_{1}

^{*}

The energy emitted by an electron in course of its transition between two quantum levels can be considered as a dissipated energy. This energy is obtained within a definite interval of time. The problem of the size of the time interval necessary for transitions is examined both on the ground of the quantum approach as well as classical electrodynamics. It is found that in fact the emission time approaches the time interval connected with acceleration of a classical velocity of the electron particle from one of its quantum levels to a neighbouring one.

The phenomenon of the electron transitions between quantum levels is basic for the quantum theory. In fact the theory began by a fit of the transition energy of a set of oscillators to the intensity of the emission spectrum of the black body examined with respect to its dependence on the body temperature [

of the same expression

where

In a further development of the theory the result of (2) has been extended to any transition energy between two quantum levels, not necessarily those belonging to the oscillator. In effect the formula

couples the energy interval

with the frequency

However an important lack of the theory which remained in it was the problem of the time interval

necessary for the process of the energy change

The aim of the present paper is to examine the problem of the transition time

on the ground of a non-probabilistic (non-statistical) approach. Therefore formally the task becomes rather opposite to a treatment which has been usually applied. Only the emission rate of energy

Formally (7) is much similar to the well-known Heisenberg principle of uncertainty for energy and time [

But the interval

so

Physical and philosophical implications of (9) have been discussed on many occasions, see e.g. [

In the first step, the aim of the formalism developed in the present paper is to demonstrate that

Evidently

The energy differences

between two neighbouring quantum states having the indices

for the hydrogen atom [

for a free particle having mass

for the harmonic oscillator having the frequency

The time periods of the electron particle circulation on the orbits are defined by a physical character of each of the above systems. They are

for the case of electron in the hydrogen atom occupying the state

for the particle of mass m being in state n in the potential box because of the relation between energy and velocity equal to

and

for all states

A characteristic property of expressions (13)-(18) is that

holds for the atomic orbit

for a free particle (free electron) in state

for the harmonic oscillator [see (15) and (18)]. A common feature of (13a)-(15a) is that

If we note that the resistance

where

we obtain

In the last step in (22) the result of (19) is taken into account.

The

Let the dissipation heat

By putting

[see (10)] we obtain from (21)-(24) the following relation

which gives

But because of (19) the formula (26) can be transformed into

which implies that

In effect of (28) the relation (26) can be presented in a more familiar form:

cf. here (7).

A comparison of the time rate of energy emission calculated according to the method presented above with the quantum-mechanical method is done in [

The physics of the test is much similar to that entering the Tolman experiment [

where the time of emission

where

Beginning with the hydrogen atom we have

is given in (16), and

On the other side, a substitution of the absolute value of

Evidently both expressions (33) and (34) are equal:

A similar operation can be repeated for the electron in the potential box. Here [see (17) and equation below of it] the velocity

so

The time period in state n is that given in (17) and

for any state n. Therefore the left-hand side of (30a) becomes

and the right-hand side is

In effect we obtain

which is a similar property to that calculated above in the case of the hydrogen atom.

The case of the harmonic oscillator is rather different than that of the electron in the hydrogen atom or the potential box because the velocity

where k is the force constant. In consequence the formalism described in (30) and (30a) is applied solely to the velocity acceleration at a single point

The electron velocity

where

By considering solely the positive sign in (43) the increment of velocity due to the change of the quantum state becomes

The length

where

so

In effect the left-hand side of Equation (30a) becomes

and the right-hand side of (30a) is

In consequence we obtain an approximate equality of both sides of (30a) represented by the relation

but not precisely the relation

An approach to the Joule-Lenz dissipation energy and its transition time can be done also on a semiclassical basis. First we note that the effective electric field

where

Here

where

can be provided by the energy difference

The efficiency of the Joule-Lenz heat

where

and

Since (see [

we obtain because of (55) the following result

By assuming that

[see (10)] the formula (60) yields evidently the result

obtained in (28).

In preceding sections the case of the neighbouring quantum states

For example for the hydrogen atom the situation (62) implies

see (13) where

Assuming that the end state of the energy emission has the index

The length of the electron path covered within the time period

see (32).

Our aim is to check the validity of the formula (30a) for the case of the emission from the state

We substitute on the left-hand side of (30a) the quantities

On the right-hand side of (30a) a substitution of

We find that the left-side of (30a) presented in (68) differs from the right-hand side presented in (69) solely by a factor of 2:

It is easy to demonstrate that a particle in the potential box and the harmonic oscillator submitted to the check given by the Equation (30a) do not satisfy this equation. For an electron in the potential box the energy

It should be noted that for the transition

This result is different from a similar product calculated in the case on

We define [

as Poynting’s vector. The time rate of the loss of energy is [

A well-known formal asymmetry of the Bohr model of the hydrogen atom is the presence of the electric field strength

in the orbit plane for any quantum state n, but this presence is combined with the absence of a similar magnetic field strength

[

With the electron circulating with frequency (74a) is associated the field strength

In effect

which is the size of a vector normal to the orbit plane.

A substitution of

Since

On the other side, the cross-section of the toroidal cylinder is dictated by the radius (see e.g. [

of the electron particle which moves along the orbit. In effect the toroidal surface is approximated by the product of (78) and the circumference of the cross-section of the torus cylinder which is

The value of the Poynting vector for a thin electron orbit can be assumed as a constant number given in (77), therefore a non-vanishing term on the right of (72) becomes equal to

This is a product of (77), (78) and (80). In effect the Equation (72) has the form

Since the emitted energy in course of the electron transition between levels

the emission time for that energy is

where the term taken in brackets is that calculated in (82).

The result of (83) should be compared with that given by the quantum-mechanical formula (29). This gives

which is a number larger by the factor of

than that of (83).

This is an expected situation because the emission rate described by the Poynting vector is not restricted to a single transition from level

It seems of interest to demonstrate that

The current

where

is the volume occupied by the electron particle and

is the cross-section area of both of the volume

Since the integral of

which is a formula identical to that given in (76).

In his derivation of the formula (3) applied in the present paper, Einstein [

The first step demonstrates that instead of (3) the Joule-Lenz dissipation energy can be applied [see (10) and (24)]. This yields an estimate of the emission time

The

It has been demonstrated that for

A good fit of

to the velocity in state n, viz.

A separate study of

A similar agreement of emission time and acceleration time is present also for

The only condition imposed on the applied formalism is that the electron states considered in a system are periodic in time. This property can be coupled rather easily with the idea of the electron orbit, for example that introduced by Bohr in the model of the hydrogen atom. In this case the electron remaining in a quantum stationary state is circulating incessibly along an orbit of a definite size within a definite constant period of time [

In general the use of the orbit idea is well known in the whole domain of the atomic, molecular and solid-state physics [

where

The Bloch model, especially of a one-dimensional crystal, is much similar to the model based on the standing-like wave functions characteristic for the electron particles enclosed in a potential box discussed in the present paper; see e.g. [

StanisƚawOlszewski, (2015) Non-Probabilistic Approach to the Time of Energy Emission in Small Quantum Systems. Journal of Modern Physics,06,1277-1288. doi: 10.4236/jmp.2015.69133