Circular Scale of Time and Its Use in Calculating the Schrödinger Perturbation Energy of a Non-Degenerate Quantum State

The paper presents a circular scale of time—and its diagrams—which can be successfully applied in calculating the Schrödinger perturbation energy of a non-degenerate quantum state. This seems to be done in a more simple way than with the aid of any other of the perturbation approaches of a similar kind. As an example of the theory suitable to comparison is considered the Feynman diagrammatic method based on a straight-linear scale of time which represents a much more complicated formalism than the present one. All diagrams of the approach outlined in the paper can obtain as their counterparts the algebraic formulae which can be easily extended to an arbitrary Schrödinger perturbation order. The calculations and results descending from the perturbation orders N between N = 1 and N = 7 are reported in detail.


Introduction
What is time? My answer is that it is a parameter which allows us to distinguish a later event from an earlier one; this distinction seems to be a fundamental property of time. On the other hand, according to Springer's "Physikalisches Handwörterbuch" [1], time is defined as an independent variable of classical mechanics. One is suggested to add here the adjective "non-relativistic" to the notion of mechanics, because the relativity-in its special picture-makes any time interval dependent on such parameters as the body velocity and light velocity. Evidently in the general relativity the dependence of time is still more ex-How to cite this paper: Olszewski, S. tended, for example due to the presence of the mass of the body [2].
In science an important problem of time became to couple its behaviour with some other physical properties than those given by classical mechanics. Perhaps the best known example is here the entropy and its connections with time. In brief we need the parameters, or effects, which can be examined parallelly with time, though they do not necessarily represent an explicit dependence on the time variable.
In the present case such example of the time connected with physics is given by a quantum perturbation effect. We assume that at some time moment-more or less accurately known-some time-independent perturbation to a quantum system is applied. For a ground state in the absence of the perturbation effect the notion of a stationary state implies an infinite duration of that state. Usually we are unable to follow in detail the history of a system changed by the perturbation, but-according to Schrödinger-we know the end of the state history equivalent to the end of the perturbation process: this is a new stationary state having a new eigenenergy, different than possessed by the system state before the perturbation was applied.
Our aim is to present the time dependence of the perturbation history-and its results-in a possibly transparent way.

Quantum-Mechanical Characteristics of the Schrödinger Perturbation Process
In fact the original characteristics of the perturbation process done by Schrödinger [3] did not involve the idea, or a variable, of time. Also in more modern treatments of the Schrödinger perturbation theory-see e.g.  (2) which is dependent only on the position variable r . By assuming-for the sake of simplicity-that the unperturbed problem is a non-degenerate one, we look now for the solution of the perturbed eigenequation ( ) ( (4) are the sets of the energy eigenvalues and eigenfunctions calculated respectively to some chosen perturbation potential (2).
The order N can be referred to the perturbation energy per E and perturbation wave function ( ) per r ψ of a non-degenerate quantum state by the formulae (see e.g. [4]): The both series, (5a) and (5b), are expressed in terms of the powers of a parameter λ . These powers of λ represent in (5a) the order given in (5) (8) and detailed values of S N are given in Table 1. But simultaneously-to the best of my knowledge-no systematic rule was provided to build up the set of individual terms entering (8), and this task becomes a much complicated one for large N.
In effect the calculation of terms (7) suitable for large N becomes a difficult task already at the stage of their construction. But a removal of this complication provides us not only with a simplicity necessary to solve the calculational problem. In fact, the importance of the perturbation methods in general can be considered as a decreasing obstacle in view of the development of the computational machinery and its technique applied to solve the physical problems. The point of importance is that an essential simplification can be attained due to the introduction of the time parameter into the perturbation theory. This introduction provides us with a suitable arrangement of the time points on the scale labelling the contact events of the perturbation potential with an originally unperturbed quantum system. The details of this idea and its use in the Schrödinger method are presented below.

Perturbation Order and a Suitable Scale of Time
Only for very small N we have The formulae (11) and (12) imply that in order to get-in average-a single

Perturbation Process along a Circular Scale of Time and Its Energy Terms
We assume that the perturbation process is a set of successive collisions of the perturbation potential (2) with an unperturbed quantum system. The collision events are extended along a topological circle characteristic for a given order N of the perturbation potential; in the next step the collisions are labelled by separate time points whose number is equal to N. Therefore the number of the time points on the scale increases gradually with the increase of N; see Figure 2 and A characteristic feature is that the set of the time points present on the scale characteristic for a given N is sufficient to represent all S N perturbation terms given in (8); moreover we obtain a one-to-one correspondence between the individual diagrams obtained with the aid of the scale and the Schrödinger energy terms entering the perturbation order N; see [9] [10] [11]. This goal can be at-   3) any other contraction of the time points than that satisfying the rules 1) and 2) should not be taken into account.
In effect, beyond the time loops indicated in Figure 1 and

Energy Terms Belonging to N = 1 and N = 2
Evidently-according to the rules 1) and 2) given above-no contraction as well as no side loop can be created for N = 1 and N = 2. The first contraction of the time points is possible for N = 3 between the points 1 and 2 represented by the In this case, beyond a non-contracted diagram for N = 3 presented in Figure 4, we obtain a new diagram connected with (13); see Figure 5. In effect we obtain for N = 3 the number of two diagrams: that of Figure 4 and that of Figure 5.
This is in accordance with the formula (8) from which we have   It is easy to check that which imply only single diagrams present for N = 1 and N = 2 in Figure 1 and The perturbation energy connected with N = 1 is represented by which is a single matrix element.
On the other hand, for N = 2 a summation process over the running states p different than n is involved: The symbols V are connected with the matrix elements in the numerator, symbol P refers to a single energy difference in the denominator.

Contractions of the Time Points on the Scale Provide us with the Side Loops of Time; Perturbation Orders N = 3 and N = 4
For N = 3 we have three time points on the scale: 1, 2, and 3. Let 3 be the beginning-end point, so the points 1 and 2 can be submitted to contraction:  Figure 4. This gives the energy term where p n ≠ and q n ≠ . On the other side, the contraction (18) (see Figure 5) gives the energy term which is a product of and V which is the term given in (16). The product (20) is taken with a minus sign.
It has to be noted that the power of the energy term in the denominator in (21) is equal to the power of P on the left of (21). The minus sign in (20) is dictated by the even number of the bracket terms present in the product in (20); an odd number of the bracket terms presenting an energy term leads to a plus sign for that term; see (16), (17) and (19).
The term V in (20) represents a contribution due to a side loop of time created by contraction (18); see Figure 5. Because of a difference of the time point indices 2 and 1 entering (18) which is equal to the side loop created by contraction (18) contributes the term entering as a multiplier in (20). In effect the total perturbation energy of N = 3 is equal to a sum: because of (23) taken into account in (20).
The energy belonging to the order N = 4 (see Figure 6) can be considered in a similar way. If the beginning-end point on the scale is labelled by 4, we have three time points where , , p q r n ≠ . Next come contractions of the points in (25): : : The contractions presented in (27)-(30) give respectively the energy terms: where , p q n ≠ and where , p q n ≠ and where p n ≠ and ( ) ( ) .
Evidently the fourth term on the right of (36) is equal to the second term because of symmetry.
The rule defining the sign of the perturbation terms is very simple: for an odd number of terms entering the product giving a perturbation term the sign of

Energy of the Perturbation Orders N = 5 and N = 6
The time scales corresponding to above N are presented in Figure 7 and Figure 8.
Here the recurrence procedure can be useful to apply, so for N = 5 we take first into account the perturbation terms of the order N lower than 5, and for N = 6 the terms of the order lower than 6, respectively.
In this way the first five terms belonging to the order N = 5 can be obtained from S 4 = 5 terms of Section 6 by introducing the time point 4 as a free point different than the beginning-end point of time. This makes any bracket contribution due to the main loop of time entering ΔE 4 [see (36)] changed by an increase equal to PV put at the end of the bracket term. The first 5 energy terms belonging to ΔE 5 are: The first term in (37) The contractions in (38) together with the side loops created by them give the following energy terms: which give respectively two perturbation energy terms: The last set of the energy perturbation terms belonging to N = 5 is given by a single contraction In this case the time points 1 and 2 present before point 3 can be either free, or contracted together. For 1 and 2 free the contraction in (42) gives the perturba- On the other hand, the contraction 1:2 combined with that in (42) gives the perturbation term ( ) In effect we obtain from (37)  .
Again, because of the presence of ΔE 3 , the sixth term on the right of (46) represents two Schrödinger perturbation terms. Evidently-because of symmetry-some terms entering (46), for example the second term and one-by-last term on the right, become equal. .
In the next step we take into account that the time point 5 for N = 6 can contract with point 1 and all points between 1 and 5. This gives the following contractions and the energy terms corresponding to them:  which is the expected result; see Table 1. A full perturbation energy of the order N = 6 is equal to a sum of the terms belonging to expressions listed above equation (69); see (47)-(61a) and (64)-(68).

Perturbation Energy Belonging to N = 7
This is the most complicated case considered in the present paper. The first S 6 = 42 terms are those connected with N = 6 because the time point 6 is now a free point of time on the scale; see Figure 9 and a list of terms below (69). The energy which give also a set of S 6 = 42 perturbation terms: In effect the number of terms due to (104)-(113) is equal to 2S 4 = 10 because the points 1, 2 and 3 can combine in S 4 = 5 ways.
The last set of contractions containing point 6 is represented by 5:6. When a combination of 5:6 with the set of free time points 1, 2, 3, and 4 is considered we The remaining combinations with 5:6 are due to contractions between points 1, 2, 3 and 4: ( ) The total number of energy terms due to (114)-(126) is S 5 = 14 which is the number of combinations due to presence of the 4 free time points, see (8) and Table 1.
In total we obtain for N = 7 the S 6 = 42 energy terms collected in (70), next also S 6 = 42 energy terms collected in (87). Another set of terms is given in the formulae from (88) to (95) which provide us with which is not only in accordance with the formula (8), but satisfies also the formula:

General Characteristics of the Energy Perturbation Terms
In general the terms of the Schrödinger perturbation energy which originate from a non-degenerate quantum state are represented by the products of the contribution due to the main loop of time and contributions due to the side loops.
This second kind of contributions is equal to definite perturbation energies Nevertheless the sum of the power exponents of P present along the scale-those which remain on the main loop as well as those which are shifted to the side loop or loops-should remain unchanged. In effect any perturbation term belonging to a given N has the same number N − 1 of the P terms and number N of the V terms, because the total number of V terms in the main loop and side loops remains constant. All S N terms give different diagrams along the Journal of Modern Physics time scale plotted for a given N, but the computational results due to several diagrams can be equal which is the effect of the diagram symmetry.

General View on Contractions of the Time Points on the Time Scales and Their Application
The way of calculating the Schrödinger perturbation energy-called sometimes also the Rayleigh-Schrödinger perturbation series-presented in the paper is rather different than procedures applied in the former approaches; see e.g. [12]. In effect of creation of the new loop of time, a new-i.e. the second-perturbation term of energy is obtained for the perturbation order N = 3 beyond the term given in (136). This is a product of two bracket terms, viz.
The first bracket term is given in (21), the second term is simply the perturbation energy of order one (N = 1); see (16).
For N > 3, say for N = 4, the free time points on the main loop can be t 1 , t 2 and t 3 , whereas the point t 4 is assumed to represent the beginning-end point on the loop; see Figure 6. In this case-beyond contractions between the neighbouring time points like (137) and (137a)-the contraction between the non-neighbouring time points is also possible. This contraction implies that the point t 2 which originally is placed between t 1 and t 3 should be shifted to a side loop. This side loop has its beginning-end point given by contraction (139) or (139a), but one free time point, namely t 2 , does remain on the loop. In effect the side loop becomes identical to that for N = 2; see Figure 3. The perturbation energy due to contraction (139) is therefore equal to product For the first bracket term in (140) see (21), for the second bracket term-see (17).
The minus sign in (138) and (140) is dictated by the even number of the bracket terms entering product.
In Table 2 we summarize the data on the time points and their contractions which give the energy terms belonging to the perturbation orders from N = 1 to N = 6.

Conclusions
The history of investigations on time is probably as old as the history of science.
In the Newtonian formulation of mechanics, the time interval is independent of any other physical parameter; in the theory of relativity, the dependence of the time interval is mainly due to the speed of the observed change.
But in many cases, including the problem considered in the present paper, the influence of the speed effect-or other physical parameters-on the time intervals can be neglected. The kind of approach proposed here is different than the Newtonian-like, namely an insight into time given by Leibniz [13] [14] [15] seems to be here of importance.  In principle the perturbation theory-linked also with the Schrödinger's authorship [3]-provides us with a method how solutions known for a simple problem can provide us with an approximate knowledge of more complicated Schrödinger's solutions. A difficulty was that a tedious procedure had to be applied in order to extract to calculations the separate kinds of energy terms belonging to a large perturbation order N. This difficulty could be much reduced when the time scale of a circular character composed of the N collision time points of a quantum system with the perturbation potential V per is assumed for each N.

S. Olszewski
The number of the allowed time-point arrangements on the scale provides us precisely with the S N perturbation energy terms characteristic for a given N. In effect the perturbation terms should not be derived with the aid of a usually tedious iterative procedure connected with solving the perturbed Schrödinger equation, but can be readily obtained by analyzing the contractions to which the time points are submitted along the scale.
It should be noted that agreement of the results for S N , as well as the energy perturbation terms for a given N, obtained in the present theory with those calculated by the Schrödinger formalism is not proved in general but has been demonstrated in the paper for the perturbation orders beginning from N = 1 to N = 7.
No time intervals or continuous time variables are considered in the paper.
The calculations are based on definite sets of the discrete time points representing the collisions of an unperturbed non-degenerate quantum system with a time-independent perturbation potential.

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