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Landé Pauli, Dirac and Spin
Published in Caio Lima Firme, Quantum Mechanics, 2022
The Dirac equation is a relativistic wave equation which describes the electron-spin. Whereas Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics can be considered as free field theory, Dirac included both the electromagnetic field and electric charge matter as quantum mechanical variables. The wave functions of Dirac equation are vector of four complex numbers, bispinors. The Dirac equation can be written as: (βmc2+c∑n=13αnpn)ψ(q,t)=iℏ∂ψ(q,t)∂t
Analysis of local fractional Klein-Gordon equations arising in relativistic fractal quantum mechanics
Published in Waves in Random and Complex Media, 2022
Ved Prakash Dubey, Devendra Kumar, Jagdev Singh, Ahmed M. Alshehri, Sarvesh Dubey
The Klein-Gordon equation presents a significant class of partial differential equations (PDEs) and is utilized frequently in field theory and relativistic quantum mechanics. This equation is very relevant for the physicist working in the field of high energy physics. Moreover, it has many applications in physical phenomena such as the behavior of elementary particles and propagation of dislocations in crystals [22]. The nonlinear Klein-Gordon equation is a second order PDE in space and time and can be expressed as [23] with initial conditions , where shows the wave displacement at time and position , and is a nonlinear force and is a fixed constant. Actually, the Klein-Gordon equation is a relativistic wave equation which identifies the nature and movement of zero spin particles at high velocities and energies resembling the speed of light. This equation admits the laws of special relativity and describes zero spin particles with relativistic energy. This equation has a number of applications in many streams of physics such as astrophysics, classical mechanics, particle physics, and high energy physics. Several works regarding the solution of the Klein-Gordon equation have been reported in past years. Khalid et al. [22] applied a new perturbation iteration scheme to solve linear & nonlinear Klein–Gordon equations. Kumar et al. [23] implemented the homotopy analysis transform method for computation of Klein-Gordon equations arising in quantum theory. Yülüklü et al. [24] investigated the nonlinear Sine–Gordon and Klein–Gordon equations via Taylor series based scheme. Recently, Malagi et al. [25] applied a computational technique for time-fractional Emden–Fowler equations. Furthermore, Prakasha et al. [26,27] investigated the time-fractional Kaup-Kupershmidt equation and Schrödinger–Boussinesq equations in an efficient way.