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The energy spectrum of the hydrogen atom has been applied in calculating the time rate of energy transitions between the quantum states of the atom. The formal basis of the approach has been provided by the quantum properties of energy and time deduced from the Joule-Lenz law. The rates of the energy transitions obtained in this way were compared with the quantum-mechanical probabilities of transitions calculated earlier by Bethe and Condon and Shortley for the same pairs of the quantum states.

The intensity of the electron transitions between quantum states in an atom met since a long time an important conceptual difficulty. Since the states are separated by indivisible quanta of energy, the problem of a gradual fulfillment of the energy gaps characterized by these quanta loses its sense and only a probabilistic approach concerning a whole of the quantum ensemble represented by a set of atoms should be considered. Such kind of the statistical reasoning has been applied to quantum states already in the framework of the old quantum theory [

In practice, an application of such a quantum-mechanical background seems to be a rather tedious task, mainly because of a possibly accurate knowledge of the wave functions necessary to calculations. But in general the appropriate wave functions are difficult to assess, and remain much less accurate than the energies. This situation made it desirable to obtain a coupling between the energy quanta and the time intervals associated with the transitions which―intuitionally―should be represented by some finite amounts of time.

Fortunately such a coupling could be provided in effect of an analysis of single electron transitions in a quantum system done with the aid of the Joule-Lenz law. We assume that both the energy amount, and the interval of time entering this law, are characteristic for the electron transition. In effect a reference between the energy and time is defined by 1) the electric resistance connected with transition and 2) the electron current present in the system [

When the electric resistance and current are combined together according to the Joule-Lenz law, we obtain readily a coupling between the energy change and that of time; see e.g. [

and both

Certainly the selection rules for electron transitions, especially those dictated by the quanta of angular momentum belonging to

But in spectroscopy also an opposite emission than from

can take place. To obtain an insight into the case of

say by

An outline of the formulae which are of use in a semiclassical theory of the energy emission in the atom has been done before [

The fundamental energy quanta are

where

(

label the quantum states of the hydrogen atom. The number

Let us note that (4) label the energy distances between the neighbouring states. But beyond of (4) the energy quanta between more distant states than neighbouring ones can also enter the calculations. In a particular case of the present paper the energy quanta which are of use become special cases of the formula

namely

The (8) give respectively for (7):

A fundamental relation does exist between

The relation is given by the quantum aspect of the Joule-Lenz law [

here h is the Planck constant.

Because of (13) the emission intensity for transitions between the neighbouring quantum levels is

If the energy distance between the levels corresponds to the next-neighbouring states we have

Because of the relations (13) as well as

and

the formula (15) can be transformed into

For the case of more distant quantum levels than

Particular ratios of the emission intensity belonging to different pairs of the electron transitions in the hydrogen atom are represented in

Certainly f in (20) should not be confused with f in (5).

A characteristic point in (20) is that the angular momentum of the beginning state

which correspond to transitions between the higher energy states having the angular momentum s

A quantum-mechanical counterpart of the semiclassical ratios of intensity considered in Sec. 2 and in

On the other hand the semiclassical intensity ratios of

in

In general the semiclassical intensity ratios presented in

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) |

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

(1) | 2.86 | |||

(2) | 8.16 | |||

(3) | 6.63 | |||

(4) | 17.6 | |||

(5) | 38.1 | |||

(6) | 13.4 | |||

(7) | 34.0 | |||

(8) | 70.1 | |||

(9) | 129 | |||

(10) | 2.86 | |||

(11) | 2.32 | |||

(12) | 6.17 | |||

(13) | 13.3 | |||

(14) | 4.68 | |||

(15) | 11.9 | |||

(16) | 24.6 |

(17) | 45.2 | |||
---|---|---|---|---|

(18) | 0.81 | |||

(19) | 2.16 | |||

(20) | 4.67 | |||

(21) | 1.64 | |||

(22) | 4.16 | |||

(23) | 8.59 | |||

(24) | 15.8 | |||

(25) | 2.66 | |||

(26) | 5.74 | |||

(27) | 2.02 | |||

(28) | 5.12 | |||

(29) | 10.6 | |||

(30) | 19.5 | |||

(31) | 2.16 | |||

(32) | 0.76 | |||

(33) | 1.93 | |||

(34) | 3.98 | |||

(35) | 7.32 | |||

(36) | 0.35 | |||

(37) | 0.89 | |||

(38) | 1.84 | |||

(39) | 3.39 | |||

(40) | 2.54 |

(41) | 5.25 | |||
---|---|---|---|---|

(42) | 9.66 | |||

(43) | 2.06 | |||

(44) | 3.80 | |||

(45) | 1.84 | |||

(46) | 2.16 | |||

(47) | 4.67 | |||

(48) | 4.16 | |||

(49) | 8.6 | |||

(50) | 15.8 | |||

(51) | 2.16 | |||

(52) | 1.93 | |||

(53) | 3.98 | |||

(54) | 7.32 | |||

(55) | 0.89 | |||

(56) | 1.84 | |||

(57) | 3.39 | |||

(58) | 2.06 | |||

(59) | 3.80 | |||

(60) | 1.84 | |||

(61) | 1.84 | |||

(62) | 3.39 | |||

(63) | 1.84 |

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) |

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

(1) | 2.86 | |||

(2) | 8.16 | |||

(3) | 6.63 | |||

(4) | 17.6 | |||

(5) | 38.1 | |||

(6) | 13.4 | |||

(7) | 4.16 | |||

(8) | 70.1 | |||

(9) | 129 | |||

(10) | 2.86 | |||

(11) | 2.32 | |||

(12) | 6.17 | |||

(13) | 13.3 | |||

(14) | 4.68 | |||

(15) | 0.81 | |||

(16) | 2.16 | |||

(17) | 4.67 | |||

(18) | 1.64 |

(19) | 4.16 | |||
---|---|---|---|---|

(20) | 8.59 | |||

(21) | 15.8 | |||

(22) | 2.66 | |||

(23) | 5.74 | |||

(24) | 2.02 | |||

(25) | 5.12 | |||

(26) | 10.58 | |||

(27) | 19.47 | |||

(28) | 2.16 | |||

(29) | 0.76 | |||

(30) | 1.93 | |||

(31) | 3.98 | |||

(32) | 7.32 | |||

(33) | 0.35 | |||

(34) | 0.89 | |||

(35) | 1.84 | |||

(36) | 3.39 | |||

(37) | 2.54 | |||

(38) | 5.25 | |||

(39) | 9.66 | |||

(40) | 2.06 | |||

(41) | 3.80 | |||

(42) | 1.84 |

If the larger of the ratios from the semiclassical and quantum-mechanical pair is considered in each examined case, all other ratios of this kind―beyond of the case (61) presented in (24)―do not exceed 1.5. In average this is evidently a much better agreement between the semiclassical and quantum-mechanical theory than attained for transitions

A different situation is represented, however, by the semiclassical data collected in

see

Because of a qualitatively different reference between the angular momenta in the beginning and end quantum states entering transitions examined respectively in

A well-known reference to the correspodence principle concerns the energy spectrum of the Bohr hydrogen atom [

Let us examine a difference which exists between the frequency of the electron motion about the atomic nucleus in state n represented by

or a similar circulation frequency in state

and the frequency associated with the electron transition from level m to level n

which corresponds to the electromagnetic wave emitted in effect of the transition. The

is the Rydberg constant.

If

and m as well and n are large numbers, we obtain from (28) the relation

Since the energy of the atom say in state n is

the action integral in the same state is

because of the electron velocity

and the orbit radius

Both quantities

which gives a result identical with (26).

For low m and n, and

which is evidently different than (36) because of the validity of (28). The idea of the correspondence principle represented by the formula (36) was that it works only on condition

The importance of the Joule-Lenz law comes from the fact that―contrary to the correspondence principle― its quantum aspects can be applied for low energy transitions concerning also the low energy states labelled by small n [

in which

and

does exist also for small n.

In fact the interval (40) is the time difference between two situations represented respectively by the neighbouring quantum values of the energy,

in only an approximate one, though it becomes well satisfied already at low n. In practical calculations, especially for the low-energy transitions, the formula (41) can be neglected but only (38) applied. This is done in the present and the former paper [

for

with the formula (38) applied separately to each step entering (43); see [

It seems useful to be noted that the orbital velocity

of the electron in the presence of the electric

and magnetic

field; cf. here [

which is identical with (34).

The time seems to be a not favourite parameter for the quantum physicists. In fact, the quantum events occupy so short intervals of time that their accurate measurement seems to be hardly possible. A similar difficulty concerns a precise definition of the beginning or end time of the quantum process. In effect the time as a measurable observable enters quite seldom the quantum-theoretical analysis or an empirical observation.

The aim of the present and a former paper [

This led to situation that the patterns of the emission intensity between the states of the hydrogen atom calculated with the aid of quantum mechanics could be compared with the semiclassical patterns of intensity data obtained from the time intervals which are characteristic for the electron transitions between the energy levels in the atom. It should be added that all transitions entering calculations have been selected according to the well- known rules of quantum mechanics concerning the electron angular momentum.

An evident similarity of the semiclassical and quantum-mechanical patterns of intensity has been attained. A better agreement between the data of the both methods occurs in case of transitions when a larger angular momentum is associated with the beginning state of emission, i.e. the state having also a larger energy, for example the transition is

Stanisław Olszewski, (2016) Semiclassical and Quantum-Mechanical Formalism Applied in Calculating the Emission Intensity of the Atomic Hydrogen. Journal of Modern Physics,07,1004-1020. doi: 10.4236/jmp.2016.79091