Simple example for "dynamic mass moment"?

Question

I encountered the concept of "dynamic mass moment" also called "the boost angular momentum" and I am looking for a simple example.

I take as definition (p.18) $$ \vec{N} = (ct)\vec{p} -(E/c)\vec{x}. $$

Is it allowed to just consider the ground state electron in the Bohr model?
$ \vec{x} = (r \cos{\omega t}, r \sin{\omega t}, 0) $
$ ct\vec{p} = ( -mvct \sin{\omega t}, mvct \cos{\omega t}, 0) $
$v=\alpha c \quad (\alpha\approx 1/137)$
$p=mv=m\alpha c$
$E/c = \sqrt{m^2c^4 + p^2c^2} /c = mc\sqrt{1+\alpha^2}$

The result is $$\vec{N}= mc \\ (-\alpha ct \sin{\omega t - r \sqrt{1+\alpha^2} \cos{\omega t}}, \\ \quad \quad \alpha ct \cos{\omega t - r \sqrt{1+\alpha^2} \sin{\omega t}}, 0). $$

I have no idea what, in this simple example, $\vec{N}$ should mean, esp. the variable $t$ which will just grow to infinity?
In the end it will become just $(ct)\vec{p}$ with an infitely large modulus?

In contrast the angular momentum $L=\vec{r}\times\vec{p} = (0,0, m\alpha c r (\cos{}^2+\sin{}^2)) = (0,0,\hbar)$ is straightforward.

Answer 1

I think the matter would be made more intuitive by considering radial and tangential components. You’d have the following (first component: radial, second component: tangential, third component: binormal): $$\mathbf{x} = (r,0,0)$$ $$\mathbf{p} = (0,m c \alpha,0)$$ Combining the two we get: $$\mathbf{N} = (mcr\sqrt{1+\alpha^2},mc^2\alpha t,0)$$ Since $r = a_0$, this can be seen as (at least in cgs-Gaussian units) $$\mathbf{N} = \left(\frac{\hbar}{\alpha} \sqrt{1+\alpha^2},\frac{\hbar c}{a_0} t,0\right) = \left(\hbar \sqrt{1+\frac{1}{\alpha^2}},\frac{\hbar c}{a_0} t,0\right)$$ in SI units you might have some extra $4\pi\epsilon_0$ terms popping up, but the juicy bits shouldn’t change.

Now, what I can gather from here is that the radial component is roughly $137 \hbar$, but indeed the tangential component keeps on growing (although it might be interesting to note that the factor multiplying time would be the energy of a photon with wavelength equal to the Bohr radius?).