On the stability of atomic orbitals and the electron spin

In the early phases in the formulation of a theory of the atom, one of the major problems that occupied the minds of its pioneers was the sudden death that the system would experience since, according to Larmor's formula, an electron that is constantly being accelerated in a circular orbit around a nucleus would, within a short period of time, radiate away its energy and eventually crash into the center. This problem is sometimes claimed to have been "solved" by quantum mechanics, but was it really? 

When Bohr postulated that the electronic orbitals must have a quantized angular momentum, that was a bold and brilliant move that led to a good quantitative prediction of the hydrogen spectrum, but did it solve the problem presented by the Larmor formula? In this post I will argue, in a heuristic way, how the electron spin might be the key to solving this problem, and thereby the stability of the atom.

The starting point in a treatise of quantum mechanics is usually taken to be the wave-function (or in some cases Heisenberg's uncertainty formula). The reason for this might be the order in which these models entered the stage historically. It should be kept in mind, though, that Schrödinger's equation does not take into account electronic spin, the spin is added only later in an ad-hoc fashion. Here I invite you to for a moment take a somewhat different view.

Assume that the quantized electronic spin is the natural starting point of quantum mechanics.

More specifically we assume that the electronic spin angular momentum is quantized in units of the reduced Planck's constant:

S = n*h_bar.

Bohr's assumption when formulating his model of the hydrogen atom was that the orbital angular momentum of the electron was quantized by the same formula:

L = n*h_bar.

How can we now add these pieces together to arrive at a solution to the atomic problem? Well, what if the spin angular momentum has the opposite sign of that of the orbital angular momentum?

L = -S.

Then we have that the total angular momentum of the system is given by

L_tot = L + S = 0.

This does not provide any rigorous proof of the stability of the orbit. From purely heuristic considerations, however, it would seem plausible that a stable system that does not constantly loose energy to the environment is characterized by a zero total of angular momentum, that is

L_tot = 0.

In the case of the hydrogen atom,  this is only possible if we introduce electronic spin, which by empirical observation is quantized.

This way of reasoning might put into question some common wisdom surrounding electronic spin. It is generally taught that electron spin can only take the values 

S = +/- 1/2 h_bar.

Again, however, does this seem "physical" to you. Take as an example a vibrating string, its frequencies are quantized, but there is no upper limit to its values.