DISCRETE PRINCIPLES OF QUANTUM CHRONODYNAMIC

© Oleg O. Fejgin

Contact to the author: tor@3s.kharkov.ua

The concept Planck’s quantum of action plays one of central roles in the modern theoretical physics. Quantum postulates are bound to fundamental structure of space - time and conservation laws that forms the periodic attempts their re-interpreting at build-up of new physical theories. The present article prolongs a cycle of examinations on formalization of a quantum nature of space - time and is interlinked to development of theoretical models on a basis locally - discrete fashions [5].

Let us consider quantum-mechanical oscillator with a discrete gang of energies of oscillations [1]:

E_i = i hn , i = 0, 1, 2, 3, n, (1)

where h-quantum of action, n -frequency. The thermodynamic probability of their embodying will make [2]:

W_i = W₀ exp (-ihn / kT), i = 0, 1, 2, 3, n, (2)

where kT-thermodynamic temperature.

Let us enter the formal definition for probability of microscopic event from the equation (2), as time localization during some allocated interval [3]:

W(t) = W₀ [exp (h_{t n} )]^-ih(e)/kT, (3)

where expression

W_t = exp (h_tn ) (4)

defines probability of time localization, and for quantity h_t from the formula (1) follows:

h_t = E_i / ih_en or h = h_e h_t; (5)

Here h_t and h_e-builders of quantum of action, in other words chrono-quantum and energy-quantum.

Let us carry out similar reasoning’s for extension of a definition an analytical view of factor W₀ from the equation (3), the given term is bound to probability of existential localization with underloadly possible energy for a viewed physical micro system. Norming W₀ on individual aggregate probability of all probable localizations gives:

W₀ = 1 – W_t^-h(e)/kT. (6)

In view of the formula (6) expression (3) can give the following view:

W_ti = W_t^-ih(e)/kT – W_i^{-(i+1)h(e)/kT}. (7)

The relation (7) can give quite particular physical sense if to take into account, that equality (4) is the trivial shape:

W_t = exp (-ih_tn ). (8)

Then the equation (7) transfers in

W_ti = W_ti^h(e)/kT – W_t(i+1)^h(e)/kT. (9)

From the received formula follows, that the probability of time localization of particular micro event is defined by a difference of localizations of previous and subsequent events in chrono-quantum gauge of their development. Transferring to a wave mechanics, we compare to the arbitrary microscopic object a wave amplitude ψ, satisfying to a canonical wave equation [3]:

Δψ + const ψ/l ² = 0, (10)

where l = const h_th_e [m (E-U)]^-0,5- a wave length of a microscopic object in mass m in energy representation. Substitution of the given expression in the equation (10) gives:

Δψ + const m (E-U) ψ (h_th_e)^-2 = 0. (11)

The received relation corresponds to the reference shape of a stationary Schrödinger equation. Hence, if to follow traditional interpretation intensity of a ψ-wave in each point of space corresponds to probability of a presence of a microscopic object in the allocated micro volume, referred to quantity of this micro volume. Thus, if to start with re-interpreting quantum relations according to equality (5) and (8) the basic principle of indeterminacy for coordinate x and an impulse p gets the following view:

Δx Δp ~ h_e h_t. (12)

At the fixed mass of a microscopic object the left-hand part of a relation (12) transfers in

Δx m Δv = mΔx Δdx/dt = mΔ²x (ih_t)^-1. (13)

Then, both parts of a relation (12) become

mΔ²x ~ h_e (ih_t)². (14)

Let us note that the shape of the equations (13) and (14) corresponds to the linear nonrelativistic case of a motion of a microscopic object. Operating with a principle of indeterminacy for coordinate, and an impulse of some micro particle, it is possible to assume velocities that from reasons of dimensionality there is a similar relation for energy E and time [4]:

ΔE Δt ~ h_e h_t. (15)

Reference interpretation of the formula (15) includes concept of indeterminacy of energy of the microscopic object, determined by time of the given energy localization and re-interpreting at quantum digitization as

Jh_e ih_t ~ h_e h_t. (16)

The relation (16) determines probability of joint localization of the normalized of energy flux ΔE = jh_e allocated conventionally in time interval Δt = ih_t. At a minimum of potential energy, U~0 for a linearized problem of a motion of a microscopic object on a restricted site probabilistic equation (11) transfers trajectories in

d²ψ / dq² + const Eψ (h_eh_t)^-2 = 0, (17)

Where the q-generalized coordinate. From the theory of a harmonic analysis well known, those solutions of the equations of a view (17) are logarithmic functions of type

ψ = ψ₀ sin [const qE^0,5 (h_eh_t)^-1]. (18)

Taking into account boundary conditions of an interval of a motion: ψ=0 at q=q₀ it is gained:

Const q₀ E^0,5 (h_eh_t)^-1 = i+1. (19)

Expression (19) defines requirements of digitization for the nonrelativistic energy of a microscopic object as a gang of i-quantum numbers:

E = const (i+1)² (h_e h_t)² . (20)

Thus, consecutive application of a principle chrono-quantum re-interpreting the basic postulates of a quantum mechanics gives in original updating trivial solutions of a canonical Schrödinger equation. It, in turn, corresponds to a new principle chrono-quantization to energy, re-interpreting as determination of energy levels on temporal sequence of chrono-quanta’s. Hence, determination of spectral energy of a micro particle in time boundaries allocated chrono-quantum may transit with the most probable quantity:

E₀ = const (h_e h_t q₀^-1)². (21)

It is necessary to note, that though values of a zero-point energy at quantum micro particles essentially depend on character of fields of forces at zero of thermodynamic temperature exists fundamental chrono-quantum interval with terrain clearance probability of localization of events.

REFERENCES

1. Aspects of Quantum Theory / Ed. A. Salam, E. Wigner. - Cambridge.: CUP, 1972.
2. Audi M. The Interpretation of Quantum Mechanics. - Chicago: USP, 1973.
3. Slater J. C. Concepts and Development of Quantum Physics. - N. Y.: DP, 1969.
4. Fejgin O.O. About an Opportunity of Build-up Universal Quantum Chronodynamic // Bulletin IPME.- No.3, 1984.
5. Fejgin O.O. DISCRETE - TEMPORAL MODEL OF UNIVERSE

Back