Form-invariant solution to quantum state on the sphere

Abstract

This paper investigates the quantum states that emerge from the transformation design of conformal mapping on the two-dimensional sphere. Three results are reported. First, the construction of form-invariant spherical harmonics labelled by the fractional quantum number through a scalar potential interaction is given. Second, the form-invariant equation of the charge-monopole system is studied. Rather than the half-integer classification of the monopole harmonics, the quantization of the monopole harmonics here can be any fractional number specified by the conformal index. The gauge equivalent condition of the vector potentials which result in the invariant equation shows that the monopole field and the quantization condition of the pole strength due to Dirac can be generalized to more general vector fields and values in the conformal space. Finally, we explore the quadratic conformal image of the charged particle coupling to the constant monopole field on the sphere. It is shown that the lowest order approximation of the image is the magnetic Hooke-Newton transmutation.