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arXiv:quant-ph/9812027

New perturbation method with the matching of wave functions

Miloslav Znojil

Published Dec 14, 1998• 0 citations
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Abstract

We propose a new approach to the Rayleigh-Schrödinger perturbation expansions of bound states in quantum mechanics. We are inspired by the enormous flexibility of solvable interactions with several (N) discontinuities. Their standard matching solution is modified and transferred in perturbation regime. We employ the global renormalization freedom of the local wave functions and derive a compact N-dimensional matrix formula for corrections. In applications, our recipe is shown non-numerical for all polynomial perturbations of any piece-wise constant zero order potential.

DOI
10.48550/arxiv.quant-ph/9812027

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We propose a new approach to the Rayleigh-Schrödinger perturbation expansions of bound states in quantum mechanics. We are inspired by the enormous flexibility of solvable interactions with several (N) discontinuities. Their standard matching solution is modified and transferred in perturbation regime. We employ the global renormalization freedom of the local wave functions and derive a compact N-dimensional matrix formula for corrections. In applications, our recipe is shown non-numerical for all polynomial perturbations of any piece-wise constant zero order potential.
We propose a new approach to the Rayleigh–Schrödinger perturbation expansions of bound states in quantum mechanics. We are inspired by the enormous flexibility of solvable interactions with several (N) discontinuities. Their standard matching solution is modified and transferred in perturbation regime. We employ the global renormalization freedom of the local wave functions and derive a compact N-dimensional matrix formula for corrections. In applications, our recipe is shown non-numerical for all polynomial perturbations of any piecewise constant zero-order potential. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 235–242, 2000
We propose a new approach to the Rayleigh–Schrödinger perturbation expansions of bound states in quantum mechanics. We are inspired by the enormous flexibility of solvable interactions with several (N) discontinuities. Their standard matching solution is modified and transferred in perturbation regime. We employ the global renormalization freedom of the local wave functions and derive a compact N-dimensional matrix formula for corrections. In applications, our recipe is shown non-numerical for all polynomial perturbations of any piecewise constant zero-order potential. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 235–242, 2000
Perturbation methods are generally used for solving wave operator equations associated with the determination of effective Hamiltonians.In many cases the standard Rayleigh-Schrödinger and Brillouin-Wigner series either converge slowly or diverge.Therefore it is necessary to modify or to renormalize the standard wave equations.For that purpose derivative and convergence superoperators within the Ralyeigh-Schrödinger and Brillouin-Wigner formalisms were introduced.A new efficient approximation for convergence superoperators is investigated in this paper.Its application to a model system of N non-interacting molecules shows that this approximation can overcome convergence difficulties.
The most physically interesting systems are not exactly solvable in quantum mechanics. For one-dimensional bound systems without exact solutions, we analytically and numerically find that the Rayleigh-Schrödinger perturbed series sensitively depends on an unsolvable integration, which leads to numerical instability in quantum mechanics. By using an exact formal solution of the non-homogeneous Schrödinger equation, we demonstrate the existence of analytically bound states and propose a simple scheme to truncate infinity so that the instability difficulty is avoided.
The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.