Numerical Methods for Determining Eigenenergies and Tunneling Rates in One-Dimensional Quantum Wells

Astronomers have long been interested in the beginning stages of star formation in cold regions of the universe. It is well established that at those low temperatures, Hydrogen atoms are the ones that are most involved in diffusive mechanisms. However, recent studies have found that diffusion of heavier particles such as oxygen can also be effective. Here we present a theoretical investigation of the efficient diffusion of oxygen-isotopes via quantum tunneling through a one-dimensional subsequent potential. Oxygen-16, oxygen-17, and oxygen-18, which were considered as point particles, were originally confined in a one-dimensional finite potential quantum well and were expected to tunnel through a finite potential barrier. This presumption was tested using MATLAB, through which graphical and numerical solutions were computed in order to obtain eigenenergies for each isotope. Results show that oxygen isotopes that diffused through a finite quantum well having a set potential height of 0.07eV and a width of 0.7Å and that tunneled through a subsequent finite barrier with a set potential height of Vo and a barrier width ranging from 0.1Å to 3Å revealed mass-independent isotopic effects, which were first discovered experimentally in 1983 by Mark H. Thiemens and Heidenreich.