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arXiv:quant-ph/9807044 · Physics Letters A

Optimized perturbation method for the propagation in the anharmonic oscillator potential

Anna Okopińska

Published Dec 1, 1998• 7 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(98)00750-6

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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.
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.