If the units of angular momentum are quantised in integer amounts of ##\hbar##, does that then imply that we have restrictions on the smallest possible radius ##r## of an orbit of a given mass ##m##, given that the speed of light is ##c##. As follows,(adsbygoogle = window.adsbygoogle || []).push({});

$$\hbar=m\bf{r}\times \bf{v}$$, where v is the velocity of the particle of mass m, if the velocity vector and radius are perpendicular then we have,

$$\frac{\hbar}{rm}=v$$

and if we account for relativity, we have

$$\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{\hbar}{r}$$

If we now take the minimum radius to be the plank length l, we have for the maximum velocity of a particle with the minimum orbital angular momentum

$$v=c\sqrt{\frac{\frac{\hbar^2}{l^2}}{\frac{\hbar^2}{l^2}+m^2c^2}}$$

Which always gives v<c, which makes sense but I'm not sure if this result is correct?

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# I Smallest possible orbit and velocity of a particle?

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