  Orbital Angular Momentum (OAM) of photons are already ubiquitously being used
for numerous applications. However, there is a fundamental question whether
photonic OAM operators satisfy standard quantum mechanical commutation
relationship or not; this also poses a serious concern on the interpretation of
an optical vortex as a fundamental quantum degree of freedom. Here, we examined
canonical angular momentum operators defined in a cylindrical coordinate, and
applied them to Laguerre-Gauss (LG) modes in a graded index (GRIN) fibre. We
confirmed the validity of commutation relationship for the LG modes and found
that ladder operators also work properly with the increment or decrement in
units of the Dirac constant ($\hbar$). With those operators, we calculated the
quantum-mechanical expectation value of the magnitude of angular momentum,
which includes contributions from both intrinsic and extrinsic OAM. The
obtained results suggest that OAM characterised by the LG modes exhibits a
well-defined quantum degree of freedom.
