# Is the motion of an electron bound into an atom inertial?

When a particle moves on a gravitational potential subject to no contact and tidal forces, the particule clearly is in inertial movement.

But what about the "motion" of an electron bound into an atom around its nucleus?

I know that in quantum mechanics we cannot strictly speak of a classical trajectory, of the electron around its nucleus. But there is motion, as many electronic atomic states have non-zero expected values for $$\hat{\mathbf{p}}$$ and even for $$\hat{\mathbf{L}}$$. So there is some type of motion.

My question is: in an isolated atom, say H, under no external forces, is the electron in inertial motion?

EDIT: To be more specific, if I could install an accelerometer on the atom's electron, would this accelerometer read zero (inertial frame) or non-zero (non-inertial frame).

Does the electron free fall in its "orbit" around the nucleus?

An planet moving around a sun is considered to be in an inertial frame because of the equivalence principle, which implies that cancellation of inertial mass and gravitational charge in the Newton's law* is exact. So, regardless of its mass, an object in a gravitational field experiences the same acceleration. This precludes detection of such acceleration by an accelerometer.

On the other hand, an electric charge moving in an electric field will have different acceleration depending on its charge-to-mass ratio. This lets one detect acceleration by some kind of accelerometer, which could contain particles of different charge-to-mass ratios and measure their relative motion to infer acceleration. This fact doesn't depend on whether we talk about a classical or quantum particle.

* Newton's law is classical, but it's a valid small-field approximation to GR.

• Thus, due to the presence of the electric field, electron motion is non-inertial, check?
– Arc
Feb 15, 2022 at 16:49
• @Arc exactly so Feb 15, 2022 at 19:56
• Ah! Now I get another point of your argument (2nd paragraph): for gravity alone we have $m a = m g$, so the mass cancel out to give $a = g$, and if the mass is gone then accelerometers measure zero. But for an electric field $m a = q E$ and thus $a = (q/m) E$, this is the charge-to-mass ratio you mention, so the mass is not gone and accelerometers measure that ratio, and thus the frame is non-inertial. Check?
– Arc
Feb 15, 2022 at 20:33
• I guess this is the reason why synchrotrons produce radiation: the accelerated charged particules are in non-inertial motion, whereas a charged particule in circular orbit with gravitation does not emit radiation because it's in inertial motion, albeit it's angular motion.
– Arc
Feb 15, 2022 at 20:34
• @Arc this is right, although I don't know of any experiment that was actually performed to check this scenario with a charged particle in a gravitational orbit. Feb 15, 2022 at 20:46

In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a constant velocity (which might be zero)—or, equivalently, it is a frame of reference in which Newton's first law of motion holds

Italics mine.

The moon around the earth is not in an inertial frame, as it is undergoing acceleration, $$d\vec p/dt$$, $$\vec p$$ the vector of momentum, going around the earth. Thus even in the Bohr model which has the electron rotating about the nucleus, the electron is not in an inertial frame.

Edit after long discussions in comments:

Arc in a comment:

theories overlap, so we should make them at least partially consistent, that's why I believe we should be able to speak of inertial frames in quantum mechanics, at least in some sense that matches classical mechanics on these boundaries

Quantum mechanics is a probabilistic theory. It can only calculate the probability of a specific event happening at $$(t,x,y,z)$$ , with the four vector at $$(E,p_x,p_y,p_z)$$ for energy momentum through the calculations of the wavefunction of the system.

The classical and quantum frames overlap for free electrons, whose "track" can be fitted with classical formulas in the detection chambers. An electron with no fields in the chamber goes in a straight line, and an inertial frame can be calculated. ( putting accelerometers is out of the question because quantum mechanics enters and the state of the electron changes by the existence of a macroscopic machine with billions of molecules.)

The theories should overlap and give the same predictions where they hold , within the limits of the variables where they are valid. Quantum mechanics theories became necessary because Newtonian mechanics was unable to describe the measurements in small dimensions, at the level of single electrons , molecules etc.

Quantum mechanics,was invented as a theory eventually , starting with the inability of classical mechanics and electrodynamics to model black body radiation, the photoelectric effect and atomic spectra.

A bound electron, if measured, at time t will have a probability of being found at rest with the laboratory, at a given (t,x,y,z) that is all. Probability. where it will be at t+dt is also a probability. That is why there are no orbits in the atom, but orbitals.

• – Arc
Feb 17, 2022 at 4:05