Square root of momentum and Dirac equation

R. Deepika, Krishnan Muthunagai

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.

Mitchel Weissbluth

Helge Kragh

Axel Schulze‐Halberg

Mohammad Javad Kazemi, Jafar Khodagholizadeh, Mohammad Hosein Barati

In this paper, we investigate the Lorentz invariance of the square root Klein-Gordon equation. This equation with interpreting $|\psi|^{2}$ as position probability density is not Lorentz invariance but we can make this equation Lorentz invariant with the help of new proper interpretation of the wave function. In this regard we consider the relation between the probability density and the wave function as$\rho=\psi \psi^{\ast}+ (\hat{D}\psi)^{\ast}\hat{D}\psi$ and show appropriate selection of the linear operator$\hat{D}$the square root Klein-Gordon equation will be Lorentz invariant. In fact the solutions of this equation is equivalent to the positive energy solutions of the Dirac equation. \end{abstract}