Finding n roots of unity

 

This leads to Quantization of θ  because there are only specific values that become possible for  θ  when we impose that after n rotations it has to return back to its same starting point. **

And the following are those values:

The problem of finding the values of θ for given values of n is more generally known as the N roots of unity.  

We will leave it as an exercise, but the following animation plots all the possible the values of θ for integer values of n=2,3,4,.. from the above equation that we found:

                            Source

And those are the roots of the equation z^n  = z i.e if you start at these points on the unit circle and make n rotations you will get back to the same point that you started with.

Have a good one!

** A more physical way to think about this is matching boundary conditions. For more insight on how boundary conditions lead to quantization, take a look at this  post.

*** Finding out the n roots of unity (Video)