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I Smallest possible orbit and velocity of a particle?

  1. May 25, 2017 #1
    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?
     
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  3. May 25, 2017 #2

    andrewkirk

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    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.
     
  4. May 25, 2017 #3

    PeterDonis

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