Smallest possible orbit and velocity of a particle?

[jamie.j1989]

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,

$$\hbar=m\bf{r}\times \bf{v}$$, 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?

 

[andrewkirk]

Relativity and quantum mechanics are not fully compatible, so one can expect to run into problems when one tries to do a calculation involving both for very very small, close objects.

I think once the distance between an 'orbiting' and an 'orbited' object becomes small enough, one would have to analyse it using pure quantum mechanics, not a mixture of that and relativity. The QM derivation of the orbitals of a hydrogen atom is an example of that.

 

[PeterDonis]

I'm not sure if this result is correct?

It's not meaningful because quantum objects don't have classical orbits.