Circular Scale of Time as a Guide for the Schrödinger Perturbation Process of a Quantum-Mechanical System

We point out that a suitable scale of time for the Schrödinger perturbation process is a closed line having rather a circular and not a conventional straight-linear character. A circular nature of the scale concerns especially the time associated with a particular order N of the perturbation energy which provides us with a full number of the perturbation terms predicted by Huby and Tong. On the other hand, a change of the order N—connected with an increased number of the special time points considered on the scale—requires a progressive character of time. A classification of the perturbation terms is done with the aid of the time-point contractions present on a scale characteristic for each N. This selection of terms can be simplified by a partition procedure of the integer numbers representing 1 N − . The detailed calculations are performed for the perturbation energy of orders 7 N = and 8 N = .


Introduction
The scale of time, which is well known in everyday life and in science, too, is a product of a long experience. As far as we can distinguish the later events from the earlier ones, we organize the idea of time as a parameter which allows us to get an insight into the degree of the past, or future, connected with our observations.
In effect a tool to classify the events, and the time distances between them, is established. Conventionally this is done with the aid of an infinite scale extended between an infinite past-say representing the negative coordinates-and a sim-How to cite this paper: Olszewski, S. In practice the Schrödinger's quantum mechanics-developed in course of 1920's [1] [2] [3] [4]-has not much to do with the intervals of time. Its main idea was rather to distinguish between the stationary states of the chosen pieces of matter. Such pieces are described with the aid of the stationary eigenenergies and eigenfunctions, both kinds of parameters being independent of time. Concretely the classical Hamiltonian function of a chosen object is transformed into its operator form, and the integration of the classical Hamilton equations is replaced by a study of a differential eigenequation of the form Here Ĥ is the Hamiltonian operator represented by a sum of the kinetic and potential operators kin potˆˆ, so-for a single particle system-( )  (5) ψ is the eigenfunction called the wave function of an object, say a particle submitted to an external field having the potential V, symbol r is the position vector, E is the energy eigenvalue.
Because of ˆˆ, , , the momentum operator in (4) is of a differential character, whereas (5) represented by a function of the particle (object) position r , is of a multiplicative nature.
The problem is that even in relatively simple physical cases the eigenequation (2) is difficult to solve. By solution we understand a set of the eigenenergies 1 2 3 , , , and eigenfunctions 1 2 3 , , , ψ ψ ψ ψ =  (8) which satisfy (2). Only in very few physical cases equation (2) can provide us with simple solutions (7) and (8). The (7) are considered to be real energies of the system's quantum states, the (8) are the wave functions suitable in calculating other physical observables than energy.
In general the case when all eigenvalues on the right of (7) are different is classified as a non-degenerate problem, an opposite case is called to be a dege- Schrödinger was certainly aware about the difficulties connected with the solution of his Equation (2); see [3]. His proposal became to calculate the solutions of a rather complicated (2) (9) is called the perturbation potential, or simply a perturbation. In order to obtain possibly accurate results Schrödinger developed a formalism in which the solutions of (2) can be expressed with the aid of solutions of (2a). In this process-beyond of the solutions of (2a)-the matrix elements of the kind (10) are also involved.
A more easy treatment of the perturbation does concern the calculation of the energies of Equation (2) with the aid of solutions of Equation (2a) obtained in case of a non-degenerate case. Nevertheless an accurate calculation of these energies requires a complicated superposition of the solutions of (2a), as well as calculation of the matrix elements in (10). In principle these calculations were performed with no reference to the parameter of time; see Sec. 2.
The aim of the present paper is to point out that an introduction of the time scale-which has, however, a nature different than the well-known scale characterized by the formula (1)-provides us with a rather spectacular simplification of the original Schrödinger's perturbation scheme.

Outline of the Time-Independent Perturbation Theory of a Non-Degenerate Quantum State
A characteristic point is that Schrödinger obtained the solution of his perturbed equation without any reference to time [3]. An outline of a more modern time-independent perturbation theory is given, for example, in [5]. In the case of a non-degenerate quantum system let the unperturbed eigenequation be considered as solved. In principle we have an infinite set of the quantum numbers n for wnich the eigenequation (11) does hold. The number of the eigenfunctions and eigenenergies of the perturbed eigenequation let be also infinite.
For 0 λ = we obtain the unperturbed problem equivalent to (11), whereas for 1 λ = we have a full perturbation problem. In principle we assume as valid give the perturbed energy of state n also with the accuracy of the perturbation order N.
By assuming the convergence of the series in (13) and (14), an increase the order number N applied in the sequence 1, 2, 3, , N  (19) improves the accuracy of solutions presented in (13) and (14).
Physically as a more easy accessible and more interesting parameter, is considered the perturbed energy (14). Huby and Tong presented the number of the kinds of terms necessary to obtain the successive components entering the Schrödinger series for the energy perturbation of any non-degenerate state n; see [6] [7]. This number is expressed as a function of N by the formula For low N the numbers N S are also rather small, for example It should be noted that the kinds of the perturbation terms entering the set of N S do not depend explicitly on state n, but they depend solely on N. Any kind of terms is, in its turn, a combination of the matrix elements of the perturbation potential with the unperturbed wave functions ( ) 0 α ψ and ( ) 0 β ψ , given in (10).
Another dependence of the terms is due to the differences of the unperturbed energy ( ) As a rule the differences (23) enter the denominators of the perturbation terms, so there should be satisfied the relations etc.; see e.g. [8] for further details.
For 2 N > numerous terms entering N S composed of (10), (23) and (24) can be submitted to infinite summations over the states indicated on the left of (24).
In practice the way of deriving the sets of N S terms necessary for the Schrödinger perturbation formalism indicated above becomes a complicated task. Concurrent methods, obtained mainly without inclusion of the time parameter, are given in [9]- [17]. The computational applications performed with the aid of these methods seem to not provide us with a complete formalism suitable for a large perturbation order N. One of the by-products of the present paper is to make the perturbation method for large N to be more simple than before.

Scale of Time Suitable for the Schrödinger Perturbation Formalism, Its Contraction Points and Side Loops
Our idea is to replace a tedious calculation of the perturbation energy attained with the aid of solving the perturbed Schrödinger eigenequation by an immediate production of the perturbed energy terms due to an application of a suita- Beyond of the maximal loop of (31), a set of the minimal loops due to contractions 1: 2, 2 : 3, 3 : 4, , 2 : can be also created. We can have still the intermediate side loops like 1: 3, 2 : 4, 3 : 5, , 3 : or other loops larger than those due to contractions in (33).
may come also into play. A general rule is that the time loops due to the acceptable contractions should not cross. This means that, for example, the combined contraction due to the pair 1: 4 2 : 5  is not admissible.
A fundamental effect is that a full set of acceptable contractions for a considered order N gives precisely the number N S of the Schrödinger perturbation terms predicted by the formula (21) for that N; no superflous neither lacking terms do occur. This is checked for the orders between 1 N = and 7 N = in the earlier papers by the author [25][26][27][28][29][30][31][32][33][34]. A full set of diagrams necessary for 6 N = is given in [25], a similar set for 7 N = enters [34]. In the present paper the perturbation energy of the order 8 N = is also examined from the same point of view giving a similar agreement of the results; see Sec. 9.

Notation Applied to Represent the Energy Perturbation Terms
Only for the perturbation orders 1 N = and 2 N = the side loops for the main loop of time do not exist. But any N S term for 2 N > is a product of energy contributions due to the main loop of time and those due to the side loops, respectively.
The contributions due to the side loops are easy to access from contractions of the time points and will be discussed first. Any contraction : α β (37) where as a rule we have provide us with the energy multiplier equal to the energy correction indicates the perturbation order of energy contributed by the side loop represented by E ∆ . In result, when the difference indicated in (39) is larger than 2, we have more than one Schrödinger perturbation term represented by the side loop, for Such loop carries N symbols V and 1 N − symbols P.
Evidently for 1 N = no P symbol enters (41) and we obtain a single term for the perturbation energy equal to For the order 2 N = we have no side loops and the perturbation energy is represented by the formula The symbol P within the brackets on the left of (43) represents a reciprocal value of the energy difference, viz.
situated between two matrix elements of per and submitted to summation process over the dummy state index p. In effect The meaning similar to the term (45) does prolongate to any perturbation term given by the main loop of time carrying no contraction points. For example This formula has two P and two dummy indices (p and q) for summation over the quantum states with exclusion of state n which is submitted to perturbation. It is easy to extend (46) to an arbitrary order N.
Evidently the side loop created by (47) does provide us with the term however our task is to present also a contribution due to the main loop of time.
In this case contraction (47) transforms the term (46)-having no contractions-to the formula The whole perturbation energy due to contraction (47) is represented by the product of (47a) and (48) taken with a minus sign: whereas the contraction (50) implies the side loop having point 2 as free on it.
This makes the energy contribution due to the side loop equal to In effect the perturbation term due to contraction (50) is equal to product of (52) and (53): The minus sign in (54) is dictated by the presence of an even number of terms entering the final product.
The notation procedure indicated above can be extended to any perturbation order N.

Time-Point Contractions on a Circular Scale and a Check of Validity of the Energy Terms Contributed by the Side-Loops of Time
Let us begin with a maximal side loop presented by the time point contraction in (31). Because the number of free points of time present on the side loop in (31) is the energy contributed by the side loop due to (31) is equal to This energy has to be joined with the energy contribution given by the main and not P alone; see (53). But beyond of (57) we note that the main loop becomes similar to the time loop characteristic for the second-order perturbation term; see (52). In effect the main loop makes the whole contribution of the contraction (31) to the perturbation energy equal to 2 2 .
A formal check of validity of the energy expression given in (58) is simple: since the perturbation energy concerns order N, it should have the total number of P in the perturbation expression equal to 1 N − and the number V is equal to N. Respectively, the perturbation energy in (51) contains the number of P equal present in (58) supplies the lacking number of P and V in the term the required number of P and V in an energy term belonging to The same reasoning can be applied to any contraction of the time points   The total sum obtained from the 14 S like terms on the right is equal to Having the contraction data in (73)-(86) it becomes easy to construct the perturbation terms belonging to 6 N = . These terms are respectively: The numbers in brackets represent the quantity of the perturbation terms in a given rows.
In a similar way the results for the perturbation terms belonging to 7 N = and 8 N = are obtained; the terms are represented in Tables 1-6.

Comparison of the Present Method with an Earlier Recurrent Approach to the Perturbation Energy [34]
In [34] we presented a formalism which makes a recurrent calculation of the Schrödinger perturbation energy possible for an arbitrary order N. The method-outlined in the present paper-is based on partitions of the number 1 N − . It seems to be more transparent and systematical than that given in [34].

S S S VP VP VPV E E
(1) Table 6.   [34]. The mentioned data of Appendix are next compared with the corresponding data due to the present method; see Table 7.
On the left-hand side of Table 7 are presented the symbols of the perturbation terms applied in the partition notation of the present paper, on the right-hand side of Table 7 the method represented in Appendix of [34] is applied.
There exists a full agreement of the data obtained in the present paper with those taken from Appendix of [34].

Summary: General Properties of the Scale of Time Suitable to Calculate the Schrödinger Perturbation Energy
One of the fundamental processes of quantum mechanics is a change of a given system upon the action of some perturbation potential which-in its character-can be independent of time, but is dependent solely on the particle coordinates. To calculate the result of such a change acting on a non-degenerate quantum system, the Schrödinger perturbation formalism-represented by the sets of energy terms labelled by orders N-is required. In principle no time approach, or time parameter, should be used to this purpose.
We are guided, however, by the Leibniz idea that a suitable arrangement of the physical events along a time scale can be helpful in an analysis of any system change, including the perturbation effect. Consequently, by assuming that a change of a system-also due to the action of a time-independent perturbation potential-requires some interval of time, a sequence and origin of the time moments entering such interval can be of importance.
In principle there exist many ways according to which the necessary sets of Table 7. Comparison of the energy terms calculated in Appendix of [34] with those obtained in the present paper: an example giving the terms belonging to the order 6 N = .
The first 14 terms presented in the right-hand side column are calculated-according to [34]-automatically on the basis of the results obtained for 5 N = . A full presentation of the perturbation terms has been done for orders 7 N = and 8 N = ; the terms of the lower N are accessible in the literature presented before [25]- [34].
At the first sight it seems that the paper has only a purely mathematical background. In fact the aim is to solve a definite Schrödinger differential equation, but the way to do that is to solve first a presumably more simple equation. Next the solutions of that more simple equation should be combined into those belonging to a more complicated problem.
Both equations are assumed to be different by a potential change independent of time. In fact the time parameter neither enters the actual perturbation equation, nor the equation representing a former more simple problem. Nevertheless the change of the potential-equivalent to the change of the Hamiltonian operator between the unperturbed and perturbed equations-occupies some time. We assume the time of the potential change as negligibly small. A much more longer time, therefore of a non-negligible size, is expected to be occupied as an effect of the original potential change. This is so because the perturbed, i.e. originally unstable system, should wait to occupy one of its stationary states. In effect the time, required to make the perturbed system equivalent to a stationary object, can be long. An estimate of the size of that interval is beyond of our ability. Mathematically however, the both states, unperturbed and perturbed one, are both accessible and can be defined without any reference to the notion of time. So a question may arise what is the role of time-if any such role does exist-in the perturbation theory?
With the absence of any time intervals in the formalism, the answer is that time is an ordering parameter. Moreover this role is rather of a gradual character World Journal of Mechanics because it does not concern the perturbation process as a whole, but is decisive in the successive steps of that process. In fact the perturbation effect can be separated into parts called the perturbation orders. Any order N is characterized by: 1) a definite number of the Huby-Tong kinds of the perturbation terms specified by the formula (21); 2) a constant number of 1 N − terms P and number of N terms V entering any perturbation term belonging to the order N.
But beyond of the number of terms characteristic for a given N, an important role plays the sequence of the "collision" events of

A Philosophical Background Concerning the Present Results
A philosophical background of the results obtained in the paper seems to be twofold. The first aim was to obtain a general look on the shape of the time scale.
A principal point becomes here to get a real relevance of the question of the direction of time, or more simply the problem of sequence of the time events, to some physical process [35]. Let us note here an opinion that the theory of the whole world time is a redundant concept-one only needs a knowledge of world's possible configurations [36].
A reply in the present case is that if the scales belonging to individual N are considered, their shape is evidently a closed line. A characteristic circular-like character of the time scale suitable to calculations of the perturbation energy for a given N seems to be not a unique property in physics; see e.g. Rey [37] and Zawirski quoted in [38]. A much discussed reference which can be cited here is In fact a total scale of time applied in the present paper-because of an increasing number of collisions with the potential per V -does increase gradually with N. This makes the second step of the time way, i.e. that due to an increase of N, similar to an infinite linear scale referred in (1).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.