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Physical Review A

Time-dependent quantum-mechanical calculation of ground and excited states of anharmonic and double-well oscillators

Amlan K. Roy, Neetu Gupta, Bimalendu Deb

Published Dec 14, 2001• 42 citations
Publisher

Abstract

Ground- and excited-state energies as well as probability densities of quantum-mechanical anharmonic and double-well oscillators are obtained by numerically solving the time-dependent Schr\"odinger equation (TDSE). The methodology employed is exact in principle and is based on solving the TDSE in imaginary time, coupled with the minimization of energy expectation value. The present method works very well for deep wells and for double-well oscillators with pseudodegenerate energy levels where time-independent methods experience difficulties. The dependence of energy on anharmonicity parameter and the splitting of pseudodegeneracy by an asymmetry parameter as well as probability-density plots are discussed. Several potentials, especially those with pseudodegenerate energy levels, are reported here.

DOI
10.1103/physreva.65.012109

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Wavefunctions, energies and selected expectation values of the low-lying stationary states of two-dimensional double-well potentials are obtained from the long-time solutions of the corresponding time-dependent Schrödinger equation. The numerical method consists of transformation to a diffusion-like equation which is then solved by an alternating-direction, implicit, finite-difference method. The method is tested for cases in which the energies have been obtained by other methods. Then the dependence of the energies and other properties on the potential parameters is discussed on the basis of results for four states of 35 different sets of potential parameters including some that lead to pseudo-degeneracies.
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.
The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.
This paper discusses some practical problems in quantum mechanics, particularly finding solutions of the Schrödinger equation for some cases, as it is a pivotal part of this area. The article focuses on using perturbation theory to calculate the approximate ground state energy for both helium atom and quartic anharmonic oscillator. For the helium atom, conventional perturbation theory is used to solve for the approximate energy, whilst for the anharmonic oscillator, a slightly different approach, that is perturbation theory incorporating a parameter, is chosen to improve the precision of the approximation. The final relative errors for the ground state energy of helium atom model are, for the first order perturbation, , and for the second order, , and that for the anharmonic oscillator is . The study of perturbation theory gives insights into understandings of how it may be modified further to pursue a better level of precision. This facilitates development of new approximation techniques for solving quantum-mechanical issues.
Abstract Energy eigenvalues, probability densities, and 〈x2r〉 values (r = 1 to 6) of one‐dimensional self‐interacting nonlinear oscillators have been obtained by evolving the time‐dependent Schrödinger equation in imaginary time, coupled with the minimization of energy expectation values. For excited states, the orthogonality constraint with lower states is maintained. Probability density plots for the ground and first three excited states are presented. A comparison of energy eigenvalues and probability densities plots is also made between oscillators with and without self‐interaction. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003