Eigenvalues of the Quantum Anharmonic Oscillator.

Mojtaba Jafarpour, Davood Afshar

We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.

Alexander V. Turbiner, Juan Carlos del Valle Rosales

Christopher C. Gerry

Achilles D. Speliotopoulos

Hilbert Spaces of anharmonic oscillators are studied. It is shown that the eigenvalue structure of all anharmonic oscillators have the same general form and are dependent on only the ground state energy of the system and a single function \\lambda(H) of the Hamiltonian H whose form depends explicitly on H. It is also found that the Hilbert Space of the anharmonic oscillator is unitarily ``inequivalent'' to the Hilbert Space of the simple harmonic oscillator, providing an explicit example of Haag's Theorem.

B. I. Madububa, N. I. Achuko, JP. C. Mbagwu +2 more

This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).

Nikolai N Rozanov, В. А. Смирнов