Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Journal of Physics

Numerical simulation of quantum anharmonic oscillator, embedded within an infinite square well potential, by matrix methods using Gnumeric spreadsheet

Aditi Sharma, O. S. K. S. Sastri

Published Jun 1, 2020• 11 citations
Publisher

Abstract

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.

DOI
10.1088/1361-6404/ab988c

Related papers

Investigating Single Quantum Anharmonic Oscillator with Perturbation Theory

Büşra Adalı, Ferhat Nutku

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.

Physics and Astronomy Reports•Dec 21, 2023•0 citations

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

Liliana Setya Nalle, Redi Kristian Pingak, Albert Zicko Johannes +2 more

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.

AIP conference proceedings•Jan 1, 2024•2 citations

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

Mojtaba Jafarpour, Davood Afshar

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.

Journal of Physics A Mathematical and General•Dec 21, 2001•61 citations

Analytic ground state energy and wave function of a quantum system in an exponential-form anharmonic interaction potential

Israel B Babalola, Oluwole Emmanuel Oyewande, Idowu H Latok +1 more

Abstract An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.

IOP Conference Series Earth and Environmental Science•Feb 1, 2021•0 citations

A QUADRATIC-QUARTIC-QUINTIC (QQQ) PERTURBATION ON HARMONIC OSCILLATOR: HAMILTONIAN MATRIX REPRESENTATION

Herry Fridolin Lalus

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.

Jurnal Fisika Fisika Sains dan Aplikasinya•Oct 22, 2020•0 citations

Towards Nonperturbative Solution of Quantum Dynamics : A Hamiltonian Mean Field Approximation Scheme with Perturbation Theory for Arbitray Strength of Interaction

B. P. Mahapatra

We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.

arXiv:1909.12508•Sep 27, 2019•0 citations
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 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.
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.
Abstract An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.
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.
We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.