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Time dilation, electrons, and quantum mechanics


Dagl1
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So I have been reading a book about relativity by Paul Davies ("About Time"). In this book he mentions that in order to fully understand the physical (or chemical) properties of heavy atoms such as gold and uranium, one has to take into account that electrons move at relativistic speeds around the nucleus. Additionally he mentions synchroton radiation (which might be not entirely related to my question). My question is that I was under the impression electrons do not actually move around the nucleus at all, but just 'are' at specific locations around a nucleus, based on the probability cloud of electrons in that specific shell. This does not really fit (for me) with what is said about electrons moving at relativistic speeds, as in that view there is no motion at all? 
He mentions as well that synchroton radiation is at higher frequencies than would be expected from the speed at which electrons orbit the machinery, because they are moving at relativistic speeds and thus the radiation pattern is at a much higher frequency. I am not sure if synchroton electrons also follow the quantum mechanical probability clouds, and thus not sure if this might not be related to my initial question.

Could some helpful members show me which assumptions are wrong or provide me with a better way of thinking about these things. I do realise that this also might be a case of where quantum mechanics and relativity don't merge yet, but I don't wanna come to that conclusion yet as it is much more likely I just don't understand the concepts well enough.

Kind regards,
Dagl

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Wave functions of stationary states work pretty much as the density/current of a stationary fluid. Consider the flow lines of a fluid in a stationary flow state. At every point in the fluid, the velocity field is well defined, has a direction and a speed, even though nothing seems to be moving on the whole.

More mathematically:

If your stationary state is represented by wave function \( \psi_{n,l,m,s}\left(\boldsymbol{x},t\right)=e^{-iE_{n,l,m,s}t/\hbar}\Psi_{n,l,m,s}\left(\boldsymbol{x}\right) \), your 'cloud' of probability would be independent of the time-varying phase factor:

\[ \varrho\left(\boldsymbol{x}\right)=\left|\Psi_{n,l,m,s}\left(\boldsymbol{x}\right)\right|^{2} \]

The whole situation would be static, and yet, it would have an associated velocity field, which mathematically is given by the Fourier transform of the amplitude,

\[ \hat{\psi}_{n,l,m,s}\left(\boldsymbol{p},t\right)=\frac{1}{\left(2\pi\hbar\right)^{3/2}}\int d^{3}xe^{i\boldsymbol{p}\cdot\boldsymbol{x}/\hbar}e^{-iE_{n,l,m,s}t/\hbar}\Psi_{n,l,m,s}\left(\boldsymbol{x}\right)=e^{-iE_{n,l,m,s}t/\hbar}\hat{\Psi}_{n,l,m,s}\left(\boldsymbol{p}\right) \]

So the distribution in momenta doesn't depend on time either:

\[ \varrho\left(\boldsymbol{p}\right)=\left|\hat{\Psi}_{n,l,m,s}\left(\boldsymbol{p}\right)\right|^{2} \]

This is only valid for stationary states.

I hope that answers your question.

It's good to see you around.

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Hey Joigus, its been a while, been lurking around here and there, busy with life^^.

Thanks for your answer!
Hmm I will have to look a lot more closely at the mathematics, which might help, but so far I don't think I completely get your explanation. Would it be possible to word it differently. If I have a stationary fluid, then the individual particles inside the fluid are all moving in random Brownian motion. So the fluid as a whole is stationary, but the particles inside are not (I know that is a bit pedantic but I think it is stopping me from understanding your explanation in more detail). How then does this relate to the electrons. Should I think of the electrons as being stationary as a whole (the system), but parts of it are moving? 
How does that reconcile the 'movement of electrons at relativistic speeds' and their probablity cloud. I think what I am asking is; do the darn things move or are they just at specific points (I think you answered this, but I just don't get it yet). If they don't move, is what is said in the book regarding the physical/chemical properties of these atoms being dependent on understanding that electrons move at relativistic speeds still true?
I suppose it is not as simple as 'they move' or,  'they don't move' ;/

Anyway I will reply later today, time to go to work and already thanks in advance for any potential answers!

-Dagl

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10 hours ago, Dagl1 said:

So I have been reading a book about relativity by Paul Davies ("About Time"). In this book he mentions that in order to fully understand the physical (or chemical) properties of heavy atoms such as gold and uranium, one has to take into account that electrons move at relativistic speeds around the nucleus

There are relativistic corrections to the energy, which do not explicitly require having motion.

The explanation that it’s relativistic speeds is pop-sci/watered-down. QM uses things like energy and momentum operators, and you solve for energy eigenstates. Which differ when the energy gets to be an appreciable fraction of the rest energy.

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1 hour ago, swansont said:

There are relativistic corrections to the energy, which do not explicitly require having motion.

The explanation that it’s relativistic speeds is pop-sci/watered-down. QM uses things like energy and momentum operators, and you solve for energy eigenstates. Which differ when the energy gets to be an appreciable fraction of the rest energy.

I hope I understand you correctly, do you mean because the energy of the electrons is higher because they are moving at relativistic speeds, there is therefore a correction necessary, however this correction is a correction to the energy? Does that mean that in the synchroton example, the higher frequency is also due to this energy correction (that feels a bit strange, as frequency is directly correlated with rotations in this case right?)?
Maybe my question is better asked this way: are corrections to the energy (due to being closer to the rest energy) different from corrections to time, due to speed? 

I still am not entirely sure how to interpret your answer in regards to a probability cloud, other than that I can mentally accept that things in this cloud may have high energies, but you do mention momentum operators. Does something which only exists at certain positions at certain times also have motion? Is that then independent of the probability? 

Sorry for not getting your answers;/
 

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1 hour ago, Dagl1 said:

I hope I understand you correctly, do you mean because the energy of the electrons is higher because they are moving at relativistic speeds,

No, I made no mention of speed. There are relativistic corrections to the energy. 

 

1 hour ago, Dagl1 said:

there is therefore a correction necessary, however this correction is a correction to the energy? Does that mean that in the synchroton example, the higher frequency is also due to this energy correction (that feels a bit strange, as frequency is directly correlated with rotations in this case right?)?

Synchrotrons are classical and you can talk about the speed of the electrons.

 

1 hour ago, Dagl1 said:


Maybe my question is better asked this way: are corrections to the energy (due to being closer to the rest energy) different from corrections to time, due to speed? 
 

No, but you don’t know the speed. The explanations talking about this are making an invalid connection, using a classical equation/concept where it’s not valid

https://en.wikipedia.org/wiki/Klein–Gordon_equation#Derivation

 

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Let me offer an alternate viewpoint ...

We are considering a quantum mechanical system, so the HUP must come into play.
The 'orbitals' of electrons are nothing more than probability distributions of where the electron is located.
The outskirts, or distant areas, of the orbital cloud correspond to lower probabilities, while the inner areas have higher pobabilities, and are usually depicted as the shape of the orbital cloud.
These higher probabilities 'constrain' the electron's location, and, according to the HUP, introduce greater 'variability' in the electron's momentum. For the positional probability that is very high, its momentum, and speed, is very indeterminate, and possibly relativistic.
Orbitals which are more tightly bound would have more indeterminate momentum, and possibly, the highest electron speeds.

 

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18 minutes ago, MigL said:

Let me offer an alternate viewpoint ...

We are considering a quantum mechanical system, so the HUP must come into play.
The 'orbitals' of electrons are nothing more than probability distributions of where the electron is located.
The outskirts, or distant areas, of the orbital cloud correspond to lower probabilities, while the inner areas have higher pobabilities, and are usually depicted as the shape of the orbital cloud.
These higher probabilities 'constrain' the electron's location, and, according to the HUP, introduce greater 'variability' in the electron's momentum. For the positional probability that is very high, its momentum, and speed, is very indeterminate, and possibly relativistic.
Orbitals which are more tightly bound would have more indeterminate momentum, and possibly, the highest electron speeds.

 

That's an interesting point of view,

How does it play out with p, d, f orbitals where the max probabilities are more remote from the centre ?

 

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3 hours ago, studiot said:

How does it play out with p, d, f orbitals where the max probabilities are more remote from the centre

Not the center of the atom, but the center of the orbital, or probability distribution, itself.
Even when they are 'lobes' about the nucleus, they have a central 'dense' area.
They are often, erroneuously depicted with sharp edges, whereas we both know, the probability has no sharp edge, but gradually falls off.

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To expand on what I said earlier: some of these explanations are leaning a bit too hard on classical physics in quantum situations.

What they are doing is trying to use a classical analogue, that a body in a circular orbit has a KE that is half the magnitude of the PE, so for an orbit close to the nucleus (i.e. using the Bohr theory, which we know isn't correct) an electron in hydrogen, which has an ionization energy of 13.6 eV, has a KE of 13.6 eV and a potential energy of -27.2 eV. Those numbers aren't actually true in the QM solution, but those are the most probable values. The improper extrapolation is to assign 1/2 mv^2 to the KE, since you can't assign a velocity to the electron. 

When you get to an atom with a large Z, some electrons have a high enough average KE that relativistic corrections are necessary. The incorrect explanation is to say you are correcting the speed, but this doesn't show up anywhere in the equations. You solve the relativistic version of the wave equation, which gives different results than the non-relativistic version (Schrödinger equation) so there is a difference in the energy eigenstates. A relativistic correction of energy, without ever invoking velocity.

I recall some years ago reading a pop-sci article on this and they linked to the paper it was based on. The pop-sci article talked about the relativistic correction of the speed, and saying that the mass of the electron increased. When I read the journal paper, none of that was mentioned. It was only the energy that was corrected, as one might expect of a rigorous paper. The pop-sci article had tried to use this classical explanation to make the effect make sense, but it made for incorrect physics.

 

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@swansont Your explanation made it quite clear what the problem with this idea about speed of electrons is.

I then do wonder why this wouldnt apply to synchrotons? Or I should clarify, when you said that synchrotons are classical and we can speak of the speed of electrons, does that mean that the book didn't water down that explanation (so a higher frequency of radiation is observed because the electrons are moving at relativistic speed and as such the frequency is higher than just the regular amount of times of orbiting per second?) That to me still seems a little illogical (and thus I probably misunderstand it), as even if space is warped, the amount of times it should go around 360 degrees should not change even if it goes at very fast speeds?

I feel either this is another piece that I have to get my head around, or my assumptions might be wrong. So to reiterate; if a electron moves 30000 times per second around an synchroton, and it is moving at relativistic speed (99% the speed of light for example) (ignoring that for such amount of orbits per second the synchroton might be necessary to be super large), does that we observe a higher frequency than 30000, because it has more energy, or because it somehow manages to pass a point (let's say at 0 degrees of the orbit) more than 30000 times because the space is contracted? For me this doesn't make much sense so I hope you or someone else can shed light on this!

Thanks everyone already for their responses!

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1 hour ago, Dagl1 said:

@swansont Your explanation made it quite clear what the problem with this idea about speed of electrons is.I then do wonder why this wouldnt apply to synchrotons? Or I should clarify, when you said that synchrotons are classical and we can speak of the speed of electrons, does that mean that the book didn't water down that explanation 

Synchrotron radiation is emitted because you are accelerating a charged particle. There’s no QM involved in solving for the dynamics of the particle (you could try, and discover the energy states are so close together that they could not be resolved). The basic rule is if Planck’s constant doesn’t show up, it’s not QM.

Here it’s the acceleration being v^2/r, and equating that with the acceleration from the Lorentz force, qv X B 

 

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Thanks, I understand that it is a classical (non QM) thing, but my questions about the synchroton are more regarding relativity and the length contraction, I suppose I might think about it a bit longer and then maybe ask it as a seperate question. Either way thanks for everyones explanations

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The basic idea in relativity is that one observer notices time dilation and another observer notices length contraction. This is how they reconcile the speed of light being the same. A lab observer sees the electron’s clock running slow, and the electron “sees” its path as being shorter. That reconciles (qualitatively, at least) each noticing more distance traveled in a span of time, and the frequency going up as the energy increases, even though the speed doesn’t change much.

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