Semiclassical treatment of logarithmic perturbation theory

Abstract

Abstract. The explicit semiclassical treatment of logarithmic perturbation theory for the nonrelativistic bound states problem is developed. Based uponh-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions for the one-dimensional anharmonic oscillator is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and exited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the harmonic oscillator perturbed by λx 6 are considered.

A perturbation theory without energy corrections

“…Perturbations of atomic hydrogen appeared in a number of works [9]. A semiclassical scheme on the basis of LPT was also put forward [10]. We later summarized the major applications of LPT and found a simple way of handling the excited-state perturbations [11], in particular, clarifying certain earlier confusions.…”

“…For a perturbative development, one has to put (10) and (11) in the aforementioned equation, along with F ¼ F 0 þ kF 1 . Then, various perturbation equations would follow, as usual, by equating coefficients of specific powers of k to zero on the basis of (12).…”

A Recursion Technique for Deriving Renormalized Perturbation Expansions for One-Dimensional Anharmonic Oscillator

“…Very recently, a new semiclassical technique for deriving results of logarithmic perturbation theory within the framework of the one-dimensional Schrödinger equation has been proposed 19 . Based upon an h-expansion this technique leads to recursion formulae having the same simple form both for the ground and excited states.…”

A new approach to perturbation theory for a Dirac particle in a central field

“…It is caused by factoring out zeros of the wave functions, which, in addition, are not the same zeros for small-and large-components of the Dirac spinor [33].On the other hand, it is known, that the radial quantum number, n r , most conveniently and naturally is introduced in consideration by means of quantization conditions, as in the WKB-approach [34,35]. However, since the WKB-approximation is more suitable for obtaining energy eigenvalues in the limiting case of large quantum numbers and the perturbation theory, on the contrary, deals with low-lying levels, the WKB quantization conditions need change.Recently, a new technique based on a specific quantization condition has been proposed to get the perturbation series via semiclassical expansions within the one-dimensional Schrödinger equation [36]. For the Dirac equation, on performing the scale transformation, r →h 2 r, the coupling constants appear in common with powers of Planck's constant,h, thus implying the possibility to obtain perturbation expansions in a semiclassical manner in this case, too.The objective of this letter is to develop the explicit semiclassical treatment of logarithmic perturbation theory for the bound-state problem within the framework of the Dirac equation and to describe a new procedure for deriving perturbation corrections through handy recursion formulae having the same simple form both for ground and exited states.The proposed technique can be regarded as a further investigation of a part assigned to a rule of achieving a classical limit in construction of semiclassical methods.…”