Understanding "photons" from electron spin and dimensional analysis

Historically, the way quantum mechanics evolved seems to be something of the following:

1. Discovery of blackbody radiation and the photoelectric effect leads to the corpuscular theory of light, that is, light consists of particles called ”photons”.

2. Bohr derives the hydrogen spectra assuming quantization of angular momentum.

3. The electron double slit experiment leads De Broglie to assume wave-like properties of matter, which in turn leads to the Schrödinger equation.

4. Oh yes, and then we have the discovery that electrons have spin which also happens to be quantized.

No shade should fall upon the pioneers of quantum mechanics that these discoveries might not have come in the correct epistemological order. But now we have the opportunity to take some steps back and revise what we have discovered. Previously, I have advocated the idea of taking the quantized angular momentum of the electron as the starting point of the analysis. If so, would it then be wisest to assume that all other quantized phenomena in nature exist independently of this first assumption or would it be wiser to try to derive other quantization phenomena based on the first assumption?

Here we will try to understand the quantized energy packets o flight, also called ”photons”, based on the quantized electronic spin. First off, what is the dimension of angular momentum?

[angular momentum] = [energy] x [time]

You could imagine the electron saying to the light:

- I can only accept or give away angular momentum in integral packets of h_bar. If it does not have this size I have to give it back to you.

Now, from a dimensional point of view there are various ways this could be accomplished, a greater amount of energy could be provided on a smaller time scale or a smaller amount of energy could be provided on a larger time scale.

Now think for a moment, does light have a natural energy scale or a natural time scale? I would say it has a natural time scale. What is that time scale? Well, the period of one oscillation T. Recall that the frequency f is given by

f = 1/T

Now we get immediately

[quantized energy] = [quantized angular momentum]/T

or with properly scaled quantities

E = h/T = hf

Experimental evidence of higher order electron spin?

In a previous post I made the conjecture that electrons can have spins that are any multiple of the reduced Planck's constant, and not just the two discrete values of +/- 1/2 h_bar which we have been taught. Is there any experimental evidence to back this up? I think there might be, have a look at the following image that was taken from an article in Nature:

The experiment performed is the well known double slit experiment with electrons. I have proposed that this experiment can be explained by the Stern-Gerlach phenomenon, that is, the more you widen the other slit the more you increase the inhomogeneity in the magnetic field produced by the moving electron and the electron trajectories are then split according to their spin. As you would expect, lower spins are more probable than higher order spins, although in the image I think I can spot spin up to the order of at least 9. It would not be surprising if the probability follows a binomial distribution.  

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.