On Angular Quantization

One of the not so obvious facts about GUCTP are that classically there is no reason to quantize the angular moment for the electron. The kind of obvious answer to this is that implicitly we must add something to the classical theories in order to do that and do that in a way that is a bit more transparent than just postulate this as we know from measurements that we should have that. I went through the post at, fundamental model for matter. And it should be clear now how the quantization comes about due to an assumption that there is a limit how large the magnetic field can be,

Consider the GUTCP/QM/Bohr-Sommerfeld angular condition that the following expression is quantize,

$$ m v R $$

As this expression is independent of \(\gamma\) we can consider this in any reference frame we like, hence we study,

$$ m_e v R $$

This is interesting especially in the frame where the only motion is from the helical motion. So, using the fact that we deduced from the force relations

$$ \frac{rv}{c} = \frac{C}{R} $$

Hence,

$$ m_e v R = m_e c (r v / c) R / r = m_e c (C / R) R / r = C m_e c / r. $$

And as in the helical frame, \(r\) is fixed, so that we are at the limit of the B field. Hence this expression is constant and quantazised. which we denote \(\hbar\). In all what happens is essentially that when you start to increase the original speed in the lab frame, there is a compensation to maintain the helical magnetic field at the limit and as a consequence we get the angular quantization in GUTCP.

$$ m v R = \hbar. $$

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