N. Santi | Searcharxiv

arXiv2004

A new general approximation scheme(NGAS) in quantum theory:application to the anharmonic- and double well oscillators

A new scheme of approximation in quantum theory is proposed which is potentially applicable to arbtrary interacting systems. The method consists in in approximating the original Hamiltonian by one corresponding to a suitable exactly solvable system (with interaction) such that the "quantum average" of both are equal, thus forcing self-consistency.The method transcends the limitations of the variational method and the perturbation theory.The results are systematically improvable by the construction of a improved perturbation theory (IPT) which automatically satisfies the condition of convergence. Uniformly accurate results are obtained for the case of the quartic-,sextic- and octic anharmonic oscillators as well as the quartic and sextic double well oscillators. The leading order results reproduce earlier results by different methods using different input assumptions.The results for the sextic oscillators agree well with exact prediction of supersymmetry. We also discuss the stability and structure of the effective vacuum state of the approximation.

quant-ph↗

arXiv2001

Generalized Hartree Method: A Novel Non-perturbative Scheme for Interacting Quantum systems

A self-consistent, non-perturbative scheme of approximation is proposed for arbitrary interacting quantum systems by generalization of the Hartree method.The scheme consists in approximating the original interaction term $λH_I$ by a suitable 'potential' $λV(ϕ)$ which satisfies the following two requirements: (i) the 'Hartree Hamiltonian' $H_o$ generated by $V(ϕ)$ is exactly solvable i.e, the eigen states $|n>$ and the eigenvalues $E_n$ are known and (ii) the 'quantum averages' of the two are equal, i.e. $< n|H_I|n>$ = $ $ for arbitrary $'n '$. The leading-order results for $|n>$ and $E_n$, which are already accurate, can be systematically improved further by the development of a 'Hartree-improved perturbation theory' (HIPT) with $H_o$ as the unperturbed part and the modified interaction:$λH^{\prime} \equiv λ(H_I-V)$ as the perturbation. The HIPT is assured of rapid convergence because of the 'Hartree condtion' : $ = 0$. This is in contrast to the naive perturbation theory developed with the original interaction term $λH_I$ chosen as the perturbation, which diverges even for infinitesimal $λ$ ! The structure of the Hartree vacuum is shown to be highly non-trivial. Application of the method to the anharmonic-and double-well quartic-oscillators, anharmonic- sextic and octic- oscillators leads to very accurate results for the energy levels. In case of $λϕ^{4}$ quantum field theory, the method reproduces, in the leading order, the results of Gaussian approximation, which can be improved further by the HIPT. We study the vacuum structure, renormalisation and stability of the theory in GHA.

quant-ph↗