Enhanced Quantum Wave function for Mathematical Modeling of Hydrogen and Helium Ionization Energies: Engineering Applications to Plasma Particle Formation

Our research developed a computational model to calculate ionization energies of hydrogen and helium atoms through an enhanced variational method with a modified trial wave function in plasma generator. We implemented this approach in MATLAB v.7.12.0 (R2011a), incorporating key physical constants, such as reduced mass, Planck’s constant, and vacuum permittivity, within the framework of the time-independent Schrödinger equation. For hydrogen, a single-electron system, our model yielded an ionization energy of 13.6 eV, matching the established experimental value precisely. For helium, as we know, electron interactions complicate calculations; our optimized variational parameter estimated an ionization energy of approximately 32 eV, compared to the experimental value of about 25 eV, reflecting minor simplifications in our assumptions. This method proves reliable and computationally efficient, making it valuable for quantum-based plasma research, AI-integrated computational science, and studies of atomic-scale phenomena like corona plasma particle formation. Our comparative analysis confirms the variational technique’s accuracy, providing a strong foundation for advancing research in plasma physics and quantum mechanics