Why can't SR explain why electrons do not crash into the nucleus?

[Andrew Mason]

In a recent discussion in this thread I wrote:

Using classical mechanics and electomagnetism, work out the speed that an electron would have to have in order to orbit a hydrogen nucleus at a distance of 10^-12 m:

F_c = \frac{m_ev^2}{r} = \frac{kq_e^2}{r^2}

v = \sqrt{\frac{kq_e^2}{mr}}

where:
r = radius of orbit = 1e-12 m
k = 9e9 Nm^2/C^2
q_e = 1.602e-19 C.
m_e = 9.1e-31 kg

v works out to 1.6e7 m/sec or about 5% of the speed of light.

Now work out what the radius of orbit could be if the electron traveled at the speed of light. This would obviously be the minimum orbital radius permitted by relativity. It would take an infinite amount of energy for an electron to get arbitrarily close to the speed of light.

I get r = 2.5e-15 m. or 2.5 Fermi units

The radius of a proton is about .5 Fermi. To reach a 2.5 Fermi radius of orbit, the electron would need an infinite amount of energy. So an orbiting electron simply can't get enough energy to crash into the nucleus!
​

I assumed that this non-QM explanation was wrong and that I was missing something obvious somewhere. I have tried to figure out why this is not at least a plausible explanation. I can't.

We know from celestial mechanics that when orbiting masses trade potential for kinetic energy, they adopt a smaller radius of orbit and speed up. Since the object that they are orbiting is large compared to the radius of orbit, a sufficiently reduced orbit radius means that they crash.

But in the case of gravity (black holes excepted) the speed of the orbiting object is not enough to change its mass appreciably (ie v<<c). So there is no minimum limit to the radius of orbit.

It is a very different matter with an electron orbiting a proton. The orbital speed at an atomic radius \approx 10^{-12} m. is relativistic, as I have shown. If it trades potential energy for kinetic, it speeds up but its mass increases as \gamma = (1 - v^2/c^2)^{-1} increases so the radius of orbit approaches a limit that is greater than the radius of the proton.

What is wrong with this argument?

AM

 

[quantumdude]

The problem with your argument is that at no point to you address the real problem, which is that the electron loses KE due to electromagnetic radiation (accelerating charges lead to EM waves, which carry away energy). So at some point, according to the classical model, the electron will simply stop moving around the nucleus and fall in.

According to GR, a similar thing happens with orbiting masses, except the waves are gravitational. But they carry away so little energy that the existence of long-lived solar systems is compatible with this prediction.

 

[learningphysics]

In a recent discussion in this thread I wrote:

Using classical mechanics and electomagnetism, work out the speed that an electron would have to have in order to orbit a hydrogen nucleus at a distance of 10^-12 m:

F_c = \frac{m_ev^2}{r} = \frac{kq_e^2}{r^2}

v = \sqrt{\frac{kq_e^2}{mr}}

where:
r = radius of orbit = 1e-12 m
k = 9e9 Nm^2/C^2
q_e = 1.602e-19 C.
m_e = 9.1e-31 kg

v works out to 1.6e7 m/sec or about 5% of the speed of light.

Now work out what the radius of orbit could be if the electron traveled at the speed of light. This would obviously be the minimum orbital radius permitted by relativity. It would take an infinite amount of energy for an electron to get arbitrarily close to the speed of light.

I get r = 2.5e-15 m. or 2.5 Fermi units

The radius of a proton is about .5 Fermi. To reach a 2.5 Fermi radius of orbit, the electron would need an infinite amount of energy. So an orbiting electron simply can't get enough energy to crash into the nucleus!
​

I assumed that this non-QM explanation was wrong and that I was missing something obvious somewhere. I have tried to figure out why this is not at least a plausible explanation. I can't.

We know from celestial mechanics that when orbiting masses trade potential for kinetic energy, they adopt a smaller radius of orbit and speed up. Since the object that they are orbiting is large compared to the radius of orbit, a sufficiently reduced orbit radius means that they crash.

But in the case of gravity (black holes excepted) the speed of the orbiting object is not enough to change its mass appreciably (ie v<<c). So there is no minimum limit to the radius of orbit.

It is a very different matter with an electron orbiting a proton. The orbital speed at an atomic radius \approx 10^{-12} m. is relativistic, as I have shown. If it trades potential energy for kinetic, it speeds up but its mass increases as \gamma = (1 - v^2/c^2)^{-1} increases so the radius of orbit approaches a limit that is greater than the radius of the proton.

What is wrong with this argument?

AM

From what I see, I think you're falsing assuming centripetal motion. Meaning, for the electron to crash to the nucleus, it isn't necessary that its velocity satisfy the relationship:
F_c = \frac{m_ev^2}{r} = \frac{kq_e^2}{r^2}

This relationship can be assumed when the electron is in a specific orbit, but while crashing down into the nucleus this relationship does not need to be maintained. There will be a spiraling motion, and I'm not sure how exactly the velocity will change as it spirals down.

I might be wrong. This is what I saw at a first glance.

 

[Andrew Mason]

The problem with your argument is that at no point to you address the real problem, which is that the electron loses KE due to electromagnetic radiation (accelerating charges lead to EM waves, which carry away energy). So at some point, according to the classical model, the electron will simply stop moving around the nucleus and fall in.

This objection is based on a questionable assumption that an electron orbiting a proton will radiate energy because it is accelerating (See: my post here). I don't believe that that has ever been proven. It seems not to be the case for a charge in gravitational orbit, although as you point out, it is very difficult to measure this because the radiation would be so small. But, given the equivalence of gravitational acceleration and inertia under GR, if an electron in a gravitational field radiated because it was accelerating it would also have to radiate if it was not moving at all, which seems to be contrary to em theory.

AM

 

[ZapperZ]

This objection is based on a questionable assumption that an electron orbiting a proton will radiate energy because it is accelerating (See: my post here). I don't believe that that has ever been proven. It seems not to be the case for a charge in gravitational orbit, although as you point out, it is very difficult to measure this because the radiation would be so small. But, given the equivalence of gravitational acceleration and inertia under GR, if an electron in a gravitational field radiated because it was accelerating it would also have to radiate if it was not moving at all, which seems to be contrary to em theory.

AM

No, it is WELL-KNOWN that an electron is a CIRCULAR motion will radiate EM radiation - refer to cyclotron and synchrotron radiation. This is NOT of the type "electron in gravitational field does not radiate" problem. There's ZERO ambiguity as far as confirming this experimentally, or else all those synchrotron centers all over the world are imagining the radiation they are using.

Zz.

 

[quantumdude]

I don't believe that that has ever been proven.

It definitely has been proven that a charge in orbit radiates. The rate at which energy is radiated is derived in Appendix B of Eisberg and Resinck's Quantum Physics, in case you'd like to look it up.

In fact it is a standard exercise in an upper-level undergraduate EM course to derive the lifetime of an atom in classical EM theory. It is about 10^-12 seconds, if memory serves.

 

[juju]

Hi,

I think that it may be possible to assume that the electron in orbit gets its energy replenished by the fields emanating from the nucleus.

juju

 

[quantumdude]

I don't see how. In the lab frame, the nuclear EM field is electrostatic and provides a central force. And as is well known, central forces don't change the speed of an orbiting body. So that begs the question, what is it in an atom that you think can replace the KE lost to EM radiation?

 

[Chronos]

I might be math challenged here, but I'm coming up short on the orbital velocity. If I take the Bohr radius, 5.292E-11, and electron mass, 9.11E-31 and do a couple quick substitutions:

r = \frac{\lambda}{2\pi} where

\lambda =\frac{h}{mv} converts to

v = \frac{h}{2\pi rm}

which yields 2.188E6 m/s.

 

[juju]

Hi,

The central electrostatic field may not change the speed of the orbiting electron, however it may be that it is this field that keeps the electron orbiting at a constant velocity by replenishing any energy lost to radiation.

Alos, it would seem that the dynamics of the central force interacting with the electron charge would have to keep the electron in a constant orbit, in spite of any EM radiation by the orbiting electron.

juju

 

[marlon]

Can't we just say this : the electron closest to the nucleus has the highest kinetic energy and the lowest (most negative) potential energy. It is most tightly bound to the nucleus but it moves around at the highest speeds compared to the other electrons in higher energy levels.

Now, i don't really know the calculations that prove this, i am sure i once saw them in a QM-course in college but i should look them up. There is no radiation because there is an equilibrium between the two energies. Isn't this correct ? Anyone knows the exact calculations that prove this.

marlon

 

[quantumdude]

The central electrostatic field may not change the speed of the orbiting electron, however it may be that it is this field that keeps the electron orbiting at a constant velocity by replenishing any energy lost to radiation.

I seriously doubt that you are talking about the same electrostatic force that is described in the Maxwell theory. If you are, then I would request that you show the mathematical form of the interaction that you think supplies the energy.

Alos, it would seem that the dynamics of the central force interacting with the electron charge would have to keep the electron in a constant orbit, in spite of any EM radiation by the orbiting electron.

I don't think so. If you treat the EM radiation as damping, then the orbital radius should shrink. And even if the radius doesn't shrink, the KE bleeds away so that the electron eventually stops. Certainly at that point it would just fall straight into the nucleus.

 

[Andrew Mason]

It definitely has been proven that a charge in orbit radiates. The rate at which energy is radiated is derived in Appendix B of Eisberg and Resinck's Quantum Physics, in case you'd like to look it up.

There seems to be controversy over this. See:
http://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9303025
http://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9711027

The disagreement seems to be over whether a uniformly accelerated charge radiates. It may be noted that in a synchrotron, cyclotron or betatron or linear accelerator, the acceleration is not uniform. Nor is it strictly a central force.

AM

 

[ZapperZ]

There seems to be controversy over this. See:
http://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9303025
http://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9711027

The disagreement seems to be over whether a uniformly accelerated charge radiates. It may be noted that in a synchrotron, cyclotron or betatron or linear accelerator, the acceleration is not uniform. Nor is it strictly a central force.

AM

So how is this different than an electron "orbiting" a nucleus? It is a "uniform" acceleration? And how is a cyclotron not "strictly a central force"? This "non-strict" property is ALL there is that would cause such a system to radiate?

We have gone through something like this a long time ago. Again, I would seriously question the validity of the assumption of an electron in an "orbit" around the nucleus. All you need to do is tell me how I can have something like that having ZERO angular momentum per the s-orbital. And this is the SIMPLEST case since I haven't yet brought up how, for example, you would explain not only the geometry of the d-orbitals, but the fact that such orbitals have a PHASE associated with different "lobes" that strongly affects how such an atom forms bonds with other atoms.

Zz.

 

[Hans de Vries]

I might be math challenged here, but I'm coming up short on the orbital velocity. If I take the Bohr radius, 5.292E-11, and electron mass, 9.11E-31 and do a couple quick substitutions:

r = \frac{\lambda}{2\pi} where

\lambda =\frac{h}{mv} converts to

v = \frac{h}{2\pi rm}

which yields 2.188E6 m/s.

Your calculation is OK. There's a shorter way to write this:

v = αc = 2187691.26 m/s

where α = 1/137.03599911


Regards, Hans

 

[learningphysics]

I might be math challenged here, but I'm coming up short on the orbital velocity. If I take the Bohr radius, 5.292E-11, and electron mass, 9.11E-31 and do a couple quick substitutions:

r = \frac{\lambda}{2\pi} where

\lambda =\frac{h}{mv} converts to

v = \frac{h}{2\pi rm}

which yields 2.188E6 m/s.

You used r=5.292E-11. Andrew used r=1E-12. Hence the discrepancy.

 

[Hans de Vries]

There seems to be controversy over this. See:
http://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9303025
http://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9711027

The disagreement seems to be over whether a uniformly accelerated charge radiates. It may be noted that in a synchrotron, cyclotron or betatron or linear accelerator, the acceleration is not uniform. Nor is it strictly a central force.

AM

You can find a derivation of the E field of a moving, accelerating charge here:

http://fermi.la.asu.edu/PHY531/larmor/

See formula 19 which is equal to Jacksons Equation 14.14. which is
also the one used in your refs.

If you now simplify (19) to that of a non moving particle then you get this:

E = \frac{q\hat{r}}{4 \pi \epsilon_0 r^2} - \frac{q}{4 \pi \epsilon_0 r} \frac{\vec{a}}{c^2} cos (\phi)

Where the first term is the usual Coulomb term and the second term
is caused by the acceleration a. Now if you go back to (19) then you'll
see that there's no way to get rid of the second term by choosing an
arbitrary speed.

Non radiating charges would need a modified EM theory at small distances.


Regards, Hans.

 

[Andrew Mason]

So how is this different than an electron "orbiting" a nucleus? It is a "uniform" acceleration? And how is a cyclotron not "strictly a central force"? This "non-strict" property is ALL there is that would cause such a system to radiate?

In a cyclotron the electron has a linear acceleration as it crosses between the Ds (and then a uniform acceleration due to the magnetic field that is perpendicular to the direction of motion).

Again, I would seriously question the validity of the assumption of an electron in an "orbit" around the nucleus.

When you go beyond one electron orbiting a proton, there is definitely a problem. I am just concerned with the simplest case of the H atom.

AM

 

[ZapperZ]

In a cyclotron the electron has a linear acceleration as it crosses between the Ds (and then a uniform acceleration due to the magnetic field that is perpendicular to the direction of motion).

So you DO have a "uniform acceleraton" from a cyclotron. Yet, you are arguing that such things do not "radiated".

When you go beyond one electron orbiting a proton, there is definitely a problem. I am just concerned with the simplest case of the H atom.

AM

And you are not bothered by this "problem"? And unless you have forgotten, such issues are ALSO relevant to H atom! You will have a hell of a time trying to explain the H spectral lines (shall we go over the selection rules?). And take 2 H atoms together and BAM! You have bonding-antibonding bonds! I'd like to see you take your orbit and explain that.

Zz.

 

[quantumdude]

So you DO have a "uniform acceleraton" from a cyclotron. Yet, you are arguing that such things do not "radiated".

Not quite. He's saying that there is a nonuniform acceleration between the D's.

When you go beyond one electron orbiting a proton, there is definitely a problem. I am just concerned with the simplest case of the H atom.

Let's not forget that the electron orbit is not going to be exactly circular. That is only the case if the nucleus is infinitely heavy.

Those papers you linked me to are a lot to go through, and I'd like to take my time with them. I think I'm overdue for my periodic review of Jackson anyway, so I'll print them out and read them alongside for comparison.

 

[reilly]

Bohr's great insight was: assume an electron in orbit does not radiate, contrary to the well established physics of that day. Rather, the radiation ocurs when an electron jumps from an orbit to a lower energy orbit. Lo and behold, crazy as it seemed, he got the hydrogen spectrum (Balmer series if I recall correctly) right. Prior to Bohr, no one had come even close. Classical theory simply could not deal with the reality of atomic spectra, because of the well established theory of radiation -- Lenard-Weichart potentials and all that.

In a sense, even more astonishing, was the success of modern QM, the Schrodinger Eq applied to the hydrogen atom. The idea of stationary states replaced Bohr's orbits, and Heisenberg, Pauli, Dirac and others incorporated E&M into QM, and thereby produced a QM theory of radiation. No orbits anymore -- except, perhaps, in the classical limit of very high quantum numbers. This is all part of mainstream physics, standard, well accepted, gives remarkable agreement with experiment, and, apparently drives a few to complete frustration. Like it or not, QM is here to stay, and is, so far, the best game in town. No orbits, not even close. So why bother to compute orbital speeds, when there ain't such things.

Ask most high enrgy physicists, particularly accelerator designers, and they will tell you, from experience, accelerating charges radiate. People do argue whether charges in free fall, under gravity radiate. But that has nothing to do with orbits - even if they existed.

Regards,
Reilly Atkinson

 

[Chronos]

And I agree with that interpretation, Reilly. It is fun to calculate the electrons orbital velocity in differents shells, but it is not meaningful. The causally challenged electron occupies every permissible location at all times [at least by my understanding]. If you 'look' for an electron at any permitted location at any given instant, it will be there - the QM version of 'if you build it, they will come'.

 

[juvenal]

People do argue whether charges in free fall, under gravity radiate. But that has nothing to do with orbits - even if they existed.

I was thinking about this issue the other day. I thought the answer would be yes, but a friend of mine who's better versed in GR wasn't so sure. Is it really controversial, as you said?

 

[Chronos]

I was thinking about this issue the other day. I thought the answer would be yes, but a friend of mine who's better versed in GR wasn't so sure. Is it really controversial, as you said?

The concept of electrons as orbital bodies... does not work. You can equivocate it to macroscopic bodies - like planets in orbit - but that is just plain wrong. The rules change at atomic scales.

Anyways, Reilly will fill you in on the details. I only took the cliff note version of QM.

 

[ZapperZ]

Not quite. He's saying that there is a nonuniform acceleration between the D's.

But that is not the plane of radiation that's emitted in a cyclotron, is it? Most of the cyclotron radiation are emitted while the electrons are being bent in the "D".

Zz.

 

[ZapperZ]

Ask most high enrgy physicists, particularly accelerator designers, and they will tell you, from experience, accelerating charges radiate.

Unfortunately, Reilly, I keep TELLING them, but they never listen! :)

Zz [who works at a linear accelerator]

 

[ZapperZ]

And as if we need ANOTHER evidence that an electron locked in a classical orbit around the nucleus will radiate, here's another kicker. A paper just published in Science has this abstract[1]:

Although an atom is a manifestly quantum mechanical system, the electron in an atom can be made to move in a classical orbit almost indefinitely if it is exposed to a weak microwave field oscillating at its orbital frequency. The field effectively tethers the electron, phase-locking its motion to the oscillating microwave field. By exploiting this phase-locking, we have sped up or slowed down the orbital motion of the electron in excited lithium atoms by increasing or decreasing the microwave frequency between 13 and 19 gigahertz; the binding energy and orbital size change concurrently.

It is VERY clear from the paper that such a Kepler-like orbit is only possible upon the supply of external energy in which the "electron" is now a superposition of many plane waves that add to a more well-defined wavepacket. This certainly is NOT the unperturbed atom that QM describes. Thus, a Kepler orbit model for an atom cannot be sustained by itself.

Zz.

[1] H. Maeda et al., Science v.307, p.1757 (2005).

 

[Andrew Mason]

The concept of electrons as orbital bodies... does not work. You can equivocate it to macroscopic bodies - like planets in orbit - but that is just plain wrong. The rules change at atomic scales.

Anyways, Reilly will fill you in on the details. I only took the cliff note version of QM.

To clarify, I am not trying to question the validity of QM. I am trying to identify the reason classical physics (including SR) fails to explain why an electron cannot orbit a proton without crashing into it.

So far the only reason seems to be that, according to classical physics, the electron would radiate all of its energy as it orbited because of its acceleration. All I am saying is that whether classical physics requires electrons to radiate in orbital motion about a proton seems to be a matter of some dispute still. So let's leave that argument to the side for the moment.

We know that electrons do not radiate energy while accelerating in the vicinity of a proton. The explanation for this is that electrons can only emit energy in packets. That is quite well proven. But it seems to me that does not explain why the electron doesn't keep getting closer to the proton and keep emitting more energy until it crashes into the proton. The evidence is that the coulomb force applies between protons and electrons down to the level of the size of the proton (approx 1 Fermi or 10e-15 m.).

My understanding is that the uncertainty principle provides the only explanation as to why the electron does not crash into the nucleus. I am just trying to see why the Special Theory of Relativity would not also come into play here.

AM

 

[ZapperZ]

We know that electrons do not radiate energy while accelerating in the vicinity of a proton.

This statement may be the source of all this problem. We DO know that if an electron is accelerating "in the vacinity of a proton", it WILL RADIATE. A proton is simply a source of E-field. Put an electron in an E-field and let it accelerate, it WILL radiate.

You cannot say that an electron IN AN ATOM in its ground state is "accelerating in the vicinity of a proton". This would NOT be correct, because I will then ask you to prove this. Show me physical evidence that there is this "electron", and it is being accelerated in an atom. That has been my whole point all along in this thread. By saying such a thing, you are explicitly assuming that an "electron" is a well-defined entity like a body orbiting a central force. This is incorrect as far as QM is concerned, and as far as what we have observed! It is why I brought up the bonding-antibonding bonds - this illustrates the fallacy of such an assumption!

If you remove the idea of a well-defined "particle" orbiting a nucleus, this whole issue is no longer relevant.

Zz.

 

[Chronos]

An unperturbed electron does not behave as if it is being accelerated. Hydrogen atoms prove a proton can get along with an electron for billions of years without getting annoyed and killing it [how many can say that of their spouses?]. The only logical conclusion is electrons are renegades - they do not obey macroscopic rules.