We revisit the non-relativistic limit of the Dirac equation in finite scalar and vector potentials and derive a Schrödinger-like equation that retains leading spin–relativistic corrections in closed form. For general central potentials, we cast the radial equation into a quadratic eigenvalue problem (QEP) using a finite-difference discretization method and develop an open-source solver to address it. We study Coulomb, harmonic oscillator, Woods–Saxon, and Yukawa potentials. We further obtain first-order energy and wavefunction corrections for the three-dimensional isotropic harmonic oscillator and Coulomb potentials via perturbation theory. This framework provides a practical bridge between non-relativistic and fully relativistic treatments, enabling accurate quantification of relativistic effects without the computational cost of full four-component calculations.

Solving the Dirac equation has played an important role in many areas of fundamental physics. In this work, we present the Dirac equation solver DiracSVT, which solves the Dirac equation with scalar, vector, and tensor nuclear potentials in spherical coordinate space. The shooting method was used with a Runge–Kutta 4 integration scheme. The potentials are parameterized in a Woods–Saxon form, which reproduce well the known single-particle states around all doubly magic nuclei and can be applied to study the shell evolution of exotic nuclei. The code can be easily extended to the study of other systems, including atomic, hadron, and molecular physics.

The distributions of the temperature excess of alphas to protons (epsilon) were studied using Parker Solar Probe measurements for Encounters 2 through 14. The distributions were mapped based on heliographic distance, Coulomb number, plasma beta, and Alfven Mach number (M (A) ). The importance of collisional effects in the thermalization of solar wind is observed for a wide range of Coulomb numbers. The distributions correlate better with N beta and NM (A) than just N. Furthermore, evidence was found for a narrow region immediately above the Alfven surface (1 < M (A) < 2) where epsilon has values much higher than the mass ratio.

It is shown that the proton formation probabilities, extracted from experimental decay half-lives, can be well reproduced by a simple multilinear formula with only three parameters. The parameters obtained by considering the standard root mean square deviation and the mini-max criteria are very similar to each other. In addition, we applied Bayesian analysis to study the uncertainties of the parameters and the model predictions. In this way we explain the systematics of proton decay half-lives. The multilinearity of the model also provides a way to classify the relative hindrance of different proton decays. All the recent experimental data agree very well with the model prediction. Our Bayesian analysis suggests that those new data do help constrain the uncertainty of the model parameters.