Emmy Noether (https://en.wikipedia.org/wiki/Emmy_Noether ) proved in the context of quantum theory that to every continuous symmetry there corresponds a conservation law. (https://en.wikipedia.org/wiki/Noether%27s_theorem ). For example, invariance to time leads to conservation of energy, invariance to space translation leads to conservation of linear momentum, and invariance to space rotations leads to conservation of angular momentum. This problem concerns related notions in the context of dielectric waveguides. Consider the eigenvalue-eigenfunction problem: = where is a linear operator on a space of functions defined over a Euclidian space, is a scalar valued eigenfunction and is the corresponding eigenvalue (not to be confused with wavelength). Suppose possess circular cylindrical symmetry, so is invariant to (commutes with) continuous translations along and rotations about an axis1 . For example, may describe the longitudinal field component or of the modes of a circular cylindrical optical fibre. Introducing a cylindrical co-ordinate system (, , ) with z-axis aligned with the symmetry axis, then the operation of translation along the axis by distance may be written: ()(, , ) = (, , + ) and operation of rotation about the axis by an angle may be written: ()(, , ) = (, + , ) Note the set of all translations and the set of all rotations from a group with a group composition satisfying:1 2 = 1+2 1 2 = 1+2 Show that the eigenfunctions of are of the form: = (; , ) exp() exp() Where is a function of the radial variable only parameterised by the parameters , that characterize the mode (c.f. quantum numbers). Explain why must be an integer and, for a lossless fibre, must be real