Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory: William E. Caswell. Department of Physics, Brown University, Providence, Rhode Island 02912
- Publisher
- Elsevier Science
- Year
- 1979
- Tongue
- English
- Weight
- 82 KB
- Volume
- 122
- Category
- Article
- ISSN
- 0003-4916
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✦ Synopsis
We introduce a generalization of Wick-ordering which maps the anharmonic oscillator (AO) Hamiltonian for mass m and coupling h exactly into a "Wick-ordered"
Hamiltonian with an effective mass M which is a simple analytic function of h and m. The effective coupling rl = X/M3 is bounded. We transform the A0 perturbation series in A into one in A. This series may then be summed using Bore1 summation methods. We also introduce a new summation method for the A0 series (which is a practical necessity to obtain accurate energy levels of the excited states). We obtain a numerical accuracy for (Epr -Eexact)/EeulLct of at least 10m7 (using 20 orders of perturbation theory) and 1O-3 (using only 2 orders of perturbation theory) for all couplings and a// energy levels of the anharmonic oscillator. The methods are applicable also to the double-well potential (DWP, the A0 with a negative mass-squared). The only change is that now the effective coupling is unbounded as h ---f 0. The series in n is, however, still summable. The relative accuracy in the energy levels for 20 orders of perturbation theory varies from lo-' for large coupling to 1 'A at h = 0.1 and to 10 % at A : .OS. We also present results for the sextic oscillator.