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Journal of Mathematical Physics

The quartic anharmonic oscillator and its associated nonconstant magnetic field

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

Published Oct 1, 1997• 3 citations
Publisher

Abstract

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.

DOI
10.1063/1.531924

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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.
Using a group-theoretic approach, we investigate some new peculiar features of a general quartic anharmonic oscillator. When the coefficient of the quartic term is positive and the potential is differentiable, we find that continuity of the derivative of the potential forces the nonexistence of an analytic wave function. For the case in which the coefficient of the quartic term is negative, we find that normalizability of the wave function requires non-Hermiticity of the Hamiltonian. Finally, we apply our method to gain some insight on the double well potential.
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.
In this paper we solve exactly the problem of the spectrum and Feynman propagator of a charged particle submitted to both an anharmonic oscillator in the plane and a constant and homogeneous magnetic field of arbitrary strength aligned with the perpendicular direction to the plane. As we shall see in the beginning of the letter, the Hamiltonian, being a quadratic form, is easily diagonalizable and the Classical Action can be used to construct the exact Feynman Propagator using the Stationary Phase Approximation. The result is useful for the treatment of quasi two dimensional samples in the field of magnetic effects in nano-structures and quantum optics. The presented solution, after minor extensions, can also be used for motion in three dimensions, and in fact it has been used for years in such cases. Also it can be used as a good exercise of a Feynman Path Integral that can be calculated easily with just the help of the Classical Action.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
Abstract The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic ( 1 2 k x 2 ) and quartic ( λx 4 ) terms is embedded within an infinite square well potential of appropriate width, ‘ a ’ and its sine eigen functions are used as basis functions ‘ N ’ for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters ‘ a ’ and ‘ N ’ are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, ‘ λ ’. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.
We take a qualitative comparative look at quantum and classical quartic anharmonic oscillators. It has been shown that the behavior of the quantum anharmonic oscillator mimics that of the classical anharmonic oscillators with the structurally same Hamiltonian in the coherent state basis of the harmonic oscillators. The associated equation of motion allows us to use Lindstet-Poincare perturbation method to compute the classical frequency of the oscillation order by order, by taking care of its dependence on amplitude and the quantum corrections. We also derive a bound for periodicity of such oscillations in both the classical and quantum cases.