Time Intervals of the Energy Emission in Quantum Systems Obtained from the Conservation Rule of the Electron Momentum

The paper presents a non-probabilistic approach to the time interval associated with the energy emission produced by the electron transition in a quantum system. The calculations were performed for the hydrogen atom and the electron particle in a one-dimensional potential box. In both cases, the rule of conservation of the electron momentum has been applied. The results, limited to the time intervals of transitions between two neighbouring quantum energy levels, occur to be much similar to those obtained earlier with the aid of the Joule-Lenz energy emission theory.


Introduction
In principle, we assume that some interval of time should accompany any quantum process in which a change of a quantum physical system does occur. In the previous approach to such processes, a probabilistic analysis accompanied any electron transition phenomenon (see e.g. [1] [2] [3] [4]), leaving unknown the corresponding interval, or intervals, of time. In general, such situation did not change in the modern quantum theory [5] [6].
A different, viz. non-probabilistic situation, took place when the classical Joule-Lenz theorem for the energy emission (see e.g. [7]) has been adapted in calculating the transition time of an electron between two quantum energy levels [8]- [14]. In this case, a very simple rule coupling the distance between two quantum energy levels with the size of the time interval for the electron transi- The electron velocity entering (1) is [17] 2 n e v n =  (4) and the electron momentum in state n becomes . n n p mv = Another approach to n p can be obtained from the quanta of the electron angular momentum is the radius of the electron circular orbit in the hydrogen atom [17].
A final result for the quanta of energy in (1) The last step in (10) is valid on condition n is a large number.
Respectively to (9) we have the momentum change ( ) which provides us evidently with a smaller electron momentum in state 1 n + than in state n.
If the momentum in states 1 n + and n should be conserved, the negative difference in (11) We postulate that the sum of (11) and (13) has to be zero, so .
This t ∆ is a time interval necessary to provide us with a conservation of momentum represented by the formula (14).

Comparison with the Joule-Lenz Law [8]-[14]
According to that law the time interval should approximately satisfy the formula In result we find that the Joule-Lenz emission time (20) differs from the time interval obtained in (17) solely by the factor of π .

Time Interval Connected with the Absorption of Energy Compared with the Time of the Emission Process
Both the absorption and emission processes are of a semiclassical nature. Therefore if in case of absorption we have a change of quantum indices the result for the time interval t ∆ becomes equal to that for the case of emission between the levels A different situation can be obtained when the emission change of states is compared with the absorption change which is for example In the case of (23) we have the momentum balance given by the condition from which we obtain the equation where the last step holds for the large n.
On the other hand, for the absorption process in (24) For large n the difference (31) becomes only a small fraction of t ∆ in (27) or (30).

Electron Particle Moving in a One-Dimensional Potential Box and Its Transition Process
A reasoning similar to that developed for the electron in the hydrogen atom can be applied also in case of the electron particle moving in a one-dimensional potential box.
Let the box has the length L. The electron quantum states for the energy are   which is the velocity of the particle. This velocity has been obtained from the electron energy in the formula (33).
The momentum balance provides us with the equation where the last step is due to (36).
From (38) we obtain the relation for t ∆ : This is a result identical to t ∆ in (39) and (39a).

Size Limits of Mechanical Parameters Entering Simple Quantum Systems
Conservation of momentum suggests to calculate the limits of mechanical parameters like energy, velocity, distance and time entering the examined simple quantum systems. These limits can be obtained in an equally simple way.
Beginning with the hydrogen atom, the relativistic limit of the electron velocity leads to requirement 137. The result in (47) is an extension of that presented in (44).
The properties connected with the radius limit of the electron orbit which for the quantum number 1 n = is equal to where the last steps hold for any large n. An alternative formula for the last step in (57) is: We obtain For very low n, say 1 n = , relation (59) for t ∆ becomes A similar reasoning can be performed for the electron particle moving in a one-dimensional potential box. In the first step, from the requirement that the kinetic energy on the quantum level 1 n = is smaller than the rest energy of the electron particle, we obtain the formula: Therefore, with the aid of the first equation given in (61), we obtain: In effect it should be The limits obtained for L and 1 v can provide us with the size of the interval t ∆ according to the formula so a maximal size of the interval t ∆ for the electron oscillation in the box be-Journal of Modern Physics comes: Another approach applies L calculated in (64) and 1 v in (63):

Summary
In the paper, the transition time between the nearest quantum energy levels is examined for the case of the Bohr hydrogen atom and the electron particle enclosed in a one-dimensional potential box. In both cases, the calculations are based on the assumption that the electron momentum in course of the electron transition is conserved.
It is found that the time intervals of the electron transitions obtained in this way are much similar to those calculated on the basis of the Joule-Lenz law for the energy emission: in the case of hydrogen, a difference between the results of both kinds is represented by a constant factor π ; for the electron particle moving in a one-dimensional potential box there exists an identity of the results for t ∆ calculated in both ways. The limiting sizes of the mechanical parameters characterizing the quantum states in the systems considered in the paper have been also calculated.
It should be noted that the electron transition time t ∆ considered in the paper does not correspond, in general, to the reciprocal time of the frequency

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