_{1}

Physically the examined perturbation problem can be regarded as a set of collision events of a time-independent perturbation potential with a quantum system. As an effect of collisions there is an expected definite change of energy of an initially unperturbed state of the system to some stationary perturbed state. The collision process certainly occupies some intervals of time which, however, do not enter the formalism. A striking property is the result of a choice of the sequence of collisions according to the applied circular scale of time: the scale produces almost automatically the energy terms predicted by the Schr
ödinger perturbation theory which usually is attained in virtue of complicated mathematical transformations. Beyond of the time scale and its rules—strictly connected with the perturbation order N introduced by Schr
ödinger—a partition process of the number
*N*-1 is applied. This process, combined with contractions of the time points on the scale, provides us precisely with the perturbation terms entering the Schr
ödinger theory.

The present paper has two aims. The first one is to provide an evident simplification of the treatment of the Schrödinger perturbation series for energy, especially at large perturbation order N. Another aim is to demonstrate that a circular scale of time can be indispensable in realization of the first aim.

According to the definition of the German “Physikalisches Wörterbuch” [

A special treatment concerning the time notion is applied in the quantum theory; see e.g. [

A general idea of the present and former papers by the author [

But in a series of former papers [

In Section 2 we present the properties of the time scale which are essential for its application. It should be noted that the time variable and the Feynman diagrams based on it have been used a long time before the present approach to the Schrödinger perturbation theory has been developed [

Time is an evidently subjective notion because it depends on the physical phenomena represented by the observed system, as well as abilities possessed by an observer. In reality the physical and philosophical properties of the notion of time were combined gradually with the experience and observation of the everyday life; science―excepting perhaps for astronomy―had, at least at its early stage, not much to do with time. A separate component of the view on time is provided by the human imagination. This second component seems to be mainly responsible for application of the time notion―with a variable degree of certainty―from the atom to universe.

It is easy to demonstrate a subjective character of time mentioned above. If we limit our “universe” to one hydrogen atom, and the observer’s ability to distinction between the atomic nucleus and electron as well as the size of the distance separating these both objects, we obtain two possibilities concerning time. The first one―created by assuming a constant nucleus-electron distance in course of the electron motion done, say, along a circle―cannot serve to establish any notion of time because no change of the system can be detected by the observer. But another situation is obtained when the distance between two mentioned particles changes systematically, say in effect of a planar motion of the electron along an ellipse. In this case the observer’s measurements are spread within the interval length equal to a double difference between the longer and shorter semi axis of the ellipse. If the motion is perfectly periodic, the observed interval of length repeats after the same period of time T. In result all time points accessible by the observations are enclosed within the interval

which repeats incessantly because no limit is imposed on the electron motion along the ellipse.

However, the everyday observations on time are evidently against the limit given in (1). The effect of these observations combined with imagination imposes a replacement of T in (1) by infinity:

Moreover, a further analysis of the contemporary situation as an effect of earlier situations combined again with imagination, provides us also with an infinite size of the interval of time concerning the past. In effect this gives a commonly admitted interval

Characteristically, the interval (3) encloses practically all possible events in nature, but it does not explain much what happens within (2) or (3). A rather simple example can be suggested by the quantum theory.

In the modern theory of atom―an object which is best penetrated by the quantum physics―we have a positively charged nucleus surrounded by the cloud of a negative electron charge. If the atom is in its lowest state of energy, called also the ground state, and no external fields or collisions act on it, it can remain―according to the present knowledge―in such a state practically infinitely long time with no change. Therefore no idea, or scale, of time can or should be applied in order to describe such atom.

But a different situation is obtained when―at some moment―the atom becomes perturbed, for example by the action of an external field. Let us assume that this field is independent of time. If the time moment of inclusion of the perturbation potential is denoted by

in any time moment

the properties of the atom are changed in comparison with those possessed at time (4). However―for a sufficiently weak perturbation potential

the atom will approach another stationary state, certainly different than that occupied at

the atom properties will be not effectively different than those possessed at

A fundamental difference in the Feynman’s and present treatment of the Schrödinger perturbation series concerns the system behaviour in dependence on time. In fact Feynman assumes that the time interval followed by a physical system on his diagrams is of an unlimited length, i.e. the interval is that of (2) or (3); see [

But these contacts are arranged practically on an equal footing: there does not exist a classification of the time moments which are more or less admissible by the system in course of its way with time. In effect a convergent result for the perturbation energy requires an enormous number of diagrams, or energy components, in order to approach a final perturbation result. This is expected to hold especially when the physical nature of the system makes the perturbation series only slowly convergent in the original, i.e. time-independent, Schrödinger perturbation formalism.

A look on the original approach proposed by Schrödinger makes it clear that the quantum system should contact its perturbation in different but specified ways, in dependence on the number of contacts. A full number of contacts of a given kind were called the perturbation order labeled by N; an increasing number of considered orders evidently increase the accuracy of a final perturbation result. Mathematically this led to special perturbation terms the number of which was strictly connected with the order N. When the perturbation of a non-degenerate quantum system was considered, the number of the perturbation terms

But the number of the Feynman diagrams

For a large perturbation order, say

On the other hand, the number of the Schrödinger perturbation terms from the formula (8) and

This means that in average

Feynman terms should be combined in order to give a contribution furnished by a single Schrödinger perturbation term for energy. In a computational practice this task could be difficult to be both programmed and performed.

The aim of the present method―instead to formulate a new computational problem―was a search to simplify both the scheme given by Feynman and that by Schrödinger. Since the atomic system and its perturbation potential remain constant, the time scale of the perturbation events became the object of an analysis. In fact we show that a suitable sequence of the system collisions with the perturbation potential done according to a circular scale of time can provide us readily with the perturbation terms given by Schrödinger and formula (8). Moreover, the individual terms of the Schrödinger series could be obtained from the proposed time scale without following their complicated derivation presented in [

As a first step we outline the fundamentals of the Schrödinger perturbation formalism specified for the case when a non-degenerate quantum state of a single particle is perturbed. Let this state be, for example, the lowest one of a set of non-degenerate quantum states being the eigenstates of the unperturbed and time-independent Hamiltonian operator

so

so for any pair of

Our aim is to calculate the eigenenrgies and eigenfunctions of a perturbed Hamiltonian

where

is a time-independent perturbation potential dependent solely on the particle position

In principle we seek for the wave-function solutions

and eigenenergies

of a new eigenproblem

but this may occur to be much more difficult than solution of an unperturbed problem (15). In general, by considering the energy alone, we look for a series

where

is the unperturbed energy and

are the correcting terms of (22) arranged according to an increasing perturbation order

In general any

is a combination of the matrix elements

where

and

are the unperturbed wave functions from the set (11).

Beyond of the matrix elements (26) also the unperturbed eigenenergies (13) enter the expressions (25). Excepting for the term

other series components

here and in (29) the bra

A total number of the kinds of terms which compose the sum of

belonging to successive perturbation orders i is equal to

where

A difficulty is in construction of particular terms entering any (31). With the aid of the Schrödinger formalism this construction becomes a very complicated task, especially for large

In general any scale of time is defined by the order in which the physical events do succeed on it. In a conventional scale of time, assumed for example in construction of the Feynman diagrams, the scale is extended from a minus to plus infinity. Therefore the time moment of the next event is compulsorily more distant from the beginning point than a former event; the events, in the perturbation theory, are represented by collisions of the perturbation potential with an unperturbed quantum system.

For the Schrödinger theory the perturbation order i, labeled usually by N, is suitable as a basis of classifying the events. For example it is convenient to assume that a separate amount of collisions belong to the order

In order to obtain a full set of the Schrödinger perturbation terms from the time points belonging to the scale of a given set of N points, the time points on

the scale should have a property to merge together in a definite way [see e.g.

This means, for example, that two points on the scale, say

can be merged together on condition that any of these points is not a beginning-end point on the scale; moreover the points represent a sequence

which is indicated by the fact that

The merging symbols representing the individual Schrödinger perturbation terms given by contractions mentioned in (33) and (34) are:

The merging process can be extended to a larger number of the time points than two. For example three time points

etc., for which the sequence relations

are satisfied. Certainly the number of the time points participating in any merging process is limited by

But there exist also combined merging processes of the time points which can be accepted by the theory. For example

can be a valid contractions pair on condition that there holds the relation

which is identical with (37a).

The merging processes listed above provide us with the side loops of time which are formed on the scale together with the main loop of time. The main loop is considered to be that loop on which the beginning-end point of the scale is present. Evidently some time points lying formerly on the main loop can be shifted to a side loop, or loops, in effect of the merging process.

By considering several sets of the merged points the rule (ii) mentioned above should be taken into account. This means, for example, that there is not allowed a crossing of the time loops on the diagram; see e.g. [

A consequent application of the idea outlined above provides us with a full set of the Schrödinger perturbation terms for energy belonging to any order N; see Section 6. These calculations, limited to

The next step to obtain the Schrödinger series is a partition of the number

which is the perturbation order N decreased by one. The −1 is a correction term due to a circular character of the time scale. The partitions done for individual N (and contractions of the time points for low N) are given in the Tables, see Tables 1-4 and

Evidently the perturbation order