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INTERRELATION BETWEEN QUANTUM AND CLASSICAL PHYSICS
Mirgorodsky Alexander Illarionovich
MECHANICAL MOTION OF THE HARMONIC OSCILLATOR OF QUANTUM MECHANICS
Earlier I already noted that the analysis of the mechanical motion of the oscillator of quantum mechanics was preceded by an analysis of the mechanical motion of the oscillator of classical mechanics. Now the path of analysis is repeated a second time in the opposite direction, at which the consideration of the oscillator of quantum mechanics precedes the consideration and description of the oscillator cycle of classical mechanics.
In the description of the Schrodinger wave equations of the cyclic action of the harmonic oscillator of quantum mechanics, essential shortcomings are revealed.
The stationary Schrödinger wave equation describes the action of a harmonic oscillator in which the particle-wave properties of mechanical motion manifest themselves:
| - | Ħ 2 | Δ ψ + V (x, y, z) ψ = E ψ | (1) |
| 2m |
The wave equation (1) describes the interaction of a wave and a corpuscle in a harmonic oscillator that is in a stationary state and in which the energy E of the mechanical motion exists in a certain space for an indefinite time in accordance with the general relation of the uncertainties of space and time.
That is why in equation (1) the dependence of the potential V on certain numerical values of the coordinates of a certain space of the oscillator is clearly shown and does not depend on the numerical values of the coordinate of indefinite time. The definite energy E of the oscillator has three spatial forms of its expression: general, singular and unit.
The solutions of equation (1) exist only for certain discrete values of the potential energy of the oscillator, which are expressed by the formula:
| E n = ħ ω 0 (n + | 1 | ) , Where n = 0, 1, 2, 3, ... are integer quantum numbers. | (2) |
| 2 |
The wave equation (1), at first glance, has an infinite set of solutions. But such solutions of the equation are only possible , but in reality in theory the wave equation has only three solutions:
| 1) E 0 = | 1 | Ħ ω; | 2) E 1 = | 1 | Ħ ω + ħ ω; | 3) E 2 = | 1 | Ħ ω + ħ ω + ħ ω; | (3) |
| 2 | 2 | 2 |
Of course, Schrödinger could not know the finite set of solutions of his stationary wave equation, since mechanical motion in a simple form can be understood only after it is already understood in a complex form. I know it from my logical analysis of the stationary state of a harmonic oscillator of classical mechanics. The energy levels of formula (2) actually follow in the reverse order. The reverse order of energy is established by mathematical analysis. The third level is the first and only level of the potential energy of the harmonic oscillator, and zero energy level does not exist at all.
In fact, there are not three levels of the oscillator energy, but three energies of the three interacting forces. The sum of three energies can be expressed in the following form:
| E = | 1 | Hv + hv + hv = | 1 | ( | H | ) + | H | + | H | = | 1 | Mv 2 | + Mv 2 | + Mv 2 | (4) |
| 2 | 2 | T | T | T | 2 |
The summands of the sum of the energies of the three forces acting and interacting in their own definite space for an indefinite time can be expressed in the form:
| E 1 = mv 2 ; | E 2 = mv 2 ; | E 3 = 0.5 mv 2 | (5) |
The terms in the sum (4) of the energy of the three forces are arranged in reverse order by the analysis.
It should be emphasized that the zero level of potential energy, the reflection on the physical meaning of which has brought headache to many physicists, does not have a harmonic oscillator of quantum mechanics. In equalities (5), the energies E 1 and E 2 can be regarded as energies satisfying the principle of identity of identical particles, and the energy E 3 can be regarded as the exchange energy of the oscillator.
The time-dependent Schrödinger equation
| iħ | ∂ ψ | = - | Ħ 2 | Δ ψ + V (x, y, z, t) ψ | (6) |
| ∂ t | 2m |
| Where | Δ = | ∂ 2 | + | ∂ 2 | + | ∂ 2 | Is the Laplace operator. |
| ∂ x | ∂ y | ∂ z |
The time equation (6) was used by Schrodinger in the analysis of the dynamic state of the oscillator after using the stationary equation (1) in the analysis of the stationary state.
On the right-hand side of equation (6), the coordinates (x, y, z) should not be in place, because the oscillator in the dynamic state exists for a certain period of time in an indefinite proper space.
The imaginary unit on the left side of equation (6) shows that the impulses of the three forces exist in an indefinite mystical, or imaginary, form, which they take for a certain time, flowing in the opposite direction, back, from the present to the past. In short, the time-dependent Schrödinger equation describes the dynamic state of the harmonic oscillator of quantum mechanics, existing in its representation in a mystical form, in which the actual relations are reversed. Inverted relations follow one another in the reverse order.
The stationary equation can be said to correspond to a cart that exists in a certain space for an indefinite time, that is, exists in a state of rest.
| From the very beginning of the existence of the cart is not empty, but has a luggage in the form of exchange energy | E = | Mv 2 |
| , | ||
| 2 |
Which appears in the Schrodinger wave as a zero-level energy. Then in the cart-wave appears the energy of one force E = mv 2 , another force E = mv 2 , the third force, etc. to infinity. Put on a cart is well connected and when moving a cart can not be lost, can not fall out of a certain space and find itself in an external uncertain space. Therefore, the wave and the amplitude of the Schrodinger wave are continuous.
A temporary equation can be put in correspondence with a horse, which is harnessed to a cart, but is behind the cart backwards. A horse exists for a certain time in an indefinite space in a state of motion. In an indefinite space that does not have an indefinite direction, the movement of the horse can not have a definite direction. All the directions of the horse's movement are imaginary, exist only in possibility, but in reality there is not one direction of movement of the horse, exactly aimed at the cart.
Therefore, the imaginary unit appears in the time equation of Schrodinger.
To move the cart with the treasure, the movement of the horse must be precisely directed to the cart. Calculate the movement of the horse, aimed at the cart, without the help of the theory of probability is impossible. The square of the continuous amplitude of the horse-wave should give the probability of finding a cart in the direction of the horse's movement. Schrödinger considered physical reality only a horse-wave, and considered the cart-corpuscle as devoid of objective physical reality.
After Schrödinger, in the description of the action of the harmonic oscillator of quantum mechanics, physics was put in the first place by the time equation (6), and by the second place by the stationary equation (1), which logically is completely justified: the horse must be harnessed in the cart and ahead of it. But if you swap the cart and the horse, leaving them in their previous position backwards, then the movement of the horse and the cart remains impossible. From the change of places of equations, the reverse order of the inverted relations described by them does not become a direct order and the imaginary form of their expression by a real form is not replaced. And yet, physicists do not spare either their strengths or their time to adapt both Schrödinger equations to a satisfactory description of the dynamic and stationary states of a harmonic oscillator, which can not in principle be completely satisfactory.
print version
Author: Mirgorodsky Alexander Illarionovich
Honored teacher of the school of the RSFSR
PS The material is protected.
Date of publication 17.11.2006гг
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