Quantum solutions for the harmonic-parabola potential system

The quantum harmonic-parabola system, whose potential energy is the negative harmonic potential, is analyzed and applied to the cases of a quantum well, barrier, and periodic lattices. The eigenstate of the quantum Hamiltonian of the harmonic-parabola system is obtained. It is shown that any function 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 eigenfunctions. The energy eigenvalues, uncertainty, and probability density for the first few states are treated for the infinite harmonic-parabola well. The energy eigenvalues and their enumeration, the energy band, and the probability density for the first few states are obtained for a well composed of the parabola and constant potentials. The transmission coefficients and each potential interval dependence are determined for this well and the barrier 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.