Quantum solutions for the harmonic-parabola potential system - art. no. 042103

The quantum harmonic-parabola system, whose potential energy is the negativ e harmonic potential, is analyzed and applied to the cases of a quantum wel l, barrier, and periodic lattices. The eigenstate of the quantum Hamiltonia n of the harmonic-parabola system is obtained. It is shown that any functio n may be expanded in terms of the eigenfunctions of the system in a finite interval, and the propagator of the system can be obtained from the eigenfu nctions. The energy eigenvalues, uncertainty, and probability density for t he first few states are treated for the infinite harmonic-parabola well. Th e energy eigenvalues and their enumeration, the energy band, and the probab ility density for the first few states are obtained for a well composed of the parabola and constant potentials. The transmission coefficients and eac h potential interval dependence are determined for this well and the barrie r of the parabola-constant potential structure. Periodic lattices composed of the parabola potential and parabola-constant potentials are constructed, and their dispersion relations, energy states, and bands are obtained.