Do electrons (individually) have definite and single value of momentum and position or do they simultaneously have multiple position (a spread) at a time?
In other words, according to the uncertainty principle, is it just impossible to measure the exact position and momentum or is it actually impossible for an electron to have a single position and momentum?
Consequently, assuming it's just impossible to measure the exact position and momentum, does this mean that in atomic orbitals the electrons actually do occupy single position at a certain time (instead of somehow having 'smeared out' multiple positions at a time).
How does velocity apply to atomic orbitals? Do electrons just randomly teleport according to the probability distribution or do they actually travel in 'normal' trajectories (going step by step instead of teleporting, for example footballs travel in step by step and don't randomly teleport AFAIK)
Edit: Also, does $\hbar$ in the uncertainty principle equation, $\Delta x \Delta p = \frac{\hbar}{2} $ arise from Planck length (From my understanding, the impossibility to measure the exact position and momentum arises from the quantization of length and time. Is my understanding correct?)
Best Answer
Here is an event with three electrons and two positrons in a bubble chamber picture.
A photon has come in from the left transferred a lot of its energy and momentum to an electron and a soft electron positron pair with the same vertex, and went along and created a second electron positron pair with higher energy, the electron ( possibly proton or nucleus) on which it interacted being a spectator and not visible in the chamber.
We know the position of the electron at the vertex with micron accuracy and its momentum from the curvature of the magnetic field with some measurement uncertainty. The delta(p)*delta(x) satisfied the HUP because h_bar is a very small number.
Were we able to measure the position with an accuracy of 10^-10meters we would be in the atomic world and the HUP would hold for the electron. with h_bar of the order of 6.5*10^_16eV*second we would still be ok with HUP for the momenta in these interactions.
The HUP is constraining when the orders of magnitude of x and p are both of atomic dimensions .
When one is in the dimensions where quantum mechanical dynamics holds one cannot think of particles as billiard ball analogues or of waves as water wave analogues. These electrons we see in the picture are "entities" described by Quantum mechanical solutions to the dynamics of the problem that we identify macroscopically as "electrons", in dimensions where h_bar is to all intents and purposes of measurement zero.
The quantum mechanical solutions tell us that electrons , i.e. entities that we can kick off as we did in the picture, are in orbitals around the nucleus of the atom, and have very specific quantum numbers and quatized energy; and these quantum numbers come from the solutions of the QM equations of the problem, and they identify orbitals for the electrons. If the atom is undisturbed there is no way to identify the location of an electron entity except by the mathematical solutions to the problem.
That is all we know, the interaction crossection probabilities, as is the case in the picture. It has no meaning to ask about an exact position of an electron, because the QM entity that manifests as an electron macroscopically can only be localized by probability functions: what is the probability to find an electron at a specific (x,y,z) if probed with a photon, for example.