Circular Scale of Time and Construction of the Schrödinger Perturbation Series for Energy Made Simple

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.

• Circular Scale of Time
• Schrödinger Perturbation Series for Energy
• Non-Degenerate Quantum State

1. Introduction

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” [1] , time is an independent variable in the classical mechanics. In fact the description of any motion cannot be done without the use of the notion of time: all fundamental mechanical parameters like the position change of a body, its velocity, acceleration, the force acting in course of the motion, apply the notion of time. With the development of the theory of relativity, a special attention has been attached to the time interval and its peculiar properties.

A special treatment concerning the time notion is applied in the quantum theory; see e.g. [2] [3] . Here the time variable does not possess its independent operator, so only the changes of the observables obtained with a change of time are considered. This kind of approach is done on the basis of commutators of the operators representing the observables with the Hamiltonian operator. In fact the energy operator becomes a central operator and energy variable becomes a central physical parameter of the quantum formalism. A characteristic point is that the notion of the time interval is practically banned out of the quantum theory in which a probabilistic approach is mainly applied in description of the effects connected with the particle transitions.

A general idea of the present and former papers by the author [4] - [9] is to point out the time importance in the use of the quantum perturbation theory. This theory has been developed [10] simultaneously with the Schrödinger wave-mechanical approach to the quantum systems [11] [12] [13] . Evidently it became clear already at the introduction moment of the Schrödinger wave mechanics that an exact application of that theory is possible only for very special physical systems. This means that in a treatment of other systems only the approximate solutions should be sought and used. Typical situations concerned an interest in the systems which differ only slightly from the systems which are easy to solve. The difference could be reduced to a rather small potential present in a more complicated Hamiltonian. Such a potential―called the perturbation potential―when combined with the wave functions and energies of a more simple system, could provide us with an approach to similar parameters of a more complicated system. The corresponding formalism―elaborated by Schrödinger and called the perturbation theory [10] ―became a purely mathematical problem in solving of which the notion of time has not been applied. Nevertheless, an accurate treatment of the perturbation formalism occurred to be an extremely complicated task. This difficulty becomes clear already at the stage of presentation of the problem.

But in a series of former papers [4] - [9] an attempt has been done to demonstrate that the introduction of the time notion, as well as application of the circular scale of time as an appropriate scale in the perturbation theory, can substantially simplify that theory and its calculations. The aim of the present paper is to outline the perturbation theory in a still more simple way.

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 [14] . The results and drawbacks of the method based on the Feynman diagrams are discussed in Section 3.

2. Scale of Time Considered in Reference to the Properties of the Observed System as Well as Abilities of the Observer

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 ―we can assume that in effect of the action of at some

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

the atom properties will be not effectively different than those possessed at . In other words the atom behaves at (7) as an unchanged object identical to that obtained at time in (6). For such an object the idea of time loses its sense. A question becomes how time is going within the interval between and .

3. Feynman’s Treatment and Present Treatment of the Time Interval Characteristic for the Perturbation Process

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 [15] . The answer of the atomic system to an applied perturbation is a set of energy transitions in different time moments done to different quantum states which are possible to be attained for an unperturbed atom. If the degeneracy with respect to energy of the system does not hold, only one of the system states is the ground state, the other ones are necessarily the excited states. In principle all excited states of the unperturbed system are admissible for the energy transitions in the perturbation process. In order to obtain a final result of the perturbation, the Feynman’s method admits that gradually more and more time moments are involved in the system contacts with the perturbation potential.

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 characteristic for a given N could be expressed by the formula [16] [17]

But the number of the Feynman diagrams , classified as belonging to the order N and necessary to obtain the energy contribution corresponding to N, was [14]

For a large perturbation order, say , we obtain from (9) the number of the Feynman terms equal to

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

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 [10] .

4. The Schrödinger Perturbation Formalism and a Suitable Scale of Time

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 . The unperturbed eigenenergies are

so is the lowest energy. The time-independent parts of the eigenfunctions of corresponding to energies (13) are

so for any pair of and there is satisfied the eigenequation

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 . We assume that the potential (17) is sufficiently small to represent the perturbation formalism by a convergent procedure.

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 entering the sum (21) are represented by expressions containing infinite sums over the quantum states. A single infinite sum is characteristic solely for :

here and in (29) the bra means and refers to any energy state beyond of that labeled by 1.

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

belonging to successive perturbation orders i is equal to

where is presented in (8) for .

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 . But this task can be evidently simplified by introducing the scale of time suitable for the Schrödinger perturbation problem.

5. Properties of the Time Scale

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 , another set of collisions belong to , still another set belongs to , etc. When the scale of time for a set of collisions is assumed to be not infinitely progressive but circular, its advantage is that the Schrödinger perturbation terms occur almost automatically from it. The basic postulate is that collisions of the external perturbation potential with an unperturbed system can be grouped together into specific sets. The total number of the time points of collisions which enter such a set does not exceed N. So we have one collision point of time in the set belonging to , two collision time points in the set belonging to , three collision time points in the set for , etc. For each set, after travelling the whole set of time points belonging to that set, the system returns to its beginning time point after which a new set of collisions can begin. This situation provides us with an unlimited number of the time scales of a circular character, each scale having time points on it. For we have only one point on the scale which represents a beginning-end point of this scale, but the scales belonging to have evidently one or more time points beyond the beginning-end point of time; see Figure 1 where diagrams for , and 3 are presented.

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

Figure 1

Diagrams denoting the Schrödinger perturbation terms for and 3

the scale should have a property to merge together in a definite way [see e.g. Figure 1(d)]. The fundamental rules concerning the merging process are two: (i) the beginning-end point on the scale describing a whole perturbation term is free from the merging process; (ii) the merging between two or more time points present on the scale can be done only in the way that a sequence of the time events represented by these points is not violated by the merging process.

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 can be also merged together. This situation is represented by the contraction symbols

etc., for which the sequence relations

are satisfied. Certainly the number of the time points participating in any merging process is limited by where the term −1 is due to exclusion of one, i.e. the beginning-end point of time on the scale.

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. [4] .

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 , have been presented already before [8] [9] . However a plain rule allowing for calculating all perturbation terms belonging to any given N was lacking. In the next Sections we develop such rule on the basis of partitions of N and contractions of the time points entering these partitions.

6. Partitions of the Time Points on the Scale

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 Table 6.

Evidently the perturbation order does not provide partitions since

Table 1

Partitions of the time points and Schrödinger perturbation terms for energy; orders N from to . are the Huby-Tong numbers of the Schrödinger terms labeled by the index i; see (8). The abbreviation symbols are represented in (41)-(41c); see also [ 8 ] [ 9 ]

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