One-dimensional Harmonic Oscillator Hamiltonian and Wave Function with Ionization Energy

1-dimensional hamiltonian with ionization energy (ξ) is shown to be exactly the same with the total energy from the standard harmonic oscillator hamiltonian, with the harmonic oscillator potential, mωx/2. Commutation relations for the ladder operators, energy at the n state and the expectation value for the potential are analyzed and shown to be consistent with the principle of ξ. We also derive 1-dimensional wave functions for Dirac delta (−αδ(x)) potential and with V (x) = 0. Formal justifications are given as to why the total energy in ξ based Fermi-Dirac statistics (iFDS) is exactly the same with the standard Fermi-Dirac statistics (FDS). Details are given on the total energy transformation from iFDS to FDS. Energy levels and Bohr radius of a Hydrogen atom are also shown to be consistent with the ionization energy concept. This concept is also applied in a free Mn atom that represent a free many-electron atom.