A New General Approximation Scheme in Quantum Theory: Application to the Anharmonic and the Double Well Oscillators

Abstract

A self-consistent, non-perturbative approximation scheme is proposed which i s potentially applicable to arbitrary interacting quantum systems. For the case of selfinteraction, 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, H 0 generated by V(φ ) is exactly solvable i.e., the spectrum of states | n > and the eigen-values E n are known and (ii) equality of quantum averages (EQA):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 H 0 as the unperturbed Hamiltonian and the modified interaction,, 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 | λ H ′ (φ ) | n > = 0 for arbitrary 'λ' and 'n'. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naïve 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 double well potentials, e.g. quartic-, sextic-and octic-anharmonic oscillators and quartic-, sexticdouble 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 super symmetry for the case of the sextic anharmonic-and the double well partner potentials. 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.

Investigations of Interacting Quantum Systems in a Novel General Approximation Scheme (NGAS)

“…(b) Moreover, the rigorously established [11] analytic structure of E(g) in the g -plane, as well as, the non -analytic dependence on the coupling strength g , are trivially contained [8,82] already at the LO-level through the solution of Eq. ( 29), which is explicitly given by eq.…”

“…(d) A further significance of ω arises from the altered ground state structure [8,50,52] due to interaction, differing non-trivially from the free field ground state. Also, the"dressed ground state" of the interacting system in the LO approximation has lower energy [8,50,52,82] as compared to the "trivial " ground state of the noninteractingtheory for any non-vanishing value of the coupling ′ g ′ . This result thus establishes the instability of the freefield ground state in presence of interaction.…”

“…This result thus establishes the instability of the freefield ground state in presence of interaction. This aspect has been discussed in detail elsewhere [8,50,52,82].…”

“…(f) Further, as stated earlier, the accuracy of the energy spectrum obtained in the LO, Eq. (3.12) is quite significant-the deviations from the "exact" -results being no more than a few percent [8,50,52,82] over the fullrange of g and n . This result ensures that the dominant contribution indeed comes from the LO as required in a perturbative-framework.…”

“…These parameters: ω, σ and h 0 are then determined analogously as in the case of the AHO, in terms of g and ξ . However, a distinct feature in the case of the QDWO is the occurrence of "quantumphase transition(QPT)" [82] governed by a "critical coupling" g c (ξ) given by the expression:…”