Fredholm integral equation for the perturbation theory in quantum mechanics
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A new approach for perturbation method, based on the Fredholm integral equation of the second kind has been introduced to theoretical physics and quantum chemistry. The method has been used in order to derive an analytical form of the \(\upsilon \)-vibrational wavefunction in a form of a continuously convergent Liouville–Neumann perturbation series and to generate consecutive perturbation corrections to the wavefunction. The second-order correction to the energy levels has been obtained. Iterated kernels and a resolvent have been constructed and employed to the calculation of the wavefunction perturbation corrections. The method proposed can be used successfully in advanced calculations of quantum chemistry and theoretical spectroscopy because of a continuously convergence of the perturbation series.
KeywordsFredholm integral equation Perturbation theory method Vibrational problem Molecular spectroscopy Theoretical physics
Perturbation theory method is one of the most widely used among the theoretical physicists and chemists, allowing them to obtain approximate analytical solutions of many physical problems and to describe the reality more properly than the so-called unperturbed model, which even though being exactly solvable, is purely mathematical. The method has been widely applied in almost every branch of physics, e.g.: nuclear physics , thermodynamics  and physics of diamagnetism , but it has the greatest impact on quantum mechanics and quantum chemistry computations (e.g. [4, 5, 6]), in which its application includes both analytical and numerical studies. Let us just mention a paper by Pople et al. . In this remarkable research the authors proposed a new method that allowed study of physical properties of molecules with polarized or distorted electronic structure via, among the others, calculation of self-consistent molecular orbital wavefunctions based on the perturbation theory method.
Even though the perturbation theory method is so widespread that a person can easily acquire a book treating about it (e.g. ), physicists still face many problems connected with this way of solving physical problems. One of the most harsh of them and probably the most difficult to cope with is the problem of convergence of power series that can be obtained with the use of the approach.
The current work presents a novel attitude towards the perturbation theory method. An efficient and continuously convergent perturbation theory method, based on the Fredholm integral equation, is presented. On the basis of this approach a new form of perturbation series has been obtained.
2 Fredholm integral equation and the perturbation theory method
3 Perturbation series convergence problem
- 7.C.M. Bender, S.A. Orszag (Springer, New York, 1999), p. 593Google Scholar
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