Could element 137 electrons become superluminal in a retrocausal atom?

[Superposed_Cat]

I found the equation for the speed of an electron today. atomic number*fine-structure constant*speed of light. rearranging we get z= v/σc. if you set v to c then you find that at element 137 the electrons become superluminal. This can be possible right? a retrocausal atom! Could someone please explain why this won't happen. Thanks for any help.

sorry, for the typo in the title.

 

[The_Duck]

This formula is a non-relativistic approximation, valid only for Z much smaller than 137. It is not true for large Z. (It is also a huge oversimplification that ignores things like the fact that different electrons in an atom have different velocities and the fact that that no electron has a definite velocity; nevertheless it does give the right order of magnitude for the electron velocity, as long as Z is much less than 137).

 

[Superposed_Cat]

what's the relativistic equation?

 

[Bill_K]

There's more involved than just "relativistic". For the nonrelativistic atom the energy of the ground state is

E = - mc²(α² Z²/2)

The corresponding relativistic formula (exact solution of the Dirac equation) is

E = mc² √(1 - α²Z²)

For αZ << 1 it reduces to the rest mass mc² plus the nonrelativistic value. However for αZ = 1 it goes to zero. This is a warning that the single-particle Dirac solution is no longer valid, since the negative energy solutions can no longer be ignored. In fact for αZ > 1 the energy becomes imaginary.

 

[Superposed_Cat]

thanks