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Journal of Physics A Mathematical and General

Calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential

Mojtaba Jafarpour, Davood Afshar

Published Dec 21, 2001• 61 citations
Publisher

Abstract

We choose the squeezed vacuum state as a one-parameter trial wavefunction, to minimize the energy of an anharmonic oscillator, with a polynomial perturbation potential. Using the optimal vacuum state obtained, we generate the optimal generalized number state basis, and set up the Hamiltonian in the normal-ordered form. Then, truncating the basis, we produce finite-dimensional matrices in a simple manner, and find their eigenvalues by standard methods. We will apply our method to the Morse potential, whose exact eigenvalues are known, as a check on the accuracy of the results that we have derived. Two models of the double well and the quartic potential are considered as examples.

DOI
10.1088/0305-4470/35/1/307

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We choose the squeezed vacuum state as a one-parameter trial wavefunction, to minimize the energy of an anharmonic oscillator, with a polynomial perturbation potential. Using the optimal vacuum state obtained, we generate the optimal generalized number state basis, and set up the Hamiltonian in the normal-ordered form. Then, truncating the basis, we produce finite-dimensional matrices in a simple manner, and find their eigenvalues by standard methods. We will apply our method to the Morse potential, whose exact eigenvalues are known, as a check on the accuracy of the results that we have derived. Two models of the double well and the quartic potential are considered as examples.
We have generalized the two-step approach to the solution of systems of NNN coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems.
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.
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).
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.
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.
This study aims to apply the matrix method to determine the energy of anharmonic oscillators containing linear, quadratic, cubic, quartic, quintic, sextic, septic, octic, nonic, and decic anharmonicities.Energy calculations were performed analytically using the matrix method with a matrix of size 4×4.The calculations were carried out by finding the Hamiltonian matrix representation of the oscillators, which was then solved using the time-independent Schrödinger equation.As a comparison, computational calculations were also conducted with Wolfram Mathematica for large matrix sizes.Results for small coupling constants and small perturbation terms such as linear anharmonic oscillators showed that energies obtained agree with the exact solution (up to 3 digits).Meanwhile, the accuracy of the analytical matrix method was found to decrease for larger coupling constants.The application of analytical matrix methods with small matrix sizes can produce relatively accurate energies at small coupling constants, while large coupling constants require larger matrix sizes to obtain reasonably accurate results.The calculation of energy by matrix method with a large matrix size will be more effectively done computationally, such as by using Wolfram Mathematica.