Is Bohr's postulate really necessary from classical perspective?

Question

Quoting the first sentence of the first postulate on wiki:

The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what classical electromagnetism suggests. (https://en.wikipedia.org/wiki/Bohr_model#Origin)

My problem is the "contrary to what classical electromagnetism suggest" part. First, i think, if that part was not true, there would be no justification to introduce that postulate in the first place.

And, yes, I do think that the quoted part is not true which, as just said, means there is no foundation/necessatiy for the postulate.

So: Classical electrodynamics predicts that only a time changing current will give rise to radiation. The fastest way to see this, is to take Maxwells equations and go the way to the wave equation for the electrical field, and dont drop J and P on the way. The result will be that you get the simple wave equation but with a source term due to the time derivative of the current (dJ/dt and d²P/dt²). So: The source of electromagnetic waves are transient (temporally not constant) currents. Currents that are constant over time dont matter.

I think that solely from a classical perspective there is no reason to assume that electrons have to produce a transient current and thus would radiate. To decide if it has to radiate or not, would need information about the classical electron charge distribution and its time dependence. That the electron has (or is associated with) some sort of classical charge distribution is also a very old idea and quite within the classical picture.

From Schrödingers equation (for which we don't need Bohrs postulate too) we finally get detailed information about some distribution associated with the electron. Even interpreting the |Eigenstates|^2 classically as charge distributions, they wont correspond to a time changing current. Even when superimposing several Eigenstates the current should be constant (i think, not rigorously done yet though)

So why is Bohrs postulate still teached besides for historical perspective? You might say it is teached solely for historical perspective, but I think it's not done in the right way (assuming what I claim is correct) since I was never told what I just wrote here (and to be honest I was afraid to ask my profs)

tldr: I think that Bohrs postulate (1st) is based on a false assumption and thus not necessary, but maybe I'm missing something important here. So: Is Bohrs postulate necessary from classical perspective?

Answer 1

A classical electron is a point charge, or at best a sphere the size of the classical electron radius. A point charge (or small sphere) moving in a circle does not constitute a steady current. For a point charge moving in a circle of radius $R$ centered at the origin in the $xy$-plane, the current is given by:

$$I(x,y,z,t)=\delta(R\sin(\omega t),R\cos(\omega t),z)$$

where $\delta$ is the 3-dimensional Dirac delta function:

$$\delta(x,y,z)=\begin{cases}\infty&\text{if }x=y=z=0\\0&\text{otherwise}\end{cases}$$

where $\infty$ is here defined such that:

$$\iiint_{\mathbb{R}^3}\delta(x,y,z)=1$$

As you can see, if you look at any particular point along the electron's path, the current will be very much not constant. Take, for example, the point $(R,0,0)$; then the formula for $I$ at that point reduces to:

$$I(t)=\delta(\sin(\omega t))$$

which is $0$ most of the time and then $\infty$ for a moment, every time the electron revolves around.