Radiation of accelerated charge in QM

[greypilgrim]

Hi,

One of the main problems of the Rutherford model is the fact that the electrons are accelerated and hence should lose energy due to radiation. Bohr's model doesn't resolve this, it only postulates that the energy levels are quantized and energy can only be emitted or absorbed by jumping between the levels.

What about the Schrödinger hydrogen atom? In many applications (for example perturbation theory) it's enough to treat the electromagnetic field classically (i.e. non-quantized). Hence if I measure the acceleration of the electron and apply Maxwell's equations, I should find a loss of energy by radiation.

Did I miss something, or do I need a full QFT description to explain why the hydrogen atom is stable?

 

[Orodruin]

Hence if I measure the acceleration of the electron and apply Maxwell's equations, I should find a loss of energy by radiation.

You are here assuming that the electron has a given acceleration, which it does not as it needs to be treated using quantum mechanics. That the energy levels are quantised means that it is impossible to treat the problem using a classical description of EM fields due to transitions between quantum states.

 

[greypilgrim]

That's why I said I measure the acceleration. Shouldn't the electron have a given acceleration at least for a short time after the measurement?

 

[Orodruin]

That's why I said I measure the acceleration. Shouldn't the electron have a given acceleration at least for a short time after the measurement?

No.

 

[greypilgrim]

Would you care to elaborate? It has well-defined acceleration at the time of measurement and since time evolution is continuous it can't get too far away from it in a short time interval.

Also, can't we measure continuously?

 

[Orodruin]

Would you care to elaborate? It has well-defined acceleration at the time of measurement and since time evolution is continuous it can't get too far away from it in a short time interval.

Also, can't we measure continuously?

No, this is wrong. It does not have a well defined acceleration as there are no acceleration eigenstates. Time evolution of the wave function is continuous. Furthermore, all of the states of the bound electron are stationary and the mean acceleration is zero. You need to stop thinking in terms of classical objects, because electrons bound to nuclei are not. You can measure continuously, or what essentially amounts to it, but not without interfering with the process. A very precise measurement of the position will require enough energy to kick the electron out of the atom.

 

[andresB]

Hi,

One of the main problems of the Rutherford model is the fact that the electrons are accelerated and hence should lose energy due to radiation. Bohr's model doesn't resolve this, it only postulates that the energy levels are quantized and energy can only be emitted or absorbed by jumping between the levels.

What about the Schrödinger hydrogen atom? In many applications (for example perturbation theory) it's enough to treat the electromagnetic field classically (i.e. non-quantized). Hence if I measure the acceleration of the electron and apply Maxwell's equations, I should find a loss of energy by radiation.

Did I miss something, or do I need a full QFT description to explain why the hydrogen atom is stable?

The electron in the Hydrogen atom doesn't radiate simply because it doesn't, the Maxwell's laws based on macroscopic classical descriptions of the motion just don't apply, they are just not the laws that rules the atom.

 

[bhobba]

The electron in the Hydrogen atom doesn't radiate simply because it doesn't, the Maxwell's laws based on macroscopic classical descriptions of the motion just don't apply, they are just not the laws that rules the atom.

QED of course does apply and that will predict the probabilities of absorbing and/or emitting photons.

It, as you correctly point out, is of course nothing like classical electrodynamics.

Thanks
Bill

 

[andresB]

Yes, I should have been clearer. I hope it will still be understandable for the OP, atoms do radiate of course, but the description of that process is not to be found in the classical theories.

 

[Demystifier]

What about the Schrödinger hydrogen atom? In many applications (for example perturbation theory) it's enough to treat the electromagnetic field classically (i.e. non-quantized). Hence if I measure the acceleration of the electron and apply Maxwell's equations, I should find a loss of energy by radiation.

Did I miss something, or do I need a full QFT description to explain why the hydrogen atom is stable?

To explain stability of the hydrogen atom, full QFT is not necessary, Schrödinger hydrogen atom is enough. To emit radiation, electron needs to jump into a lower energy state (otherwise energy would not be conserved). But Schrödinger quantum mechanics explains why the energy of the hydrogen atom is quantized, and consequently why there is no lower energy state to jump into it. Consequently, it cannot radiate.

 

[Vanadium 50]

Boy, we have really gone down the rabbit hole here.

Classically, an object radiates when it has a changing dipole (technically, multipole) moment. For example, if I have a charge -1 object orbiting a charge +1 object, I have a spinning - and thus changing - dipole. An accelerating charged object is one example of a changing multipole.

In QM, an atom in a stationary state (i.e. an energy eigenstate) has constant multipole moments. Since they aren't changing, there is no radiation. You don't need to know anything about the internal dynamics of the atom.