Computational Challenges in Solving the Time-Dependent Dirac Equation for Self-Interacting Fermions

One of the most significant concerns related to numerical algorithms to solve quantum mechanics differential equations is maintaining stability. In this project, the time-dependent Dirac equation is solved using the MSD2 method to determine the wavefunction and various operator expectation values of an electromagnetically self-interacting fermion. The resulting non-linearity of the equation may become a source of instability. The latter is controlled by checking the norm and verifying Ehrenfest's equations. The spatial lattice used in the algorithm introduces a momentum Fourier cutoff whose effect is eliminated by choosing appropriate initial conditions. The use of dynamic memory allocation accommodated the large memory demands of the algorithm. There is an inherent tradeoff between the accuracy imposed by the spatial lattice and the run time and computer memory as stability requires smaller time-steps. Finite results for the renormalized mass and charge of the fermion have been obtained proving the validity of the method.