Abstract
This study aims at implementing a truncated matrix approach based on harmonic oscillator eigenfunctions to calculate energy eigenvalues of anharmonic oscillators containing quadratic, quartic, sextic, octic, and decic anharmonicities. The accuracy of the matrix method is also tested. Using this method, the wave functions of the anharmonic oscillators were written as a linear combination of some finite number of harmonic oscillator basis states. Results showed that calculation with 100 basis states generated accurate energies of oscillators with relatively small coupling constants, with computation time less than 1 minute. Including more basis states could result in more correct digits. For instance, using 300 harmonic oscillator basis states in a simple Mathematica code in about 8 minutes, highly accurate energies of the oscillators were obtained for relatively small coupling constants, with up to 15 correct digits. Reasonable accuracy was also found for much larger coupling constants with at least three correct digits for some low lying energies of the oscillators reported in this study. Some of our results contained more correct digits than other results reported in the literature.
Highlights
Quantum anharmonic oscillators have long been used to test the power and shortcomings of new approximation techniques proposed to solve the Schrödinger equation of quantum systems
Some of the approaches include an algebraic method based on the ladder operator [9], analytic quasilinearization method [10], Lie algebra [11,12], the PoincareLinstedt method [13], multiple-scale perturbation theory [14], Wick’s normal ordering technique [15], examination of polynomial solution [16], quantum Monte Carlo method [17], and pertur-bation theory
Matrix representation of the Hamiltonian of the anharmonic oscillators was performed by solving equation (8) using equations (9) and (10)
Summary
Quantum anharmonic oscillators have long been used to test the power and shortcomings of new approximation techniques proposed to solve the Schrödinger equation of quantum systems. They are often used in testing computational approaches originally designed for systems with many fermions [1]. Anharmonic oscillators can be used to represent various challenging potentials, such as the double-well potential, which is very often used in theoretical physics studies [2] They were found to be useful in modeling many phenomena in nuclear physics, solid-state physics, atomic and molecular physics, and laser theory [3,4]. Many other approaches have been developed to calculate the energies of the systems
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