[habbababba]

I have stumbled upon this article and the phrase 'The First Image of a Hydrogen Atom's Orbital Structure' sounded to me misleading. How can orbitals be observed if they are mathematical constructs? And how does this suggestion fit into the Heisenberg uncertainty principle to start with?

[Corribus]

Orbitals are constructs in that they are (with the exception of hydrogenic atoms) mathematical approximations of where electron density is in space on average. They are derived from first principles considerations. Hydrogenic wavefunctions/orbitals are exact analytical solutions to the Schrodinger equation (or equivalent), but again they are still based on first principles considerations. Molecular orbitals are more artificial, based as they are on even more approximations, but they still offer good agreement with experimental data in most cases. It shouldn't be surprising, based on the success of QM over the last century, that an actual experimental measurement of electron density in space around an atom should be in good agreement to theoretical predictions of electron density. You can take Newton's Equations and predict where a cannon ball will land when fired from a hill at a certain angle, and indeed the results will match well with the predictions.

[Borek]

IOW: the electron density around nucleus is a real, measurable quantity. Statement "orbitals are mathematical constructs" means "it is our way of finding out how the electron density looks like" - and it works quite good.

Heisenberg principle is in no way violated. It says there are limitations to how precisely we can know where the electron is and how fast it moves, but it doesn't in any way prohibit us from measuring - quite precisely - electron density, as it is an averaged quantity.

[habbababba]

With all appreciation to your contributions, my question has not yet been addressed appropriately.

As you said, orbitals are the solutions to the Schrodinger equation. But they are just that! Not sure if this is too loosely put, but those solutions stand for the possible positions occupied by the electron. This arises from the limitations set by the Heisenberg uncertainty principle. So the probable nature of the position of the electron inside of the atom is an effect of the Heisenberg uncertainty principle, not the other way around. You mentioned, Borek, that the electron density is an average quantity. If I'm to find the average velocity of a cannon ball, I would need precise velocities of the cannon ball at certain times. The same idea applies to this experiment, if orbitals, as you say, are average quantities, then 'measuring' or 'observing' those orbitals means that precise, individual data were actually measured, collected and then computed to get this average quantity you mentioned, and therein lies the violation to the Heisenberg uncertainty principle: those precise individual data could not have been measured if the uncertainty principle were to be respected and hence our orbitals cannot be observed.

[Enthalpy]

Do you like these orbitals? I find them cute.
http://education.mrsec.wisc.edu/SlideShow/slides/scanning/pentacene.html
http://www.nature.com/nchem/journal/v3/n4/fig_tab/nchem.1008_F3.html

In other words, orbitals are observed and measured, which QM calls "real".

One very useful lesson from these observations is that the atomic force microscope interacts all the time with the same pair of electrons, all over the probe's path over the molecule. This is important to meditate, since it debugs the very (most?) common misunderstanding of QM brought by the two-slit experiment, which would want wavefunctions to be observable only as statistics over many particles. It shows as well that interactions (here between the pentacene molecule's HOMO electrons and the probe's tip electrons) don't necessarily destroy particles nor disrupt completely the wavefunction.

Instead of "the possible positions of the electron" I prefer to say "the electron is the wavefunction" and "its shape is 2p". So did Schriffer by the way (from the BCS theory). The only drawback I've seen up to now is that the electron doesn't repel itself.

A drawback of "the possible positions of the electron" is that interactions are never points. Interactions span the whole volume common to both particles, and due to the finite energy, this volume is never a point. When an electron absorbs a photon in a semiconductor, the electron stretches over many 1000 atoms before and after the absorption, and the absorption itself stretches over such a volume - this is observed in superlattices and quantum wells, where for instance the polarisation of an emitted photon results from the orientation of the electron's wavefunction over the device, which a single atom couldn't explain.

So because wavefunctions are observed while point particles aren't and can't be, I like less the "the possible positions of the electron" which suggests false interpretations, and prefer "the electron is a wave" with the caution that its charge doesn't repel itself.

[Enthalpy]

About the uncertainty: the size of an orbital (I say: of an electron) results from the electron's kinetic energy. The orbital's size is just a position uncertainty or delocalization. Heisenberg's limit is attained by Gaussian wavefunctions; being no Gaussian functions, orbitals don't attain this limit, but are not far from it.

As a consequence and as you pointed out, an electron with the same kinetic energy as the trapped one could not measure a position more accurately than to one orbital (or atom size). Better accuracy needs a heavier particle or a more energetic electron.

Beware, though, that Heisenberg's uncertainty applies to one measure only. If you measure over many events, the statistics reduces the uncertainty, so you get a better observation. Take a telephone modem for instance: when the signal is clean (=many particles) it can transmit 56kb/s over only 3100Hz bandwidth, which violates Heisenberg's energy-time uncertainty telling that it takes 1/2pi seconds to distinguish two signals spaced by 1Hz. That would have been with a bad signal-to-noise (=one single particle).

Now, take a scanning electron microscope. They see individual atoms presently, but imagine a better one in the future, with a resolution better than one atom. It sends electrons with 100keV or 1MeV (not 10eV as the kinetic energy in a hydrogen atom) to the atoms in a solid target. A magnetic lens focusses each electron to such a small area.

Thanks to the higher energy, the probing electron can be smaller than the target one. Sometimes they interact, what we observe at the detector because the sensing electron was deflected. This happens only if both electrons were near enough to an other because the energetic electron needs much force to deviate, hi Heisenberg. Then we can say "this time the wavefunction of the target electron reduced its volume to that little around the rather well known position of the sensing electron" and for instance the target electron has been ejected from a position more accurate than an atom's volume.

Over many sensing electrons and target electrons (replenished by conduction at the solid target) we can reconstruct a probability for the target electron to concentrate around any position: a |psi|².

The interesting part of the idea of a particle is that some electron's properties like the charge are kept as integer numbers when the wave (I say the electron) changes its size and shape in an interaction.

The less useful and potentially misleading part in the particle concept would be to say "electrons are points" or "possible positions" because we have no means to observe an electron as a point.

Electrons are points in the limited sense that at any energy (TeV) accessible to humans, they still behave like elementary particles, and if they concentrate to the corresponding volume, they keep their usual attributes unsplit. An other argument is that electrons' Landé factor hints to an elementary particle.

And it remains that, whether you imagine electrons as diffuse or not, in Schrödinger's equation you need q²/d from the distance to the nucleus, but you need no q²/d where d would be the distance between varied positions in the electron's shape and q a charge density around these positions. So some mystery remains.