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AIP conference proceedings

Determination of quantum anharmonic oscillator’s energy using a simple matrix method

Liliana Setya Nalle, Redi Kristian Pingak, Albert Zicko Johannes, Zakarias Seba Ngara, Minsyahril Bukit

Published Jan 1, 2024• 2 citations
Publisher

Abstract

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.

DOI
10.1063/5.0200254

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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.
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.
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).
Abstract This paper discusses a one-dimensional harmonic oscillator system subjected to simultaneous quadratic-quartic-quintic perturbation. The main objective of this paper is to calculate the matrix representation of its Hamiltonian. To keep more generally, these three perturbation terms use different small parameters. The method used in this paper is the standard algebraic method using Dirac notation with the bases that have also been shown. The results of the analysis that has been carried out, it is obtained in the form of the Hamiltonian matrix representation of this system which contains the three different small parameters. If the three parameters are chosen to be zero (without perturbation), the matrix form will be reduced to a standard harmonic oscillator matrix.Keywords: QQQ perturbation; harmonic oscillator, Hamiltonian matrix representation.
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.
This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2 q 2 + lambda q 4 ). As a lowest-order approximation an extremely simple formula for energy levels, E i (0) = (i + 1/2)1/4(3/alpha i + alpha i ), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha i is the real positive root of a cubic equation: y i alpha i 3 + alpha i 2 - 1 = 0, with y i = 6lambda(2i 2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda 4 terms, as it should.