_{1}

Quantum aspects of the Joule-Lenz law for the transmission of energy allowed us to calculate the time rate of energy transitions between the quantum states of the hydrogen atom in a fully non-probabilistic way. The calculation has been extended to all transitions between p and s states having main quantum numbers not exceeding 6. An evident similarity between the intensity pattern obtained from the Joule-Lenz law and the corresponding quantum-mechanical transition pro-babilities has been shown.

Since the very beginning of quantum theory the transition rate of energy connected with the occupation change of quantum states has been considered on a combined probabilistic-and-statistical footing [

However, the problem could obtain a new approach if the quantum background coupled with the Joule--Lenz law of the energy emission is taken into account. In this case the quantum aspects of that law discovered recently [

In effect we can calculate the transition time of a single electron particle between two quantum states. Further connection of such time with the energy rate of radiation becomes a simple task. An auxiliary component of this theory is the fact that it can be compared with the quantum-mechanical calculations giving a rather satisfactory assessment for the new, i.e. non-probabilistic, results.

This is so because the ratio of the intensities of two spectroscopic lines gives in fact the ratio of transition probabilities [

In the present calculations of the changes of quantum states only the energy and time are involved. Therefore there is no reference to the selection rules of transitions given by such parameters like, for example, the orbital angular momentum.

A characteristic step of a later Bohr’s approach to the atomic spectra was a poposal of the Fourier analysis of the displacement vector associated with the position change of the particle submitted to transition [

In effect, because any elementary interval of energy has its corresponding interval of time, these elementary time intervals can be added together into full intervals necessary to be considered in description of a given quantum process. In result the rate of the electron transitions between rather distant quantum levels could be calculated as a function of the elementary intervals of energy. In the present paper we do such time analysis and apply it in calculating the rate of electron transitions in the hydrogen atom. Before we do that, the elementary properties of both energy and time entering the transitions will be represented.

Elementary transition is that between two neighbouring quantum levels, say

in general

But any transition energy

etc., in general

Because a complementarity relation deduced from the Joule-Lenz law does exist between the elementary energy interval and transition time interval we have

Evidently the

But still another kind of relations―similar to that introduced by Kramers and Heisenberg [

Here

Certainly the same relation can represent

From the formulae in (4) it is evident that any transition coefficient

Simple properties of

etc. In effect

etc., where

are respectively the intensities, or energy rates, of transitions from level 2 to 1, from 3 to 2, from 4 to 3, from 5 to 4, etc.

We see that the ratios of the coefficient squares are equal to the ratios of intensities.

The emitted energy intensity of transition between a paricular pair of quantum states is sometimes called a component of the spectral line [

where

The aim of the present paper is to join, as far as possible, the calculations of the emission rate of single transitions given by the present theory with the former theory of the line spectra, or obtained from experiment. Since it is difficult to make an absolute comparison between the theory, or theories, or experiment, the calculations are referred mainly to the relative intensities of the spectral lines.

In fact we shall demonstrate that the Bohr energies of electron transitions in the hydrogen atom applied in the present theory can give a rather satisfactory approximation for the ratios of the transition probabilities between the atomic states given by the quantum-mechanical theory. To this purpose the transitions from the atomic states

to the states n_{(s)}s where

are considered because all results of calculations can be compared with the quantum-mechanical data listed in [

In examining the intensities due to the present framework we take into account the ratios

where

and

Both the numerator (labelled by

which are positive quantities, first in view of (14) and (15), second because of the energy components equal to

But the intensities entering (13) are represented by the formulae

and

which contain in general the time intervals different than the elementary intervals presented in (2). For example we can have

or

However such intervals can be decomposed into elementary ones, so we obtain

or

The last steps in the Formulaes (26) and (27) come from (3). Evidently any term entering (16) and (17) can be decomposed into

and

On the right of (28) and (29) the indices s and p could be omitted because of the lack of dependence of the right-hand side of the formulae in (18)-(21) on s

It should be noted that for transitions between levels

A time ago Einstein has remarked that the time of transitions between deep-lying quantum states should be very small [

This is the time period of the first quantum state in the hydrogen atom [

For large quantum numbers n and m there is evident the formula [see (6)]:

whereas the ratio of the time periods of the hydrogen atom is also:

Evidently the values of

The Formulaes (8)-(9) indicate a reference between the coefficients squares

In fact we find that there exists an evident correspondence between the ratios of the quantum-mechanical transition probabilities

calculated for different pairs of transitions given by the

calculated with the aid of the present method. In

In fact the ratio of two intensities obtained with the present theory referred to the corresponding ratio of the quantum-mechanical probabilities rather seldom exceeds number 2, although the ratios entering the calculations vary between the numbers being evidently smaller than unity [cases (61), (65), (86), (91)] to the numbers equal to several thousands [cases (13) and (14)].

The ratio equal to 2 is exceeded by the cases (6), (10), (31), (35), (56), (65), (86) and (91) where respectively there is obtained

but only in the case (99)

exceeds 2.

A time ago Ornstein and Burger [

see items (51), (78) and (100) in

They have found respectively the following quantum-mechanical ratios for the transition probabilities:

The experimental ratios of the intensities were found equal to [

see also [

so they are closer to the experimental data in (37) than the data given in (36).

The intensity

The term

Evidently on the basis of (39) we have

No | Case | Formula for the intensity ratio and the value of that ratio | |
---|---|---|---|

(1) | |||

(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) | |||

(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) | |||

(64) | |||

(65) | |||

(66) | |||

(67) |

(68) | |||
---|---|---|---|

(69) | |||

(70) | |||

(71) | |||

(72) | |||

(73) | |||

(74) | |||

(75) | |||

(76) | |||

(77) | |||

(78) | |||

(79) | |||

(80) | |||

(81) |

(82) | |||
---|---|---|---|

(83) | |||

(84) | |||

(85) | |||

(86) | |||

(87) | |||

(88) | |||

(89) | |||

(90) | |||

(91) | |||

(92) | |||

(93) | |||

(94) | |||

(95) | |||

(96) | |||

(97) |

(98) | |||
---|---|---|---|

(99) | |||

(100) | |||

(101) | |||

(102) | |||

(103) | |||

(104) | |||

(105) |

No | Case | Quantum-mechanical ratio | Intensity ratio from | |
---|---|---|---|---|

(1) | 5.4 | |||

(2) | 29.2 | |||

(3) | 17.1 | |||

(4) | 83.3 | |||

(5) | 238 | |||

(6) | 43 | |||

(7) | 193 |

(8) | 514 | |||
---|---|---|---|---|

(9) | 1110 | |||

(10) | 90 | |||

(11) | 390 | |||

(12) | 987 | |||

(13) | 2045 | |||

(14) | 3765 | |||

(15) | 5.4 | |||

(16) | 3.2 | |||

(17) | 15.4 | |||

(18) | 44.1 | |||

(19) | 7.98 | |||

(20) | 35.8 | |||

(21) | 95.4 | |||

(22) | 206 | |||

(23) | 16.6 | |||

(24) | 72.2 | |||

(25) | 184 | |||

(26) | 379 |

(27) | 697 | |||
---|---|---|---|---|

(28) | 0.60 | |||

(29) | 2.86 | |||

(30) | 8.16 | |||

(31) | 1.48 | |||

(32) | 6.63 | |||

(33) | 17.6 | |||

(34) | 38.1 | |||

(35) | 3.08 | |||

(36) | 13.4 | |||

(37) | 33.9 | |||

(38) | 70.1 | |||

(39) | 129 | |||

(40) | 4.76 | |||

(41) | 13.7 | |||

(42) | 2.47 | |||

(43) | 11.1 | |||

(44) | 29.5 | |||

(45) | 63.6 | |||

(46) | 5.14 |

(47) | 22.3 | |||
---|---|---|---|---|

(48) | 56.8 | |||

(49) | 117 | |||

(50) | 216 | |||

(51) | 2.86 | |||

(52) | 0.52 | |||

(53) | 2.32 | |||

(54) | 6.17 | |||

(55) | 13.2 | |||

(56) | 1.08 | |||

(57) | 4.68 | |||

(58) | 11.9 | |||

(59) | 24.6 | |||

(60) | 45.2 | |||

(61) | 0.18 | |||

(62) | 0.81 | |||

(63) | 2.16 | |||

(64) | 4.67 | |||

(65) | 0.38 | |||

(66) | 1.64 |

(67) | 4.16 | |||
---|---|---|---|---|

(68) | 8.59 | |||

(69) | 15.8 | |||

(70) | 4.49 | |||

(71) | 11.9 | |||

(72) | 25.8 | |||

(73) | 2.09 | |||

(74) | 9.05 | |||

(75) | 23.0 | |||

(76) | 48.4 | |||

(77) | 87.4 | |||

(78) | 2.66 | |||

(79) | 5.74 | |||

(80) | 0.46 | |||

(81) | 2.02 | |||

(82) | 5.12 | |||

(83) | 10.6 | |||

(84) | 19.5 | |||

(85) | 2.16 | |||

(86) | 0.175 |

(87) | 0.76 | |||
---|---|---|---|---|

(88) | 1.93 | |||

(89) | 3.98 | |||

(90) | 7.32 | |||

(91) | 0.081 | |||

(92) | 0.35 | |||

(93) | 0.89 | |||

(94) | 1.84 | |||

(95) | 3.39 | |||

(96) | 4.34 | |||

(97) | 11.02 | |||

(98) | 22.8 | |||

(99) | 41.9 | |||

(100) | 2.54 | |||

(101) | 5.25 | |||

(102) | 9.66 | |||

(103) | 2.06 | |||

(104) | 3.8 | |||

(105) | 1.84 |

The lifetime of the excited state p is represented by a sum of

performed over all possible transitions from state p to states q which are lower than p (see [

In the hydrogen atom the lowest possible state q is represented by

for

and

for

etc. In the last steps of (46)-(46b) we applied the partition of the transition times into their component intervals similar to those applied in Section 3.

In general the lifetime of state p is

where q are the states lower than p so they all satisfy (42). Next any reciprocal value of

We have found before [

where for large n the formula

is fulfilled with a good accuracy: the

for the level

the lifetime for the level

The procedure outlined above can be extended to an arbitrary n. We obtain [

so for large n

Evidently the present calculations do not take into account the quantum numbers other than n.

The quantum-mechanical calculations done for the lifetimes of the excited levels in the hydrogen atom are represented in [

In considering the transition times of electrons in the hydrogen-like atom a situation when the electron is moving in the field of the nucleus having the charge

In fact both

because

see e.g. [

This is an easy task if we note that the Joule-Lenz law gives

where

is the resistance of the current i induced by the energy transition

equation (61) becomes approximately

so

holds irrespectively of the size of Z.

It is easy to show that

see also [

we obtain

where

Equation (59) together with (68) implies that

which is in accordance with the well-known result; see e.g. [

This implies that

or

which is the result given in (66).

In effect because of (65)

so the Formula (57) remains unchanged upon the change of Z.

But the result of (73) has an important consequence concerning the emission intensity which is

so we obtain that the intensity of transitions in the hydrogen-like system having

In the paper a semiclassical approach to the transition intensities between p and s quantum levels of the hydrogen atom is compared with the quantum-mechanical transition probabilities for the same pairs of levels. An evident convergence between the sets of the data calculated by the both methods is obtained.

The present method is fully a non-probabilistic one. This is so because the idea of probability became unnecessary to apply as far as we do not ask when (or why) the system is going to change. In fact we look for a definite change of the occupation of quantum states in the system and the energy connected with it. In this case there is no uncertainty, or search, in the system to obtain the interval of time necessary for transition. Formally the changes of the quanta of energy and time remain on an equal footing. A difference-especially evident in the case of the hydrogen atom-is mainly connected with the computational practice: The quanta of energy are easy to calculate (with the aid of the fundamental constants of nature taken into account), but we are unable to do the same thing with the intervals of time. In effect first the intervals of energy have to be obtained, next they serve us as a background for calculating the intervals of time.

Once the system “decides” to change its definite population into another one, the time necessary to perform the transition process is defined-together with the energy change connected with transition-by the complemen- tary relation (3), or a superposition of (3). A single (3) is adequate for an emissive transition between two neighbouring energy levels. On the other hand, if for some (unknown) reasons, the atom “decides” to choose the energy change (emission) corresponding to a larger distance between the levels than described by a single Formula (3), the transition time should necessarily fit to this requirement. In this case the individual formulae (3) serve also to calculate the components of the whole time interval necessary for transition; see formulae (26) and (27).

Computationally this makes the semiclassical approach much more simple than the quantum-mechanical one. For example we readily obtain that the ratio of the intensities

and we have (see

For a reason similar to (77) the intensity of

and we have

It should be noted that when we consider space and time as elements of a common space-and-time system, the quantum theory “selects”the time variable to a treatment based on a fully different footing than it may concern the intervals in space. Because of Equation (3) the time is divided into portions, or quanta, similar to those of their energy partners

In result a whole of the time interval between two events-which are the beginning and end of the emission-is divided into portions, or quanta, similar to those of the energy partners entering (3). In effect the time interval between two events is either elementary, i.e. defined by a single

A simple example of an application of the Formula (30) can be given also for some cases of the ratios of the s-p transition intensities which have been not yet considered in the present paper. For example we have for the ratio

wheras the quantum-mechanical ratio of transition probabilities is

for the intensity ratio

whereas the quantum-mechanical ratio of transition probabilities is

and for the intensity ratio

whereas the quantum-mechanical ratio of transition probabilities is

Stanisƚaw Olszewski, (2016) Emission Intensity in the Hydrogen Atom Calculated from a Non-Probabilistic Approach to the Electron Transitions. Journal of Modern Physics,07,827-851. doi: 10.4236/jmp.2016.78076