Generalization of adding angular momenta and circular potential in quaternionic quantum mechanics

Complex numbers were created by introducing the imaginary unit i to represent the square root of -1, allowing solutions to equations that involved square roots of negative numbers. Complex numbers were further extended to quaternionic numbers by introducing additional imaginary units, namely "j" and "k". Quaternions are non-commutative and involve four components, with each component being a real number or a multiple of an imaginary unit. Usually complex numbers are used to represent wave functions in quantum mechanics. Solutions using quaternions to square well, spin and angular momentum, Dirac equations have been obtained by many researchers. In this article, we have made use of quaternions to study the generalization of adding angular momenta, the digital signal processing of a quaternionic function and circular potential of a particle in real Hilbert space and have obtained quaternionic solutions in terms of Bessel functions.