_{1}

^{*}

The Schrodinger perturbation energy for an arbitrary order N of the perturbation has been presented with the aid of a circular scale of time. The method is of a recurrent character and developed for a non-degenerate quantum state. It allows one to reduce the inflation of terms necessary to calculate known from the Feynman’s diagrammatical approach to a number below that applied in the original Schrodinger perturbation theory.

The conceptual background of quantum mechanics seems to be inadequate when the perturbation methods are considered; see e.g. [

On one side a complicated Schrödinger formalism should be applied, [

The aim of this paper is to address these highlighted problems by applying a circular scale of time with diagrams. Several steps in this direction have previously been achieved [

The present contribution combines the results obtained from both the circular scale and the partition properties of

Time scale is represented by a sequence of events (time points) with different intervals between them.

If we are travelling by train for example, we do not really need to rely on a watch to observe time lapse as it is sufficient to study the station names. By analysing the stations we are able to estimate not only the geometrical distance from the point of our departure, but get also an idea of time progression. The only knowledge we need is a sequence of stations extending along our trip. If we do not insist on the accuracy of the intervals of time, we can satisfy ourselves to know that “nearer” means an earlier station in time, whereas a “further” station means a later moment in time.

Obviously this kind of parallelism holds true irrespective of the type of track along which the train is travelling, although the tracks can vary in terms of their shape. One type, for example, is that we are systematically further from our departure point, but it could also be a different type of track―perhaps circular in nature. This would mean that after travelling through several stations the train arrives at its original departure point. Then comes an important aspect of the journey. If the track crossing the departure-end station is the only one, its effect on the train―if it preserves its motion (including the motion direction)―is to repeat its trip along the same set of stations as before.

Another scenario may occur however, if we assume that there is a possibility for the train to choose to travel upon a differing track at its original departure end point. Without going into details, a natural conclusion is that a journey along an alternative track will be different than that of a former one.

This simple example depicted above connecting time with a track can raise a question concerning our concept of time in general. From childhood we are accustomed to a notion of time which is associated with change. If daily events were to be essentially the same, we could need a watch to track each day as otherwise no distinction between them could be realised.

In practice, however, a completely different picture is inscribed in our memory―nothing, or almost nothing, repeats itself completely. This means that in all our observations we can find occurrences that do not repeat themselves. Therefore this supports a conclusion that time behaves like a train as in the first example quoted above. No station is repeated, time is progressing from a start point to an arbitrarily distant moment in the future. As it is easy to understand that our start point can be considered by other observers as a rather advanced point in time, we can easily also relate to a concept of the past. This notion―considered in an opposite direction to our everyday experience of progressing time―can classify events at an arbitrarily greater distance from us. Quantitatively, the past distance becomes similar to the notion of a distant future.

In effect, if we numerate the events, we obtain a sequence of the form

for the future, and

for the past.

Certainly the scale of time, being a result of our observational knowledge and imagination, is not the only one which can exist. Let us consider an atom which is not influenced by any external or internal factor. For example an atom of a non-radioactive element which in its ground state can remain static for an infinitely large set of measurements undertaken by an observer. This set can be represented, for example, by a sequence of constant numbers, say

since the effect of the first measurement does not vary with the next ones, on condition the measurements do not influence the atom.

A slightly more complicated illustration is that of a periodic mechanical system, for example represented by the electron moving in a hydrogen atom described by the Bohr-Sommerfeld atomic theory; see e.g. [

The set can be made infinitely long because no change of two special electron positions in the atom mentioned above can be attained in the absence of the system perturbation. Obviously more complicated sets of repeated numbers than (4) can represent more complicated periodic systems.

A characteristic point is that not only in classical physics, but also in quantum theory, we follow the scale of time of (1) and (2) which is typical for everyday life. According to this scale the events labelled by time do not repeat themselves. Our aim is, in the first step, to point out that an application of such scale in the quantum theory is not really justified. To this purpose a time analysis of the quantum-mechanical perturbation problem within the formalism introduced by Schrödinger will be carried out.

Schrödinger based his approach to the quantum theory of the atomic systems on the property of duality represented by the particle-like and wave-like pictures of the matter. The Hamiltonian operator on which the approach is based is in fact similar to that of a moving particle. However the operator of the particle momentum is chosen in the way that it gives an equation for a wave associated with the examined particle. A well-known fact on such an approach is that it gave very satisfactory practical results. When solved accurately, the Schrödinger wave equation could produce very accurate data on numerous observables from the atomic world. This especially concerned the electron energy of atoms.

The point raised by Schrödinger himself was that, in general, his treatment becomes a very complicated mathematical task for relatively simple systems. Such a situation exists when a rather simple atom is influenced by an external field, for example of an electric or magnetic character. To avoid the difficulty of solution of his equation also being valid in this case, Schrödinger proposed to consider the potential of an external field as a small perturbation of the potential governing the atom in the absence of any external influence. In consequence, the perturbation formalism operates with the wave functions describing the states

and energies

of an unperturbed quantum system together with the perturbation potential

characteristic for an external physical effect.

On many occasions it can be assumed that (7) does not depend explicitly on time but is a function solely of the position vector

In fact, in order to solve the perturbation problem, Schrödinger developed a complicated formalism which does not apply the notion of time at all. This was possible because both the beginning state of an unperturbed system (atom) and the state obtained in effect of the perturbation, are assumed to be stationary and therefore independent of time. When limited to the calculation of the perturbation energy,

on condition we assume that the series is convergent and N is large enough to give a good approximation for deltaE. In practice the complication of

More than twenty years after Schrödinger’s method, Feynman classified quantum events according to their dependence on time. In effect a formalism suitable to calculate the sum

However this calculation did not follow, in general, the series of

(see [

whereas the number

Thus (10a) and (10b) imply that on average approximately about

Feynman terms should be combined in calculating an individual Schrödinger perturbation term entering

However, the scale of time applied by Feynman does not in fact difffer, from the conventional scale presented in (1) and (2): there is no limitation, or constraint, imposed on the time variable applied in that scale. Our aim is to present in Sections 4 and 5 an approach where time is classified according to a scale which is different than that characterized by (1) and (2). This is a scale represented by a topological circle, or is composed of a set of such circles. Efficacy of the scale obtained in this way in calculating the Schrödinger perturbation terms becomes evident.

In general this kind of scale is circular in its nature, since it is based regularly on a loop of time being a topological circle. A single circle, called the main loop of time, is characteristic for any

The details of the scale depend on the order number

An important point is that the total number of scales, or diagrams representing them, which can be constructed is equal to

Examples of diagrams for

For any

This is a sufficient condition to obtain all

Because any diagram represents one kind of the

One kind of the energy terms, namely that corresponding to a single, called main or uncontracted, loop of time is very easy to calculate. For example for

The number of P’s is equal to

whereas the P’s entering (12) are subsequently the reciprocal energy terms:

For an uncontracted diagram all powers of P are equal to 1 (see (12)), but for contracted ones at least some of the powers of P, as well as those of the terms entering (14), can be larger than 1.

Evidently the summation process over the indices

One of the points of time on the loop (the beginning-end point) is free of manipulations like summation or contraction. Let us label it by 7 and attribute to state n. Consequently the points 1, 2, 3, 4, 5, and 6 on the same loop can be attributed respectively to

The partitions of 6 into components, beginning with the lowest terms is represented in

Partition index | Partition | Index referring to the perturbation terms in Tables 2-4 | |
---|---|---|---|

1 | 1 + 1 + 1 + 1 + 1 | 1 | (1) |

2 | 1 + 1 + 1 + 1 + 2 | 1 | (2) |

3 | 1 + 1 + 1 + 2 + 1 | 1 | (3) |

4 | 1 + 1 + 2 + 1 + 1 | 1 | (4) |

5 | 1 + 2 + 1 + 1 + 1 | 1 | (5) |

6 | 2 + 1 + 1 + 1 + 1 | 1 | (6) |

7 | 1 + 1 + 1 + 3 | 2 | (7), (8) |

8 | 1 + 1 + 3 + 1 | 2 | (9), (10) |

9 | 1 + 3 + 1 + 1 | 2 | (11), (12) |

10 | 3 + 1 + 1 + 1 | 2 | (13), (14) |

11 | 1 + 1 + 2 + 2 | 1 | (15) |

12 | 1 + 2 + 1 + 2 | 1 | (16) |

13 | 2 + 1 + 1 + 2 | 1 | (17) |

14 | 1 + 2 + 2 + 1 | 1 | (18) |

15 | 2 + 1 + 2 + 1 | 1 | (19) |

16 | 2 + 2 + 1 + 1 | 1 | (20) |

17 | 1 + 2 + 3 | 2 | (21), (22) |

18 | 2 + 1 + 3 | 2 | (23), (24) |

19 | 1 + 3 + 2 | 2 | (25), (26) |

20 | 3 + 1 + 2 | 2 | (27), (28) |

21 | 2 + 3 + 1 | 2 | (29), (30) |

22 | 3 + 2 + 1 | 2 | (31), (32) |

23 | 2 + 2 + 2 | 1 | (33) |

24 | 3 + 3 | 4 | (34) - (37) |

25 | 2 + 4 | 5 | (38) - (42) |

26 | 4 + 2 | 5 | (43) - (47) |

27 | 1 + 1 + 4 | 5 | (48) - (52) |

28 | 1 + 4 + 1 | 5 | (53) - (57) |

29 | 4 + 1 + 1 | 5 | (58) - (62) |

30 | 1 + 5 | 14 | (63) - (76) |

31 | 5 + 1 | 14 | (77) - (90) |

32 | 6 | 42 | (91) - (132) |

The sum of 32

which is a product of

In order to obtain the energy terms, it can be noted that the component numbers

For example for

This result is associated with the first order

which enters as a factor multiplying the result obtained from the main loop of time given in the form of the product of V and P enclosed within the brackets

On the other hand, for

are

and

The energy terms multiplying the contribution coming from the bracket term

for partition (19), and

for partition (20). Similar partitions can be done for

which come from

A characteristic point is that the exponents of

Also the

A general rule for the exponents of

Index of the perturbation term | Perturbation term | Contractions present on a circular scale of time |
---|---|---|

(1) | No contraction | |

(2) | ||

(3) | ||

(4) | ||

(5) | ||

(6) | ||

(7) | ||

(8) | ||

(9) | ||

(10) | ||

(11) | ||

(12) | ||

(13) | ||

(14) | ||

(15) | ||

(16) | ||

(17) | ||

(18) | ||

(19) | ||

(20) | ||

(21) | ||

(22) | ||

(23) | ||

(24) | ||

(25) | ||

(26) | ||

(27) | ||

(28) | ||

(29) | ||

(30) | ||

(31) | ||

(32) | ||

(33) | ||

(34) | ||

(35) | ||

(36) | ||

(37) |

Index of the perturbation term | Perturbation term | Contractions on the circular scale |
---|---|---|

(38), (39) | ||

(40) | ||

(41) | ||

(42) | ||

(43), (44) | ||

(45) | ||

(46) | ||

(47) | ||

(48), (49) | ||

(50) | ||

(51) | ||

(52) | ||

(53), (54) | ||

(55) | ||

(56) | ||

(57) | ||

(58), (59) | ||

(60) | ||

(61) | ||

(62) | ||

(63) - (67) | ||

(68), (69) | ||

(70) - (71) | ||

(72) | ||

(73) | ||

(74) | ||

(75) | ||

(76) | ||

(77) - (81) | ||

(82), (83) | ||

(84), (85) | ||

(86) | ||

(87) | ||

(88) | ||

(89) | ||

(90) |

(a) The remaining four contractions are:

Indices of the perturbation terms | Perturbation term | Contractions on the circular scale |
---|---|---|

(91) - (104) | ||

(105) - (109) | ||

(110) - (114) | ||

(115), (116) | ||

(117), (118) | ||

(119), (120) | ||

(121), (122) | ||

(123), (124) | ||

(125) | ||

(126) | ||

(127) | ||

(128) | ||

(129) | ||

(130) | ||

(131) | ||

(132) |

(a) Other 13 contractions are:

in case of presence of the product of two energy terms

etc. The maximum increase of the exponent is dictated by the number submitted to partition which is 5 in the examined case, so the exponent of

Let us note that in ^{6} inside. This correlates with a single number

In Appendix the energy terms for

The sign of an energy term is dictated by the number

More briefly, the term is taken with a positive sign for an odd

In the paper we presented a way of calculating the Schrödinger perturbation energy by a method which can be easily extended to large perturbation orders N. A non-degenerate quantum state and its perturbation by a time-independent potential are examined.

The method, in its original shape, has applied a circular scale of time with contractions of the time points present on the scale. By contraction of all time points on the scale (beyond the beginning-end point) in the way that no contraction lines drawn between the points can cross together, the number of kinds of the energy perturbation terms known from the Schrödinger theory, as well as simple formulae for calculating these terms, could be obtained [

are of use. An advantage of the algebraic approach is that it can be readily extended to an arbitrary order N.

A similar time scale and algebraic method based on it have been applied before [

The perturbation energy of orders

and

for notation see Section 1 of the main text.

The perturbation energy of order

The terms are respectively

The perturbation energy of

The terms are

It can be noted that partition (A.10) gives two terms entering (A.14). Henceforth the number of kinds of the perturbation terms represented by a given formula is indicated in brackets

For

They give respectively

Partitions (A.19) and (A.20) give each two kinds of terms since

Partition (A.22) gives five kinds of terms since

A characteristic point concerning the numbers given in brackets from (A.23) to (A.35) is that their sum is equal to

The calculation of

The partitions provide us with

Let us note that any partition having 3 as its component [(A.44)-(A.46) and (A.49), (A.50)] provides us with two kinds of terms

A characteristic point is that if some number