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A NEW GENERAL APPROXIMATION SCHEME IN QUANTUM THEORY: APPLICATION TO THE ANHARMONIC AND THE DOUBLE WELL OSCILLATORS

B. P. MAHAPATRA

UGC-Academic Staff College, Sambalpur University, Jyoti Vihar 768019, India

On leave from: Department of Physics, Sambalpur University, BURLA 768019, India.

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N. SANTI

Department of Physics, BJB College (Autonomous), Bhubaneswar 751014, India

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, and

N. B. PRADHAN

Department of Physics, GM College (Autonomous), Sambalpur 768004, India

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https://doi.org/10.1142/S0217751X05022408Cited by:4 (Source: Crossref)

Abstract

A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction H_I(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H₀ generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues E_n are known and (ii) equality of quantum averages (EQA): 〈n|H_I(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and E_n are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H₀ as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(H_I(ϕ) - V(ϕ)), as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that 〈n|λH′(ϕ)|n〉 = 0for arbitrary λ and n. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naive perturbation theory where the original interaction λH_I(ϕ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic and the double well potentials, e.g. quartic-, sextic- and octic-anharmonic oscillators and quartic-, sextic-double well oscillators. Uniformly accurate results for the energy levels over the full allowed range of λ and n are obtained. The results compare well with the exact results predicted by supersymmetry for the case of the sextic anharmonic potential and the double well partner potential. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.

Keywords:

PACS: 03.65.-w.02.90.+p.31.15.Md

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