Electron orbital velocity in hydrogen atom?

[carrz]

Are there any actual numbers for electron orbital velocity in hydrogen atom?

 

[sophiecentaur]

No, because you cannot model an electron, bound to a nucleus as an orbiting object. The Bohr atom model is not a valid one - which is why it was replaced some time ago by the model which treats the electron, when bound to an atom, as a wave or a probability density function. That is why the energy states are nowadays referred to as 'orbitals' and not 'orbits'. The term 'orbital' should. perhaps, not even be used but, for the sake of Chemists, it seems to be.

 

[carrz]

No, because you cannot model an electron, bound to a nucleus as an orbiting object. The Bohr atom model is not a valid one - which is why it was replaced some time ago by the model which treats the electron, when bound to an atom, as a wave or a probability density function.

Is there some velocity associated with electron in Bohr atom model?

That is why the energy states are nowadays referred to as 'orbitals' and not 'orbits'. The term 'orbital' should. perhaps, not even be used but, for the sake of Chemists, it seems to be.

But if we can measure electron orbital magnetic moment then I should be able to work out some velocity from Biot-Savart law, right? Can we measure electron orbital magnetic moment in hydrogen atom?

 

[sophiecentaur]

Is there some velocity associated with electron in Bohr atom model?




But if we can measure electron orbital magnetic moment then I should be able to work out some velocity from Biot-Savart law, right? Can we measure electron orbital magnetic moment in hydrogen atom?

You can get a number but what meaning would it have if the electron is not in one particular place at any time? You can only have a velocity if there is a displacement / position.

 

[carrz]

You can get a number but what meaning would it have if the electron is not in one particular place at any time?

I think it can only mean "rate of change in position", in units distance per unit time. But whatever the meaning it should be related and proportional to its momentum. Wouldn't you agree?

I know we can not quite measure it, but are you sure QM actually forbids electrons to move in continuous trajectories? And why is that again, uncertainty principle, radiation maybe? It was radiation problem Bohr atom model suffered mostly, wasn't it?

You can only have a velocity if there is a displacement / position.

Isn't that true for a magnetic moment as well? Both the magnitude and orientation of a magnetic field is defined by the charge velocity vector, isn't it?

 

[sophiecentaur]

I think it can only mean "rate of change in position", in units distance per unit time. But whatever the meaning it should be related and proportional to its momentum. Wouldn't you agree?

I know we can not quite measure it, but are you sure QM actually forbids electrons to move in continuous trajectories? And why is that again, uncertainty principle, radiation maybe? It was radiation problem Bohr atom model suffered mostly, wasn't it?


Isn't that true for a magnetic moment as well? Both the magnitude and orientation of a magnetic field is defined by the charge velocity vector, isn't it?

My take on this is:
When an electron is in a bound state, it doesn't have a defined position so what meaning can velocity (= rate of change of position) have? If you stick the electron in a vacuum and fire it in a direction, so it is not bound, then you can assign it a position and momentum (subject to Heisenberg) and predict when it will hit a target, for instance. When it's bound, the momentum is known accurately, because of its defined energy state so its position is not knowable to finer than a fuzzy region, given by the wave equation solution.
I don't think there's much point in trying to use the Bohr model for anything else but interesting History discussions. We know it's flawed.
As for magnetic moment, there are a whole range of orbits and velocities for a classical orbiting charge which will produce the same field.

 

[Nugatory]

I know we can not quite measure it, but are you sure QM actually forbids electrons to move in continuous trajectories? And why is that again, uncertainty principle, radiation maybe? It was radiation problem Bohr atom model suffered mostly, wasn't it

For bound states, yes, QM says that the electron does not have a continuous trajectory. This is neither the uncertainty principle nor the answer to the radiation problem (which appears in the Rutherford model not the Bohr model), although both of these results fall out of the solution of the Schrodinger equation for the bound states of the hydrogen atom, along with the lack of a continuous trajectory.

Isn't that true for a magnetic moment as well? Both the magnitude and orientation of a magnetic field is defined by the charge velocity vector, isn't it?

Not in QM. This is most clear in the case of the intrinsic magnetic moment of subatomic particles, which is associated with the (sadly misnamed - nothing is spinning) quantum mechanical property of spin, but it's also the case for the magnetic moment associated with the (equally misnamed) orbital angular momentum of the electron.

There are a number of sites (including wikipedia) that have pretty decent explanations of what the solutions to the Schrodinger equation for hydrogen look like.

 

[carrz]

When it's bound, the momentum is known accurately, because of its defined energy state so its position is not knowable to finer than a fuzzy region, given by the wave equation solution.

That's even better, I'll try to find that momentum number of an electron in hydrogen atom and then I'll work out some velocity from there.

Not knowable position doesn't really mean "cannot move in continuous trajectory". What precisely is the equation, theory, or experiment that is supposed to forbid electrons to move in continuous trajectories when bound to an atom?

 

[carrz]

For bound states, yes, QM says that the electron does not have a continuous trajectory. This is neither the uncertainty principle nor the answer to the radiation problem (which appears in the Rutherford model not the Bohr model), although both of these results fall out of the solution of the Schrodinger equation for the bound states of the hydrogen atom, along with the lack of a continuous trajectory.

Are you saying it is the Schrodinger equation which somehow implies electrons can not move in continuous trajectory when bound to an atom? Can you explain?

There are a number of sites (including wikipedia) that have pretty decent explanations of what the solutions to the Schrodinger equation for hydrogen look like.

I don't see how a probability cloud exclude continuous trajectories. It seems to me Schrodinger equation can be applied to classical systems, like solar system, just the same.

 

[Nugatory]

Are you saying it is the Schrodinger equation which somehow implies electrons can not move in continuous trajectory when bound to an atom? Can you explain?

Look at the solutions of the Schrodinger equation for the bound electron in a hydrogen atom. Calculate the expectation value of the position as a function of time in these solutions. Does this describe a continuous trajectory? No.

Now look at the solutions of the Schrodinger equation for an unbound particle moving through a cloud chamber or similar device. Calculate the expectation value of the position as a function of time for these solutions (this is the "Mott problem" - google for it). Does this describe a continuous trajectory? Yes.

It seems to me Schrodinger equation can be applied to classical systems, like solar system, just the same.

It can be. However the solution for a collection of 10⁵⁰ atoms traveling in close formation (that's what a planet is) in the gravity well of a mass 10¹² meters distant doesn't look anything like the solution for a bound electron. Instead, it predicts pretty much exactly the same thing as classical mechanics does.

 

[Matterwave]

You can easily calculate the "velocity" in the Bohr model.

In the Bohr model, the angular momentum is quantized $L=n\hbar$, classically (the Bohr model is a semi classical model) the velocity is related to the angular momentum $L=mrv$. Therefore very simply we have:

$$v=\frac{n\hbar}{mr}$$

So for the ground state, the radius is the Bohr radius $r=a_0$ and n=1 so:

$$v=\frac{\hbar}{ma_0}\approx 2*10^6 m/s$$

What this value means...probably nothing...

 

[sophiecentaur]

Are you saying it is the Schrodinger equation which somehow implies electrons can not move in continuous trajectory when bound to an atom? Can you explain?




I don't see how a probability cloud exclude continuous trajectories. It seems to me Schrodinger equation can be applied to classical systems, like solar system, just the same.

From your remarks here and elsewhere, I can see that you are too much wedded to the Classical model to make any significant progress with this. You cannot ignore QM and, to get on with QM you just have to 'think different'. This was a problem everyone had a hundred years ago but there was a sea change in opinions and they managed to bring us to where we are today.
If you insist on a classical approach, you may well get a numerical answer but it will have no significance. You may as well include the base dimensions of the Great Pyramid in your calculations. It just cannot be as simple as you are making out or there would never have been the need to go to QM and beyond.

 

[carrz]

Look at the solutions of the Schrodinger equation for the bound electron in a hydrogen atom. Calculate the expectation value of the position as a function of time in these solutions. Does this describe a continuous trajectory? No.

You can replace Coulombs potential with gravity potential and Schrodinger equation will describe the same thing for planetary orbits, and you can also get their harmonic oscillator or wave function, even though planets move in continuous trajectories, or maybe better to say, because of it.

 

[sophiecentaur]

You can do lots of Maths and get lots of different, mathematically correct, answers but do they mean anything in the real world? I am not sure what you hope to get out of this. You asked a question at the start yet you seem to be telling us 'your' answer, which, afaics, is counter to accepted Physics. All the replies are telling you this, it seems.

 

[Nugatory]

You can replace Coulombs potential with gravity potential and Schrodinger equation will describe the same thing for planetary orbits, and you can also get their harmonic oscillator or wave function, even though planets move in continuous trajectories, or maybe better to say, because of it.

You can indeed make that replacement and solve for the energy eigenstates of an orbiting planet under the assumption the planet represents a single quantum object in the same way that an electron does. However, unlike the electron, the planet consists of an enormous number of individual particles with no particular coherency relationship between them. Thus, this assumption is an unrealistic oversimplification; it's only useful for showing that the energy eigenstates are so close to one another that we can treat them as continuous (again, unlike the electron). The classical trajectory that we observe for classical objects like planets emerges when we do not make this oversimplification.

 

[carrz]

You can do lots of Maths and get lots of different, mathematically correct, answers but do they mean anything in the real world?

I was thinking the same thing. I know what F=ma or v=s/t mean, there is a reason or causality implicit in those equations, but what does Schrodinger's equation mean? That electrons vanish and then pop up into existence someplace else, for no reason, without causality? I'm not convinced, and I don't think Schrodinger's equation carries any such meaning.

It's like an equation to tells us the odds of getting a 6 out of 10 dice rolls. But it can not tell us which side will turn up on each try, and similarly it is unreasonable to expect of Schrodinger's equation to be anything more than just statistical. It tells us general odds, but nothing about specific time or specific location, and so I conclude there is plenty of room for any kind of trajectories in that equation, including continuous one.

I am not sure what you hope to get out of this. You asked a question at the start yet you seem to be telling us 'your' answer, which, afaics, is counter to accepted Physics. All the replies are telling you this, it seems.

I was thinking to calculate orbital magnetic moment or electron's momentum and see how it compares with experimental measurements. Would you be surprised if got some velocity from measured orbital magnetic moment via Biot-Savart law, then used that velocity to calculate momentum, and it turns out it matches experimental measurements?

 

[carrz]

You can easily calculate the "velocity" in the Bohr model.

In the Bohr model, the angular momentum is quantized $L=n\hbar$, classically (the Bohr model is a semi classical model) the velocity is related to the angular momentum $L=mrv$. Therefore very simply we have:

$$v=\frac{n\hbar}{mr}$$

So for the ground state, the radius is the Bohr radius $r=a_0$ and n=1 so:

$$v=\frac{\hbar}{ma_0}\approx 2*10^6 m/s$$

What this value means...probably nothing...

I'd say that's a reasonable number. By the way, do you know in what units would electron orbital magnetic moment be measured?

 

[carrz]

You can indeed make that replacement and solve for the energy eigenstates of an orbiting planet under the assumption the planet represents a single quantum object in the same way that an electron does. However, unlike the electron, the planet consists of an enormous number of individual particles with no particular coherency relationship between them. Thus, this assumption is an unrealistic oversimplification; it's only useful for showing that the energy eigenstates are so close to one another that we can treat them as continuous (again, unlike the electron). The classical trajectory that we observe for classical objects like planets emerges when we do not make this oversimplification.

Ok. Do you know if electron orbital magnetic moment can actually be measured? I imagine it would be very difficult to decouple from its spin magnetic moment and that of the proton. Also momentum, can we actually measure it, and how?

 

[Nugatory]

It's like an equation to tells us the odds of getting a 6 out of 10 dice rolls. But it can not tell us which side will turn up on each try, and similarly it is unreasonable to expect of Schrodinger's equation to be anything more than just statistical. It tells us general odds, but nothing about specific time or specific location, and so I conclude there is plenty of room for any kind of trajectories in that equation, including continuous one.

You may not have noticed, but you've subtly changed the subject. It was "What does the Schrodinger equation and the formalism of QM say?" and now you're suggesting that whatever they say, there's more to the story than that.

You are describing a hypothetical ("there is room for" means "cannot be excluded") hidden variable theory in which the position of the particle is a hidden variable.

If you're not already familiar with the DeBroglie-Bohm pilot wave interpretation, you may want to spend some time reading about it.

However, you'll find yourself in the dead end that all discussions of interpretations lead to: There is no way of extracting from a non-local hidden variable model (Bell/Aspect experiments have rejected the local hidden variable possibility) any prediction that will be different from the statistical predictions of quantum mechanics.

 

[Nugatory]

I'd say that's a reasonable number. By the way, do you know in what units would electron orbital magnetic moment be measured?

It's a reasonable number in the same way that 75 kilograms is a reasonable number for the weight of a large unicorn - unicorns don't exist, but if they did I'd expect them to weigh within a decimal order of magnitude or so of what a good-sized antelope weighs. However, just as I have no idea what good an estimate of the weight of a non-existent unicorn is, I'm not sure what you can do with that reasonable number.

As for the units of magnetic moment... This is one of the things that wikipedia wil get right: http://en.wikipedia.org/wiki/Magnetic_moment#Units

 

[carrz]

If you're not already familiar with the DeBroglie-Bohm pilot wave interpretation, you may want to spend some time reading about it.

I didn't know that works as an atomic model theory, I only know about it for the double-slit experiment. Is it one of accepted "mainstream" explanations, and it has continuous trajectories?

 

[carrz]

As for the units of magnetic moment... This is one of the things that wikipedia wil get right: http://en.wikipedia.org/wiki/Magnetic_moment#Units

Thanks.

Only, it doesn't seem to match Biot-Savart law:

I get joules per meters squared, and I thought I was supposed to get tesla, but tragedy is that neither match those "magnetic moment" units. I can not compare unless they have the same units. What am I missing here?

 

[Drakkith]

It's like an equation to tells us the odds of getting a 6 out of 10 dice rolls. But it can not tell us which side will turn up on each try, and similarly it is unreasonable to expect of Schrodinger's equation to be anything more than just statistical. It tells us general odds, but nothing about specific time or specific location, and so I conclude there is plenty of room for any kind of trajectories in that equation, including continuous one.

I don't think this can fit with observed phenomena such as quantum tunneling.

Edit: I've moved this thread to the Quantum Physics forum where it belongs.

 

[bhobba]

Is there some velocity associated with electron in Bohr atom model?

You need to step back a bit and think in terms of the formalism of QM. The Bohr model was simply a stepping stone to that. And it had flaws eg:
http://nsb.wikidot.com/pl-9-8-1-6

Bohr's biggest contribution in his model was to introduce quantum principles to classical physics, but his model had a few limitations

Spectra of Large atoms:
The Bohr model could only successfully explain the hydrogen spectrum. It could NOT accurately calculate the spectral lines of larger atoms.

The model only worked for hydrogen-like atoms
That is, if the atom had only one electron.

Relative Spectra Intensity
Bohr's model could not explain why the intensity of the spectra lines were NOT all equal. This suggests that some transitions are favoured more than others.

Hyperfine spectral lines
With better equipment and careful observation, it was found that there were previously undiscovered spectral lines These were named Hyperfine lines and they accompanied the other more visible lines. Bohr's model could not explain why this was the case due to the lack of equipment and development in quantum physics. The reason for these lines is actually because of a hyperfine structure of atoms. Solved through developments into Matrix Mechanics

The Zeeman effect
It was found that, when hydrogen gas was excited in a magnetic field, the produced emission spectrum was split. Bohr's model could not account for this Solved by accounting for the existence of a tiny magnetic moment of each electron.

Stationary states
Although Bohr stated that electrons were in stationary states, he could not explain why.

Mixture of Sciences
The Bohr model was a mixture of quantum and classical physics. This is an issue because it was thought that quantum physics was completely irrelevant and different to classical physics.

In QM, which was necessary to fix the flaws above, objects do not posses properties until observed. You can't speak of things like velocity, position, momentum etc of electrons in atoms - forget about them.

Thanks
Bill

 

[bhobba]

Not knowable position doesn't really mean "cannot move in continuous trajectory". What precisely is the equation, theory, or experiment that is supposed to forbid electrons to move in continuous trajectories when bound to an atom?

Since QM is a theory about observations and is silent about what going on when not observed you can hypothesise all sorts of things and never be able to be proven wrong experimentally. That's because you can only ever know about something by observing it.

If you disagree - fine - I won't get into an argument about it. Simply produce your calculations making predictions that can be checked experimentally based on electrons moving in a continuous trajectory etc.

Thanks
Bill

 

[carrz]

I don't think this can fit with observed phenomena such as quantum tunneling.

Can you point out the problem more specifically?

 

[carrz]

Since QM is a theory about observations and is silent about what going on when not observed you can hypothesise all sorts of things and never be able to be proven wrong experimentally. That's because you can only ever know about something by observing it.

Exactly. But, on the other hand, classical predictions can be proven wrong, and I am yet to see that is indeed the case for continuous deterministic trajectories. I'm not saying QM got it wrong, 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.

If you disagree - fine - I won't get into an argument about it. Simply produce your calculations making predictions that can be checked experimentally based on electrons moving in a continuous trajectory etc.

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 equations?

 

[carrz]

You need to step back a bit and think in terms of the formalism of QM. The Bohr model was simply a stepping stone to that. And it had flaws eg:
http://nsb.wikidot.com/pl-9-8-1-6

Bohr's biggest contribution in his model was to introduce quantum principles to classical physics, but his model had a few limitations

Bohr's model must be flawed because it does not even attempt to model neither Biot-Savart nor intrinsic spin magnetic moment. But I'm not concerned with any general theory, I'm specifically only talking just about "velocity", and whether QM really forbids trajectories to be continuous, or not.

Spectra of Large atoms:
The Bohr model could only successfully explain the hydrogen spectrum. It could NOT accurately calculate the spectral lines of larger atoms.

The model only worked for hydrogen-like atoms
That is, if the atom had only one electron.

Relative Spectra Intensity
Bohr's model could not explain why the intensity of the spectra lines were NOT all equal. This suggests that some transitions are favoured more than others.

Hyperfine spectral lines
With better equipment and careful observation, it was found that there were previously undiscovered spectral lines These were named Hyperfine lines and they accompanied the other more visible lines. Bohr's model could not explain why this was the case due to the lack of equipment and development in quantum physics. The reason for these lines is actually because of a hyperfine structure of atoms. Solved through developments into Matrix Mechanics

The Zeeman effect
It was found that, when hydrogen gas was excited in a magnetic field, the produced emission spectrum was split. Bohr's model could not account for this Solved by accounting for the existence of a tiny magnetic moment of each electron.

Stationary states
Although Bohr stated that electrons were in stationary states, he could not explain why.

Mixture of Sciences
The Bohr model was a mixture of quantum and classical physics. This is an issue because it was thought that quantum physics was completely irrelevant and different to classical physics.

Thanks. That's all very interesting. Can you tell me more about "stationary states"? This is the first time I hear about it and from what I know I don't see it would be so.

 

[Nugatory]

Thanks. That's all very interesting. Can you tell me more about "stationary states"? This is the first time I hear about it and from what I know I don't see it would be so.

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

But I have to caution you... Some of the questions you've asked suggest that there are gaps in your study of the basics of QM; closing these gaps one question at a time is tremendously inefficient. If you can get hold of a decent QM textbook (you'll find many recommendations here) and work through it (people here will happily help you through any hard spots) you will find yourself much better equipped to take on the problems you've been asking about here.

 

[tom.stoer]

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

There are numerous reasons why the classical description of point-like electrons in terms of position, velocity and trajectories fails. This was one of the basic reasons to abandon the Bohr model and to construct quantum mechanics. Of course one can ignore these facts and do calculations based on trajectories. For the reasons mentioned above the results of these calculations are completely meaningless!

The basic lesson from quantum theory is that electrons should be described using wave functions. One has to give up the idea of point-like electrons, position, velocity and trajectories. Nevertheless it is possible to define expectation values for physical observables. This can be done for the velocity as well (but not in a straightforward way).

Denoting a quantum state of an electron in a Hydrogen atom by |nlms\rangle the simplest idea us to calculate the expectation value for v². Instead of the velocity v we use the momentum-operator p and define v = p/m. Then we get

\langle v^2\rangle = m^{-2} \langle nlms| p^2 |nlms\rangle

This is the correct QM equation to define the velocity squared. It works for the Hydrogen case and for other more complicated atoms as well. The equation tells you that you can calculate the expection value of the velocity squared; but it does not tell you that the electron has exactly this velocity squared!

Now it's your choice: you can stay with the Bohr model and continue to calculate meaningless things; or you can start with the basics of QM.

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!

 

[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 Schrodinger 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 Schrodinger 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.