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Heliyon

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

R. Deepika, Krishnan Muthunagai

Published Feb 1, 2024• 4 citations
Publisher

Abstract

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.

DOI
10.1016/j.heliyon.2024.e25597

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The introduction of complex numbers marked a significant leap in mathematics, introducing the imaginary unit i to represent the square root of -1. This innovative concept proved invaluable in solving equations involving square roots of negative numbers. The extension to quaternions involved introducing additional imaginary units denoted as j and k . A quaternion in an interesting concept that extends complex numbers to 4-D.The manuscript is about using quaternions to calculate wave packets in one-dimension, and anti-hermitian operators to obtain the results in quaternionic form including expectation values of position, momentum and energy. The results are compared to the exisiting results on complex wave packets in one-dimension.
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