Circular Scale of Time Applied in Classifying the Quantum-Mechanical Energy Terms Entering the Framework of the Schrödinger Perturbation Theory

The paper applies a one-to-one correspondence which exists between individual Schrödinger perturbation terms and the diagrams obtained on a circular scale of time to whole sets of the Schrödinger terms belonging to a definite perturbation order. In effect the diagram properties allowed us to derive the recurrence formulae giving the number of higher perturbative terms from the number of lower order terms. This recurrence formalism is based on a complementary property that any perturbation order N can be composed of two positive integer components a N , b N combined into N in all possible ways. Another result concerns the degeneracy of the perturbative terms. This degeneracy is shown to be only twofold and the terms having it are easily detectable on the basis of a circular scale. An analysis of this type demonstrates that the degeneracy of the perturbative terms does not exist for very low perturbative orders. But when the perturbative order exceeds five, the number of degenerate terms predominates heavily over that of nondegenerate terms.


Introduction
As soon as quantum mechanics in its wave-mechanical form was developed, the perturbation problem of the eigenenergies and eigenstates came into consideration [1].A necessity of the perturbation theory was dictated by the fact that only very few systems on the atomic level could be examined in an exact quantum-mechanical way.In an overwhelming part of physical problems, the approximate methods had to be developed, and one of them was the Rayleigh-Schrödinger (RS) perturbation framework [2].Rayleigh's name was involved because a similar perturbation method was applied by that author in a treatment of the acoustic waves; see [2].More developed approaches to the perturbation theory than [2] are in [3][4][5][6][7][8].
In fact, perturbation theory is a complicated formalism already at the level of a one-particle (one-electron) system that occupies a non-degenerate unperturbed state [1,2].Essentially the method is based on the calculations of the matrix elements between the unperturbed wave func-tions p , q , (1) and the potential function V which represents a perturbation of an originally unperturbed energy Hamiltonian.If any matrix element is considered as a result of a single particle scattering on the potential , the number of the RS terms required to calculate the perturbation energy increases dramatically, especially when many different scatterings on V are taken into account.In principle, a full RS series is obtained when the number of the different types of scatterings on is allowed to go to infinity.In practice, however, we consider only the perturbations of some finite order , where defines the number of factors of the type entering a product of terms (2) forming a single perturbation term.In general, any of these factors is based on the eigenstates of the kind of (1) of an unperturbed Hamiltonian.
The RS formalism of the perturbation of a non-degenerate state strictly limits the number N S of kinds of perturbative terms characteristic for a given .This number is represented by Different values of N S for different numbers are listed in Table 1.

N
The derivation of ( 3) is based on a complicated combinatorial formalism [9,10].But, in many cases, we look for a recurrence formulae that allows us to calculate N S from the lower-order terms , , , , , The aim of the present paper is to provide such formulae.This treatment is based on a circular time scale.
The idea of introducing the time variable as a tool to place the perturbative terms in order according to the size of came from Feynman [4,11].The scattering events with the perturbation are arranged along a straightlinear time scale and all diagrams connecting a given number of events should be taken into account.However, the number Consequently, a combination of N P energy terms into N S terms becomes an uneasy task.But the difficulties of the Feynman formalism can be avoided when its straight-linear time scale is replaced by a circular time scale.In the latter case, a one-to-one correspondence exists between the diagrams obtained on the circular scale and the RS perturbation terms [12][13][14][15][16].In effect, any component term entering the set of N S terms corresponds to a separate diagram contributing a definite formula of the RS perturbation energy of order .This result is attained by applying an appropriate contraction rule for the scattering events on the circular scale.Any contraction prescribed by this rule is different, and the whole number of diagrams obtained in this way becomes exactly equal to . Moreover, an analysis of all contractions for a given gives precisely the same energy terms, as they are provided by the RS theory.
In Section 3 we present the recurrence formulae for N S attained on the basis of a graphical analysis applied in [12][13][14][15][16].In fact, these formulae represent the complementary relations for N S obtained on the basis of N a and . The property of complementarity becomes evident if we note that any component of for which there is satisfied the relation: for any pair of integer numbers and b .The changes of a N N N S due to the change of are reported in Section 3.

N
Another advantage of the circular scale is its use in detecting the degeneracy of the Schrödinger perturbation terms.In fact, any diagram representing the perturbation term is either symmetrical for itself, or asymmetrical with respect to another diagram; see Section 4. This property provides us with a simple rule that there is no degeneracy of energy for a symmetrical diagram, but a twofold degeneracy is connected with any pair of asymmetrical diagrams.No other kind of degeneracy is obtained on the basis of the symmetry analysis of the diagrams.Sections 4 and 5 demonstrate that the twofold degeneracy due to a circular character of the time scale holds for the most part of the perturbative energy terms belonging to a given on condition .N 5 N 

Recurrence Formulae for N S and Their Complementary Character
In the first step, we point out that the recurrence formalism for N S can be obtained without an analysis of all contractions occurring on a circular scale for a given .If all scattering events are arranged on a line-see e.g. Figure 1 for -we can separate successively points on that line.
The Formula ( 11) is a very simple result that is checked in Table 2 up to the order .10 N 

A Change S of N S
Figure 2 presents an example of such separations for performed in each case with the aid of a single vertical line.
In many cases we like to calculate the number N S N of some perturbative order from This calculation can easily be performed following the diagram on Figure 3 with the case as an example.From Figure 4, we obtain: According to the results from previous studies [12][13][14][15][16], any set of separated points can be considered as lying on some special loop of time of the circular scale characteristic for a given .In the next step, such a loop has its characteristic number of contractions leading to a corresponding number of diagrams for that loop.In effect, the number of the RS terms corresponding to any separation of the kind represented in Figure 2 labeled by i s is dictated by the size of the component integer numbers that satisfy the relation

S S S S S S S S S S S S S S S S
The total number of N S is therefore equal to  from the recurrence formulae of ( 10) and (11).

S S S S S S S
S S  

S S S S S S S S S S S S S S S
The result of ( 14) agrees with the Huby-Tong formula of 28.
The difference (13) added to gives: Calculations similar to those presented pe  13) and ( 14) rformed for N not exceeding 10 are presented in Table 3.The di rences other than (15), such as those between (3); cf.here the data in Table 1 .

S S S S S S S S S S S
9 3 2 14 28

S S S S S S S S S S S S S S
28 9 6 5 42 90    diagram (b).Either of symmetry mentioned above is characteristic for any diagram of order N , so the diagrams can be classified as either symmetric, or asymmetric with  on the main loop of time with a point that is the most distant from  .For even N , the second point is labele by a point d 2 N on the loop.However, for odd N , the most distant point from  does not coincide with a cattering event on the time scale, but is half of the time interval between the scattering points, labeled by s  

S S S S S S S S S S S S S S S S S S S S S S S S S S
this is no point on the loop beyond the beginning-end point r  and there are no contractions on that loop; see Figure 6.
The symmetry behavior of the dia s the energy: the symmetr iagrams are nondegenerate, which means that the perurbative ter corresponding to such diagram is different from all o grams affect perturbation ic d m ther terms.But the asymmetric diagrams give pairs of degenerate energy terms.This degeneracy is only twofold because it is due solely to the property of asymmetry of diagrams entering given pair.According to the rules applied earlier [12][13][14][15][16] the diagrams presented in Figure 5 provide us with the following perturbation terms: This factor corresponds (apart of its sign) to the secon non-d ntum state d-order perturbation energy   because the side loop of diagram (d) has two points on it.In the Formulae ( 16)-( 18a) n E is a egenerate energy of an original (unperturbed) qua n .The sym la bols , , bel unperturbed energies of states , , p q r  The formula for the matrix element pq U between states p and q is given in (2).The repetition of the indices , , , p q r  in the numerator of the energy expressions given in ( 16)-(18a) implies a summation over p, q, r, 5, we give a s cti ··· In Section ate perturbation ele e for the degener terms of a given m pply on rul order N .

Algebra of the Time Loops in Diagrams and Selection of Degenerate Perturbation Terms
Graphical presentation of the perturbation terms is a rather tedious way to select the degenerate terms, but an algebraic expression of the diagrams can be developed to implify the selection problem.As an exa ple, we a s this kind of algebraic expression for diagrams of order 6 N  .
Any diagram can be represented as a product referring the numb er of points lying on the loops in that dia-to gram.The side loops are considered in chronological order, which means that the loops containing earlier scattering points are represented before those containing later points.An exception is the loop having the beginningend point  : this loop is presented regularly as the end factor of any product.For example, the loops in Figure 5 have the following notation: The non-degenerate energy terms are listed in Table 6.The diagrams on a circular scale corre onding to the listed terms are presented in [12].
  is given in [12]; a list of their algebraic represe tions is presented in Table 4.When one of two eq loops is near to the point        The calculations make reference to the properties of the time contractions characteristic for a circular scale of time along which the perturbative effect of a quantummechanical system is developed.
The property of asymmetry of the time loops on a circular scale is applied in examining the degeneracy of the perturbation terms.An absence of gener erturbative orders is und.However, a twofold degeneracy of most of perturbative terms can be detected when the perturbative order N becomes larger than 5.

Figure 1 .
Figure 1.Fundamental pattern of scattering points for calculating the number N S in the RS perturbation theory; see (3).The pattern for the perturbative order 6 N  is taken as an example.
Here is the same constant number for all terms in (8) and(9).A full set of for is given in Figure2.The number of terms the set is evidently equal to . 1 N  Because the loops,with their possible further contractions, behave independently each of other, any i s term of the set characterized by(9) provides us with the number of perturbative terms equal to

Figure 2 .
Figure 2. Separations of a fundamental pattern of scattering points in Figure 1 useful in calculating N S .The perturbative order 6 N  is taken as an example.The effect of separations is presented in (12).

Figure 3 .
Figure 3.The fundamental pattern of scattering points for calculating the increment 1 N N S S S     in the RS perturbation theory; see (13) and (15).The perturbative order 6 N  is taken as an example.

Figure 4 .Figure 3
Figure 4. Separations of a fundamental pattern of scattering points in Figure 3 useful in calculation of 1 N N S S S     for 6 N  ; see (13).        1 14 5 1 5 2 2 2 1 5 1 1 14 topological symmetry of energy diagrams on a circ A scale can be easily demonstrated by an example.Suppose we have the energy terms of the perturbative order 6 N  .In Figure5we present four diagrams of 6 N  ; a ful D l set of the 6 42 S  diagrams is given in [1 iagram (a), also cal ain time loop, has no contractions; it is symmetrical with respect to the line joining the beginning-end point of the loop with point 3, which is the most distant point from 2]. led the m  .The line divides the loop into halves.Diagram (d) has a similar symmetry that is characteristic by the time contraction 2:4.On the other hand, diagrams (b) and (c) represent a different kind of symmetry: a mirror reflection with respect to the line joining points  and 3 gives diagram (c) from (b), and similar reflection of (c) gives

Figure 5 .
Figure 5. Two symmetric and two asymmetric diagrams for energy terms belonging to .Symmetric diagrams (a) and (d) represent non-de ergy term ymmetri diag of

Figure 6 .
Figure 6.Diagram of the perturbative order N   ) give nd the be repre senting the side loops of diagrams (b) and (c n in Figure 5.There is no scattering point beyo ginningend point -

Table 1 .
Consequently N S ; see

Table 2 . The N S numbers calculated for 2 10
N  

Table 5 . A list of tw ld degenerate contributions to the perturbation energy of order . Their algeb e- presentations are g ven in T
Nevertheless, a sequence of L in degenerate products may be different, with the exception the last term of the product, which should be the same for a give egenerate pair.The degenerate pairs are tified in Table5, and the non-degenerate energy terms are listed in Table6.A characteristic point is that no kind of degeneracy b ngement of scattering points on the time scale is de-

Table 6 . Non-dege rate terms contributing to the pertur- bation energy of o er 6 N ne rd  . Their algebraic rep enta- tions are given in Table 4, for t e diagrams see [12].
Table 4 give degenerate te a non-degenerate contribution.