Relabeling spin or angular momentum operators

Main Question or Discussion Point

Spin or angular momentum in my book is formulated in the basis of eigenstates of the operator that measures the angular momentum along the z-axis. But in principle I guess this could just as well have been done in the basis of eigenstates of Ly or Lx. Will that change anything in the equations? For example for spin where all vectors and matrices are written in basis of the eigenstates of Sz.

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Spin or angular momentum in my book is formulated in the basis of eigenstates of the operator that measures the angular momentum along the z-axis. But in principle I guess this could just as well have been done in the basis of eigenstates of Ly or Lx. Will that change anything in the equations? For example for spin where all vectors and matrices are written in basis of the eigenstates of Sz.
The physics is the same in all directions. First we pick a direction that we'll measure the spin along, then we choose a set of coordinate axes. For convenience and because it simplifies the calculation, then we choose the axes so that one of them points along the chosen direction, and then by convention we label that axis z.

Khashishi