Solving Hydrogen Atom Orbital Puzzles: Nodes & Electron Movement

[justsof]

Hello,

probably a simple question but here it is.
Considering the orbitals of a hydrogen atom, let's take a spherical one. Now the radial wave function can have several nodes. (n-l I believe) So if I get this right, there are spherical shells around the nucleus where there is no chance of finding an electron when measured.
Since there is no chance of finding an electron there I would assume an electron does not cross this spherical shell.
So if one measures the electron to be a certain distance from the nucleus it couldn't move to another distance across a such a shell?
or could it because it is no real thickness...
I hope for some enlightning answer :).

Have a good day,
Arno

 

[Matterwave]

The electron doesn't really follow a trajectory like in classical physics. All you know about electrons is that you can find them at one place, or you can find them at another, if the wave-function is non-zero at those places.

But anyways, as far as I know, the radial wave-function should be exponential decay or polynomial multiplied by exponential decay, and should not be oscillatory...Can you show the radial wave-function you're looking at?

EDIT: Ahh, I'm sorry I didn't realize that even with polynomials and exponential one can get nodes.

In this case, refer to point 1 above and disregard point 2.

 

[epenguin]

You are right, different excited states have these nodes and successive spherical shells. The radial 'nodes' are actually spheres were the electron density is 0. But these are of infinitesimal thickness, and they are not potentials, not barriers. In fact the electron density is something (non-zero) everywhere except in these infinitesimal nodes. It goes out to infinity, it is just concentrated mainly though in one of these shells between a couple of nodes - far from the nucleus it gets very tenuous. It is usually pictured as a fuzzy cloud that is darker in a shell etc. - surely you have seen such a picture?

A way to think of it, and I stand to be corrected by proper physicists, is an electron is a wave. But you think of a wave as a thing that is spread over a long distance like sea waves, not much like some of the properties of electrons that you know, e.g. the ones making a spot on your TV screen. What makes it more like a lump of stuff though not a point particle is when the electron is confined by a force. The first such confinement they give you in textbooks (on order to start with the simplest one) is the (1 dimensional) infinite potential square well. That means no force on the electron except an infinite repulsive force at two points, so it has to stay between them. Reasonably intuitively or with fairly simple math you predict the wave function has 0, or 1, or 2, or 3,...(as the energy increases) regularly spaced nodes with bands of electron density i.e. probability of there being an electron between them (described by sines as in classical vibrating strings which habve nodes where the string is stationary, amplitude zero).

The atom is just the same with a less simple and drastic force that confines the electrons (now an inverse square law that tries to keep them near the nucleus) + 3 dimensions to think about. You have the same sort of waves but they are not so symmetric and are concentrated towards the atom's nucleus.

Not every student (and TBH not every textbook) makes the connection between the result of these two types of confinement, but the simpler situation can help in understanding why atoms are the way they are.

 

[varga]

1.) The electron doesn't really follow a trajectory like in classical physics.

2.) All you know about electrons is that you can find them at one place, or you can find them at another, if the wave-function is non-zero at those places.

This is what I wanted to talk about but my thread was closed, so let me point out my concern here. If 2. is true, than you really DO NOT KNOW what kind of trajectories they have, and there is no obvious reason why would 2. exclude the possibility of 1.


A.) Continuous trajectories, "SLIDING", any new location follows after old location in spatial and chronological sequence defined by the velocity vector at previous position.

B.) Non-continuous trajectories "APPEAR-DISAPPEAR", any new location(s) may be spatially separated from the previous location. "Velocity vector", if any, will NOT define new position and the chronological sequence of these locations may not be in direction of displacement.


I you are to suggest electrons do not move by "sliding" in continuous trajectories, then they can not have any real velocity as 'derivative of position' (as we know it), yet there has to be very specific geometric relation of the velocity vector, i.e. chronological displacement sequence, if electrons are to form orbital magnetic moment. Is electron velocity "real"? If yes, then what kind of velocity is that if it has no 'velocity vector'?

 

[Matterwave]

This is one reason why we usually don't refer to velocities in QM, we just refer to momentum.

But I cannot exclude the possibility of a true trajectory. All I can say is that in standard QM there is no well defined true trajectory. Perhaps you can take your argument up with someone who ascribes to the de-Broglie-Bohm interpretation of QM.

 

[PhaseShifter]

A.) Continuous trajectories, "SLIDING", any new location follows after old location in spatial and chronological sequence defined by the velocity vector at previous position.

You are still assuming the electron has a "position" and "velocity," and follows a trajectory. This model, as was explained several times in your thread, is not compatible with quantized energy states. The reasons for this had been known nearly 50 years before the Bohr model was ever proposed--it's just that the Bohr model conveniently fit the spectroscopic data until a better model came along.