How is the energy of an electron lost in a classical hydrogen atom?

[jfizzix]

Consider the following.
You have an electron of negative charge orbiting a proton of positive charge at some distance R (i.e. a classical hydrogen atom).

I understand the hydrogen atom is unstable under classical physics because the accelerating electron loses its kinetic energy as electromagnetic radiation.

My question is set up as follows.
All you have is the proton acting on the electron.
In the rest frame of the proton, the electric field is constant, extending radially outward, so that it can do no work on the electron,
In the rest frame of the proton, there is no magnetic field on the electron due to the proton either.

What field is slowing down the electron?

Any comments would be appreciated, as this has puzzled me for quite come time.

 

[Drakkith]

Why wouldn't the electric field be able to do work? I'm pretty sure a static electric field will easily perform work on charges.

 

[jfizzix]

the electric field can't do work on the electron because the field on the electron is always perpendicular to the direction the electron is moving, making the dot product of force with instantaneous displacement zero.

 

[kevinferreira]

What about the electric and magnetic fields created by the electron?

 

[Jano L.]

I understand the hydrogen atom is unstable under classical physics because the accelerating electron loses its kinetic energy as electromagnetic radiation.

It is usually thought so on the basis of Larmor's formula for radiated energy and the idea that the radiated energy has to come from the internal energy of the atom. This argument does not work with forces, and attempts to introduce such radiation reaction forces acting on the microscopic particles did not lead to a consistent theory.

However, neither of the above two assumptions is necessary; the derivation of the Larmor formula is not valid for point particles and even if the system of charges radiates, the energy does not need to come from inside the system - it can come from other sources. For example, it can come from the work external forces do on the atom.

An antenna is a good example daily life - it radiates but it does not lose energy.

Approximate numerical calculations of the trajectory of the electron in fluctuating electromagnetic field seem to support this picture. For example, see the paper: D. C. Cole, Yi Zou, Quantum mechanical ground state of hydrogen obtained from classical electrodynamics, Physics Letters A 317 (2003), p. 14–20.

http://dx.doi.org/10.1016/j.physleta.2003.08.022

 

[jfizzix]

If we assume the hydrogen atom is isolated, the energy can only come from either the electron or the proton, or the fields generated by them. Maybe if we consider the particles to be of nonzero size, we can discuss the effect of one part of the electron on the rest of it as it orbits the nucleus, but it's not clear to me how this would work out mechanically.

An antenna loses energy as quickly as it's replaced by an electric power source.

So if we allow the electron to be a fuzzball or sphere of nonzero size, what force would then be responsible for the recoil the electron experiences?

 

[Jano L.]

If we assume the hydrogen atom is isolated, the energy can only come from either the electron or the proton, or the fields generated by them.

Yes. But then again, the atom is impossible to isolate; there are always electromagnetic fields present.

Maybe if we consider the particles to be of nonzero size, we can discuss the effect of one part of the electron on the rest of it as it orbits the nucleus, but it's not clear to me how this would work out mechanically.

Yes, but it would be very difficult in general - we would have to introduce non-electromagnetic forces. There are attempts based on approximations of "rigid sphere". See the book by Yaghjian (there is a possibility to view few pages).

http://www.springer.com/physics/optics+&+lasers/book/978-0-387-26021-1

So if we allow the electron to be a fuzzball or sphere of nonzero size, what force would then be responsible for the recoil the electron experiences?

In a sense, the force of one part of electron on another. The Lorentz-Abraham expression for "radiation reaction force" is an approximate result of this idea.