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.

Shi-Hai Dong

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.

Alto Osada, Rekishu Yamazaki, Atsushi Noguchi

Thomas W. Allen, Costas A. Anastassiou, W. H. Klink

Quantum mechanical anharmonic oscillators and Hamiltonians for particles in external magnetic fields are related to representations of nilpotent groups. Using this connection the eigenfunctions of the quartic anharmonic oscillator with potential Vα(x)=(α+(x2/2))2 can be used to determine the eigenfunctions of a charged particle in a nonconstant magnetic field, of the form Bz=β2+β3x. The quartic anharmonic oscillator eigenvalues for low-lying states are obtained numerically and a function which interpolates between α≪0 (a double harmonic oscillator) and α≫0 (a harmonic oscillator) is shown to give a good fit to the numerical data. Approximate expressions for the quartic anharmonic oscillator eigenfunctions are then used to get the eigenfunctions for the magnetic field Hamiltonian.