Electron orbital velocity in hydrogen atom?

[DrDu]

Last but not least a small appetizer: of course one can study the velocity as a vector instead using the velocity squared. The interesting result is that the expectation value of the electron velocity vanishes for the ground state of the hydrogen atom. This sounds weird. Welcome to the quantum world!

I don't consider this to be so astonishing. Classically a state with vanshing angular momentum corresponds to the electron falling in a straight line through the nucleus until it reaches a turning point, falls again through the nucleus until it reaches the turning point on the other side. Clearly the expectation value of the velocity vanishes at any point.

 

[bhobba]

Exactly. But, on the other hand, classical predictions can be proven wrong,

Sure - but clasically you can in principle observe anything all the time - you can't do that in QM.

I'm saying QM doesn't really exclude continuous trajectories, and if it actually does, then I'd like to see some more convincing evidence.

You are misunderstanding the situation.

QM is silent about what's going on when not observed. We have all sorts of interpretations and some such as Bohmian Mechanics have exactly that ie a well defined position, velocity and momentum. That's not the issue - the issue is there in no way to PROVE it's like that. If you can you will get an instant Nobel prize and recognition as being up there with the greats like Feynman and Einstein.

That's what I'm talking about. I propose to calculate velocity from measured orbital magnetic moment via Biot-Savart law, then use that velocity to calculate momentum via p=mv, and finally then compare that result with experimental measurements. What do you think, would you be surprised if the result matched experiments? Is there really any reason to believe those relations would not be preserved after passing through classical equation

You will fail because QM explicitly says you can't do that. Specifically since the electron is bound to the atom we know its position to a fair accuracy. By the uncertainty principle since we have a reasonable idea of position, you can't know the momentum, or to be more precise it has greater variance.

And if you continuously measure it you run into the quantum Zeno effect:
http://en.wikipedia.org/wiki/Quantum_Zeno_effect

But write it up and submit it to a journal and see what they say.

Post it back here - it should prove interesting.

Thanks
Bill

 

[tom.stoer]

I don't consider this to be so astonishing. Classically a state with vanshing angular momentum corresponds to the electron falling in a straight line through the nucleus until it reaches a turning point, falls again through the nucleus until it reaches the turning point on the other side. Clearly the expectation value of the velocity vanishes at any point.

Coming from the Bohr model and circular electron orbits a vanishing \langle\vec{v}\rangle sounds weird!

 

[carrz]

Google for "stationary state quantum mechanics" and you'll find some good explanations.

I see. But that's not surprising at all even for Bohr's model. In chaos theory those states are called "strange attractors". You can see solar systems and galaxies settle in "stationary states" as well. It's just a natural way to conserve and balance the energy of the system, through the pathways of the least resistance. Natural laws are amazingly efficient, maximally so, for some reason. And the only reason there can possibly be is always only action and reaction, cause and effect. We can perhaps say that position of electrons in atomic orbitals is random, but if it was not caused, it could not be bound.

 

[DrDu]

Coming from the Bohr model and circular electron orbits a vanishing \langle\vec{v}\rangle sounds weird!

That's the main point I don't understand. Given the OP's question, why does everybody talk about Bohr's model?
An orbital is a solution of the 1 particle Schroedinger equation, and the question about velocity makes sense in QM, too. So no need to invoke Bohr.

 

[Nugatory]

That's the main point I don't understand. Given the OP's question, why does everybody talk about Bohr's model?
An orbital is a solution of the 1 particle Schroedinger equation, and the question about velocity makes sense in QM, too. So no need to invoke Bohr.

OP introduced the Bohr model back in post 3

 

[jtbell]

QM is silent about what's going on when not observed. We have all sorts of interpretations and some such as Bohmian Mechanics have exactly that ie a well defined position, velocity and momentum.

In Bohmian mechanics, the trajectories of the electron in a hydrogen atom are very non-"Bohrian". In fact, for states with a specific energy "level" and zero angular momentum, (the 1s, 2s, 3s etc. states), the electron is stationary!

http://www.bohmian-mechanics.net/whatisbm_pictures_hydrogen.html

(I haven't found examples for states with angular momentum, e.g. 2p.)

 

[carrz]

That's the main point I don't understand. Given the OP's question, why does everybody talk about Bohr's model?
An orbital is a solution of the 1 particle Schroedinger equation, and the question about velocity makes sense in QM, too. So no need to invoke Bohr.

I agree, although if it is directly related to velocity then we should consider it, whether the idea originated with Bohr's atom model or wherever else, like radiation problem for example.

I think many are looking in the wrong direction. Let's look at the "velocity" itself: v=s/t. It only says "rate of change in position", it does not actually say anything about successive locations being right next to each other, or not.

Pixels on a computer screen don't really move with any velocity, they teleport from their current position to a new location without existing anywhere in between, just as electrons supposedly do, and yet, even as such, pixels can still have "velocity". But what does it mean? Obviously it is related to how long those pixels exist in one location before they teleport to another, and it is also related to whatever time interval may be during this teleportation when they can be nowhere at all or at both locations at once. The point is there is more to "velocity" than just "continuous trajectory".

 

[sophiecentaur]

I agree, although if it is directly related to velocity then we should consider it, whether the idea originated with Bohr's atom model or wherever else, like radiation problem for example.

I think many are looking in the wrong direction. Let's look at the "velocity" itself: v=s/t. It only says "rate of change in position", it does not actually say anything about successive locations being right next to each other, or not.

Pixels on a computer screen don't really move with any velocity, they teleport from their current position to a new location without existing anywhere in between, just as electrons supposedly do, and yet, even as such, pixels can still have "velocity". But what does it mean? Obviously it is related to how long those pixels exist in one location before they teleport to another, and it is also related to whatever time interval may be during this teleportation when they can be nowhere at all or at both locations at once. The point is there is more to "velocity" than just "continuous trajectory".

Your analogue of moving pixels is an interesting one and you can take it further. I think it actually shows where your ideas are 'not entirely correct' and it's interesting that you introduced it. The pixels don't "teleport", the value of one changes as the value of the next one changes. The pixels are not virtual (real LEDs perhaps) but the image is only shown by the pattern they form. If you look at models of the Schrödinger solution, they show fuzzy areas (displayed on pixels. They don't show anything moving (except for the different shape of each solution as the energy changes. You would find it hard to identify which of those changing (not moving) pixels was the real one which described the bound electron in the real atom. This is a direct parallel with the probability distribution function and the idea of an electron actually 'being' somewhere.

 

[tom.stoer]

The point is there is more to "velocity" than just "continuous trajectory".

Right!

But all what you are saying is related to classical mechanics and therefore becomes inapplicable, meaningless or wrong when applied to quantum systems.

For example v=ds/dt does no longer make sense.

 

[Nugatory]

Let's look at the "velocity" itself: v=s/t. It only says "rate of change in position", it does not actually say anything about successive locations being right next to each other, or not.

On the contrary, it does. The definition of the derivative includes a requirement that the function be continuous at the point where we're differentiating, and the method by which we calculate it doesn't work without that continuity assumption.

 

[carrz]

We know from observation of atomic spectra that there are discrete energy levels for the bound-states of electrons. From the energies of the photons we can calculate the energies of these bound states. The Bohr model reproduces (more or less accidentially!) the energy levels of the Hydrogen atom. But the Bohr model fails for a couple of reasons:
1) according to classical electromagnetism an orbiting charge must radiate with a continuous spectrum; this forbids stationary orbits; but the atoms do not have continuous but discrete spectra; and they have stationary ground states which do not radiate at all
2) the Bohr model fails completely for atoms with more than one electron
3) even for the hydrogen atom we observes corrections to the energies calculated with the Bohr model

I'll only talk about the 1st one as I don't see other two are related to velocity or continuous trajectories. -- It is only "accelerating" charges which supposedly must radiate. The question here is whether orbiting with uniform velocity is really that kind of acceleration that is supposed to produce radiation. I could say for example that electrons are really moving in a straight line and its only the space-time which is curved around there, but this wouldn't hold true for the "real acceleration", so there is definitively some kind of differences between the two, and I guess that one of those differences could also be that "fake acceleration" does not really produce charge radiation.

 

[Nugatory]

You can see solar systems and galaxies settle in "stationary states" as well.

You may have misunderstood what a quantum mechanical stationary state is. Neither solar systems nor galaxies are ever in a stationary state, and it is precisely because they aren't in stationary states that classical notions of position and velocity emerge.

We can perhaps say that position of electrons in atomic orbitals is random

We can say that, just as we say can saypretty much anything we please about things that don't exist: "All eight-legged egg-laying horses are carnivorous" is a true statement, and doesn't even conflict with the equally true statement "No horses are carnivorous".

 

[carrz]

That's not the issue - the issue is there in no way to PROVE it's like that.

I'm not sure what it would prove, if anything, but if I take experimental measurements of electron orbital magnetic moment, pull it through classical magnetic and momentum equations, and arrive back at experimentally measured value, it would imply velocity still has some meaning, whatever that meaning is.

I don't really care whether the result would match or not, I'm just curious, and it would be convincing either way. So if it turns out the result does not match, then I'll start considering all the craziness of QM might actually be really true.

You will fail because QM explicitly says you can't do that. Specifically since the electron is bound to the atom we know its position to a fair accuracy. By the uncertainty principle since we have a reasonable idea of position, you can't know the momentum, or to be more precise it has greater variance.

I don't care about position at all, just momentum, and orbital magnetic moment. Fail or not, I can't even begin if we can not actually measure those. Can we?

 

[sophiecentaur]

I get the feeling that you are so desperate not to be 'wrong' that your position is getting too entrenched for you to see the logic of all this. You don't need our approval to do those sums that you propose so just do them. It is really a waste of all of our time to pursue this as you don't want to follow the advice you've been given (by many of us).
You must either be convinced we are all wrong or you really don't understand what you are being told. Read elsewhere and see what we are saying is pretty well universally accepted.

 

[carrz]

Your analogue of moving pixels is an interesting one and you can take it further. I think it actually shows where your ideas are 'not entirely correct' and it's interesting that you introduced it. The pixels don't "teleport", the value of one changes as the value of the next one changes. The pixels are not virtual (real LEDs perhaps) but the image is only shown by the pattern they form. If you look at models of the Schrödinger solution, they show fuzzy areas (displayed on pixels. They don't show anything moving (except for the different shape of each solution as the energy changes. You would find it hard to identify which of those changing (not moving) pixels was the real one which described the bound electron in the real atom. This is a direct parallel with the probability distribution function and the idea of an electron actually 'being' somewhere.

I don't see any difference. Are you saying that for some specific instant in time electron is not actually anywhere, or that it is a little bit of everywhere at once?

 

[bhobba]

Are you saying that for some specific instant in time electron is not actually anywhere, or that it is a little bit of everywhere at once?

Neither.

Its a meaningless question in the QM formalism - its silent about it.

Thanks
Bill

 

[carrz]

For example v=ds/dt does no longer make sense.

Pixel moved 10 pixels in 20 seconds: v= 10/20 = 0.5 pixels/per second. If the distance between pixels was 4 millimeters it would be 2 millimeters per second, and you can have any distance between individual locations in the sequence, they do not need to be "continuous" at all, and velocity still works and has the same meaning.

In fact, if you don't accept the universe is digital at some level, rather than being analog or continuous, then you have a problem to solve Zeno's paradox and explain turtles running faster than Greek heroes.

 

[Nugatory]

Pixel moved 10 pixels in 20 second: v= 10/20 = 0.5 pixels/per second. If the distance between pixels was 4 millimeters it would be 2 millimeters per second,

Except that nothing is moving here at all. I have two LEDs, stationary and 2mm apart, and first one lights and then the other.

 

[micromass]

I think we can lock this thread here.