserve momentum). [1]
Electrons are posited to be fundamental, and e-e e-e<sup>+</sup> collisions haven't, as far as I know though well outside my original field of study, produced any data suggesting internal structure.
This doesn't mean that they don't have structure, we simply have no theory (that I know of, but then again, I'm a former solid state guy) that predicts structure, nor do we have sufficiently powerful colliders to get us to a point to see such structure.
I found the article uninformative - nowhere in it could I find a picture of this "geometry". The incomprehensible diagrams were nice, but only because I collect that sort of thing.
