Time Quanta Present in Simple Quantum Systems

Differences of the time periods in two independent quantum systems are examined on a semiclassical level. The systems are the electron in the hydrogen atom and a free-electron particle moving in a one-dimensional potential box, respectively. It is demonstrated that in both systems the relativistic correction to the time interval can be expressed as a multiple of the same quantum of time. The size of the quantum is proportional to the ratio of the Planck’s constant and the rest energy of the electron particle.


Introduction
The problem of the time calculation seldom discussed in the quantum theory can remain actual even for very simple quantum systems. These systems are often of the periodic character, so we need the time periods of the electron motion as parameters. This kind of parameter is usually approached in a semiclassical way. Perhaps the best known example concerns the time periods of the electron orbital motion in the hydrogen atom [1] [2].
Other well-known examples of a semiclassical motion are the free particle in a potential box and the harmonic oscillator. In the last case a difficulty concerning time is provided by an unstable size of the velocity connected with a moving particle.
In order to make the problem of calculations of the time periods to be very simple, only the electron in the hydrogen atom and the particle moving in a one-dimenssional potential box are considered exactly in the paper. The time quanta in the harmonic oscillator are calculated only in an approximate way.
The main aim of the paper was to point out that a relativistic change of a difference between two oscillation periods of time in a quantum system can be obtained with the aid of the time quantum which has its size independent of the examined system.

Electron
so the time period T n connected with the electron circulating along the same orbit is deduced from the formula The task is to calculate the difference

Relativistic Difference of the Time Periods Given in (6)
In order to improve (6) we apply the relativistic theory in the sense that the components entering (6) should be multiplied respectively by the factors (see e.g. [3] [4]) In effect we get a proper relativistic change of the time periods connected with the difference (6). Two terms in each of the power expansions (7) and (8) We obtain: At the end of (12) appeared a supplement having a dimension of time and being proportional to the difference  )   2  2  3  2  2  2  2  3  3  3  2  1  2  2 1  1  2  1 2  2 1  2 1  2 1  1  2 1 . n n n n n n n n n n n n n n n n n n − + These results make (6) proportional to both (13) and (15).

Electron Oscillation in a One-Dimensional Potential Box and Its Properties
For a constant potential within the potential box of length L we have the electron energy states [5] therefore by putting (16) equal to (17) we obtain the electron velocity in the potential box: For a given n this velocity can be considered as having a constant value for the whole length L.
Due to the boundary conditions at the box ends, the electron behaves like a free particle oscillating along the length for each oscillation time period T n . A non-relativistic approach to T n gives from (18): Our aim is to examine the difference The assumption of 2 1 n n > implies a negative time difference in (21); see (20).

A Relativistic Difference of the Time Periods Entering (21)
A constant velocity (18) where the term in brackets in (22) is a positive number; see (24). We have A sum of results obtained in (23) and (24) (taken with a minus sign) gives ( ) A supplement obtained at the end of (25) is identical to that entering (12) calculated for the hydrogen atom. This enables us to consider the term having obtained either for the hydrogen atom, or in the potential box.

Approximate Approach to the Time Quanta in Case of the Harmonic Oscillator
In this case we assume-for the sake of simplicity-that the time periods T belonging to different quantum states are equal giving the circular frequency where k is the oscillation constant and m is the mass of the oscillator [6]. All energy states are given by the formula The velocity n v of the oscillating particle between the limits n n a x a − < < has its maximum at a central particle position 0 x = .
In calculating the relativistic correction to the time period T we take into account-for the sake of simplicity-the average particle velocity In effect the relativistic correction to T becomes: But because of (30) we have

Summary
We examined the differences of the time periods possessed by the electron either in the case of its circulation in the hydrogen atom, or when the electron is enclosed as a particle moving in a one-dimensional potential box. The approach to the time properties of each of these physical systems is modified by the relativistic theory.
It is found that contribution given by the relativistic term of time for both systems is the same. This enables us to consider the relativistic corrections introduced to the differences of the time periods as representing the quanta of time: the size of quanta is proportional to the difference of the quantum numbers which define separately any of the time periods present in each system. Therefore, in order to obtain the correction, the quantum numbers multiply solely one term independent of the physics of the examined system. This term is a constant ratio which is not much different than unity.