The physical meaning of the de Broglie wavelength and the Heisenberg uncertainty relation.

 

 

Saynyuk Nikolay

 

 

The successes of quantum mechanics in many areas of atomic physics are obvious. It is difficult to find another theory that with such tremendous accuracy would be consistent with experimental data. But, despite the almost century-old existence of this theory the basic principles underlying it, as wave-particle duality, Heisenberg uncertainty relation essentially remained unappreciated and not substantiated. But these basic principles lies a lot of interesting information to shed light on some of the problems faced by modern fundamental physics. Consider these principles in more detail.

 

The wave properties of the elementary particles.

All calculations are performed in quantum mechanics, made the assumption that elementary particles possess this mysterious property, as the wave-particle duality, are the point. That is not true, known to many physicists. Modern ideas, in particular in string theory, based on the fact that at very high energies there, having a length of string, and the fluctuations that determine the diversity observed in nature of stable and unstable particles. In order to understand the question, what is actually determined by the wave properties of microscopic objects, this paper will also be postulated that elementary particles are not point and have a more dense core, but unlike the physical concepts of string theory the diameter of the core determined by the Compton wavelength of the particle.

     (1)

Each particle having rest mass, in addition to the postulated core is also a potential field that decreases at infinity. If the particle is charged, it is electric and gravitational fields. If the particle is neutral, it is only the gravitational field.

Consider a particle diameter equal to the Compton wavelength, which has a rest mass and its own statistical field. Displace the particle at a distance d. In a place with a particle and its potential to shift the field. And in the remote location of the potential value of the field change. But this does not happen instantly, but after some time, due to the fact that the velocity of propagation of the field cannot exceed the speed of light. From this it follows that at the time of the accelerated motion of a particle, a wave perturbations of its own potential field, which propagates with the speed of light.

 Consider the passage of the same particle with the size of the Compton wavelength with speed , through a narrow slit. Time passing through a gap defined by the expression:

         (2)

Due to its potential field, the particle will interact with the walls of the slit, and experience some acceleration. Let this acceleration will be small and the velocity of the particle after passing through the slit, as before can be considered equal. The acceleration of particles will cause a wave of indignation own field, which will be distributed with the speed of light. During the passage of a particle, this wave will extend the gap at a distance:

   (3)

With regard to (1) and (2) we get:

    (4)

Thus, the introduction of quantum mechanics in a non-zero particle size allows you to automatically obtain an expression for the de Broglie wavelength. The physical meaning of the de Broglie wavelength becomes clear. This is not an aggregate function, as is commonly believed, but a real wave, which arises in the potential field in the accelerated motion of a particle having a core equal to the Compton wavelength. The first conclusion from this can be done, that the Copenhagen interpretation of the wave function is false and is only an approximation to reality.

Second. Since the wave properties of particles are determined by the potential field, the wave properties of macroscopic body can have a strong potential field, for example, the planets and stars. Substituting the expression (2) and (3) instead of the postulated in this paper a particle diameter size or macroscopic body l get a more general expression for the de Broglie wavelength:

 

                                                        (5)

 

As you can see the de Broglie wavelength for the armatures differs from the expression (4) and can have a wavelength depending on their size is much larger than it had been assumed previously, focusing only on the formula (4). Disclosure of the physical meaning of the de Broglie wavelength provides a key for constructing a quantum theory of gravity. All you have to do this, it is a necessity given that Planck's constant, which is widely used in quantum mechanics, for the gravitational field can have a different meaning. To determine this value, we use the expression (1), which will be considered more fundamental than Planck's constant. From (1) follows:

     (6)

As seen from (6), Planck's constant contains three parameters: the size of the core particle, mass and velocity of light. This allows you to write an expression for the Planck constant, which can be used for the armatures having the potential field, if the expression (6) instead of the diameter of the core particle and its mass to substitute the appropriate diameter and mass of macroscopic body:

    (7)

That in turn allows us to formulate the Schrodinger equation for the motion of the planets in the central gravitational field of the Sun:

            (8);

 

Where m - a mass of the planet;

          M  a mass of the Sun;

          G  the gravitational constant.

The procedure for the solution of equation (8) is no different from the procedure of solving the Schrodinger equation for the hydrogen atom. This avoids the cumbersome mathematical calculations, and immediately writes down the solutions of this equation:

                                          (9)

 

Where      

Since the presence of the trajectories of the planets moving in orbit around the Sun is no doubt, the expression (9) is convenient to transform and present it through the radiuses of the quantum orbits of the planets. Consider that in classical physics, the energy of the planet in an orbit is given by the expression:

 

                        (10);

Where

      -      the average radius of the orbit of the planet.

Levelling (9) and (10) shall get:

 

 

                         (11);

Quantum mechanics does not allow an unequivocal answer, in which the excited state may be related to the system. She just lets you know all the possible states and the probability of finding each of them. Equation (11) shows that for every planet there are an infinite number of discrete orbits, where it can be. Therefore, we can try to determine the principal quantum number of the planets by comparing calculations made by the formula (11) with the observed radii of the planets. The results of this comparison are presented in Table 1.

 

                            The Table 1.

 

 

Planet	Actual radius of the orbit R mln. km.	The Result of the ccalculations   mln. km.	n	Mistake mln. km.	Percentage error %
Merkury	57.91	58.6	12	0.69	1.2
Venus	108.21	122.5	7	14.3	13.2
Earth	149.6	136.2	7	-13.4	-8.9
Mars	227.95	228.2	17	0.35	0.15
Jupiter	778.34	334.3	1	-444	-57
Saturn	1427.0	920	2	-507	-35
Uranus	2870.97	2816	8	-54.9	-2
Neptune	4498.58	4888.4	10	+389	8
Pluto	5912.2	5931	256	18.8	0.3

 

As can be seen from Table 1, each planet can be attributed to what is the principal quantum number. And these numbers are quite small compared to those that would be obtained if the Schrodinger equation instead of the minimum quantum of action, defined by the formula (7) would be used Planck's constant, commonly used in quantum mechanics. The discrepancy between the calculated and observed values of the radii of the orbits of the planets are considerable. Perhaps this is because the derivation of equation (8) was not taken into account the mutual influence of the planets, leading to a change in their orbits and, consequently, the quantum numbers. But the main show - the orbits of the planets in the solar system are quantized, just as is the case in nuclear physics. This is supported by astronomical observations. The ancient astronomers noticed that the positions of the planets in the solar system is not chaotic, but is subject to certain laws, which is expressed by the Titius-Bode rule. These data clearly show that quantum effects occur in gravity.

Thus, it can be argued that a theory of quantum gravity is possible, but it should be noted that the elementary particles have a size greater than zero and the minimum quantum of action for the armatures is given by (7).

The second consequence, which can make the assumption that elementary particles have a core is equal to the Compton wavelength associated with the existence of quarks.

 

Consider the interaction of two identical particles of the nucleus equal to the Compton wavelength of the rest mass, moving in opposite directions with speed. From the beginning of the collision and to a complete stop of the particles will take some time due to the fact that the rate of momentum transfer within the particles can not exceed the speed of light. During this time, the kinetic energy of the particles will move in the potential energy due to deformation. At the time of the stoppage of a particle of its total energy will consist of the sum of - the rest energy and potential energy stored during the collision. Later, when the particles begin to move in the opposite direction of the potential energy can be expended on the excitation of natural oscillations of the particles. The simplest type of vibration at low energies, which can be excited in the particles can be represented in the form of harmonic oscillations. The potential energy of a particle at a deviation from equilibrium on the value is on the form.

         (12)

                                              k- the coefficient of elasticity of the string

 The Schrodinger equation for stationary states of a harmonic oscillator can be written as:

          (13)

 

The exact solution of equation (13) leads to the following expression for the discrete values

 :

, где n = 0, 1, 2, …   (14)

In the formula (14) unknown coefficient of elasticity of elementary particles is k.

It can be approximately calculated from the following considerations. The collision of the particles when they stop all the kinetic energy is converted into potential energy. Therefore, we can write the equation:

                                                   (15)

If the momentum is transferred inside the particle with the highest possible speed equal to the speed of light, from the beginning of the collision and before the divergence of the particles pass the time  necessary in order to spread the momentum of a particle diameter of the whole is equal to the Compton wavelength:

 

           

           (16)

       During this time, the deviation from the equilibrium state of a particle due to deformation can be:

         (17)

In view of (17), expression (15) can be written as:

          (18)

Whence:

          (19)

Substituting (19) into (14) we obtain an expression for the possible values , suitable for practical calculations:

       where       (20)

Tables (2, 3) show the values of  for electron and proton, calculated on formula (20). In the tables is also specified energy, liberated at disintegration of the agitated conditions when turning

  and full energies of the particles in agitated condition .

     The Table 2.Oscillatory spectrum of the electron е (0,5110034 МэВ.)

Main Quantum Number   n	МэВ	МэВ	МэВ
0	0,041		0,552
1	0,122	0,081	0,633
2	0,203	0,162	0,714
3	0,285	0,244	0,796
4	0,366	0,325	0,877
5	0,448	0,407	0,959
6	0,529	0,488	1,040
7	0,610	0,569	1,121
8	0,692	0,651	1,203
9	0,773	0,732	1,284

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The Table 3.Oscillatory spectrum of the proton P (938,2796 МэВ )

 

Main quantum number n	МэВ	МэВ	МэВ
0	74,69		1012,97
1	224,26	149,58	1162,50
2	373,47	298,78	1311,75
3	522,67	447,97	1460,95
4	672,8	598,11	1611,08
5	822,0	747,31	1760,28
6	971,2	896,51	1909,48
7	1120,4	1045,72	2058,68
8	1270,54	1195,85	2208,82
9	1419,74	1345,05	2358,02
10	1568,94	1494,25	2507,22
11	1718,14	1643,45	2656,42
12	1867,34	1792,65	2805,62
13	2016,53	1941,84	2954,81
14	2165,73	2091,04	3104,0
15	2315,87	2241,18	3254,15
16	2465,1	2390,38	3403,38
17	2614,27	2539,58	3552,55
18	2763,47	2688,78	3701,75
19	2912,67	2837,78	3850,95
20	3062,8	2988,11	4001,08
…	…	…	…

 

Let us consider the spectrum of excited states for the proton (Table 3). As can be seen, the energy of some of the transitions is comparable with the energies of the rest of the particles, which are observed in experiments. For example, the energy released in the transitions between adjacent levels of a harmonic oscillator (149.58 MeV) is comparable with the rest energy of the charged pions (139.57 MeV). The discrepancy is 7.2%. Therefore, we can assume that the decay of the excited states of the proton can form pi mesons. What is actually happening, according to the results of experiments in inelastic collisions of protons. A characteristic feature of the vibrational spectra is that the decay of the excited states is predominantly cascade, that is, energy is released between two adjacent levels. So excited to high energy protons, moving in the ground state, will form many identical particles, the rest energy is comparable with the energy between two adjacent levels. A similar phenomenon is observed in experiments with high-energy collisions of protons and pionization called hadron jets, since most of the secondary particles produced in collisions of protons, pi mesons are. And it can serve as a confirmation that this manifested vibrational spectra.This effect can be further tested in elastic collisions of electrons at the same time if you study the emission spectrum of electrons in collisions. In this case, the production of new particles occurs. But, as seen from the table (2) line emission spectrum of electrons close to the 0.081 MeV, has a good show through. More detailed information can be found in the work (www.mtokma.narod.ru / string.doc), which shows that excitation of the vibrational spectra of the particles may be formed all unstable particles observed in experiments, and the need for the existence of quarks, there is no . Thus, for the construction of all open in the nature of the involvement of unstable particles of quarks is not required. Therefore, the future of physics might come to believe that quarks do not exist, and this is just a successful mathematical model to explain the existing patterns at this level in the structure of hadrons.

 

Heisenberg uncertainty relation.

In 1927, Heisenberg, as a result of numerous thought experiments came to the conclusion that it is impossible to accurately measure both the position and momentum of the particle. This conclusion is given by:

    (21)

Niels Bohr showed that a similar relation holds for the energy uncertainty and uncertainty since the interaction of the object with the measuring instrument:

   (22)

In addition to these relations in a microcosm, there are other additional values to each other. These relations are confirmed by numerous experiments. But what they are due remains unknown. Let's draw for the explanation of this phenomenon is made in this paper, the assumption that elementary particles have a core equal to the Compton wavelength. Let's see if it meets the Compton wavelength of the Heisenberg uncertainty relation. In order to travel a distance equal to the speed of light takes time:

       (23)

Substituting (23) into (1) and noting that  - an energy rest particles, get:

 

   (24)

As can be seen in this case the Heisenberg uncertainty relation is exactly satisfied. Thus, it becomes clear that this ratio is due to the presence in the microcosm of elementary particles of the nucleus equal to the Compton wavelength. The fact that numerous experiments have consistently confirmed the formula (24) clearly indicates that in nature there is no point objects. And the energy at which the elementary particles already have a volume much lower than those of energies, which are discussed in string theory. At the beginning of this paper, the existence of such a core particle was simply postulated, without giving any justification. But these studies are available. And they follow from the general theory of relativity (GTR). The Einstein equations for the gravitational field have the form:

 

                  (25)

Where

      -  the Einstein tensor;

 

       -   the gravitational constant;

 

     -   energy-momentum tensor;

 

     -    indexes running values from 0 to3

 

Cosmological term in (25) is omitted because of its small value, and for analysis in this paper it is needed. A few months after the publication of a German scientist GR Schwarzschild [1] was the first solution of equations (25). This solution describes the gravitational field of a spherical mass in the surrounding area. If the radius of the sphere in which the concentrated mass, the gravitational radius coincides with the solution (25) has the form:

                                                                          (26)

Where:

R  the radius of the curvature space; G  the gravitational constant; M  the spherical mass.

 

Let us check what happens if the equation (25) instead of the gravitational constant G  substitute any other value of the same dimension .

Successively repeating all the steps for solving equations (25) carried out by Schwarzschild at one and the same value of energy-momentum tensor can come to the same solution as (26).

                                                                   (27)

In contrast to (26) in equation (27), the Schwarzschild radius will have a different meaning, and the curvature of space-time is also different. In essence, this is a completely different universe, having its four-dimensional space-time continuum, whose properties may differ significantly from the universe, which is determined by the gravitational field. Since on value  was not assessed any restrictions, then it can take any importance in interval . From this it follows that, substituting in equation (25) each time a new value , we will be getting a new universe and these universes may be infinite. How this may be true? In formulating his theory, Einstein's curvature tensor of space-time is proportional to the energy-momentum tensor through the coupling constant G.  And that is uniquely tied them to the gravitational field. But the energy and momentum have other physical fields existing in nature. This fact allows the use of equations of the theory of general relativity with the same success in this case. Consider how will be (27) if we make the coupling constant    so large that it will comply with the forces that act on the level of elementary particles. Then the value  will be:

                                                                              (28)

Where

    -   the constant Plank,

   - the mass of any elementary particle.

Justification of (28) is given in (www.mtokma.narod.ru/kvant.doc).

Substituting (28) in (27) shall get:

              (29)

With such a large value of the coupling constant, which is given by (28), the Schwarzschild radius in the Einstein equations is minimized to the size of the Compton wavelength of elementary particles.

As can be seen, the Compton wavelength is one of the solutions of the equations of general relativity (25), with a coupling constant (28). And it allows a completely different way to look at the structure of elementary particles. This is not a string, and collapsed due to the strong interaction to the size of the Compton wavelength of the micro black hole.

From the above we can draw several conclusions:

1. The modern physical concepts of string theory that high-energy particles are variations of some strings or branes may be incorrect. At the same time obtained in the framework of this theory, the findings of the possible existence of an infinite number of universes (the so-called problem of the landscape) can be confirmed, although modern theorists make every effort to resolve this infinity.

2. Quarks do not exist. Standard model - this is only an intermediate step in understanding the world.

3. Wave properties possessed by all objects with the potential field, not just the elementary particles.

4. The minimum quantum of action is not a universal constant is suitable for all known interactions in nature.

5. The Copenhagen interpretation of the wave function is false.

6. The orbits of the planets in the solar system is also quantized, as well as the energy states of electrons in atoms. Mathematical formalism of quantum mechanics can be fully applied in gravity, provided that the Planck constant in this case will be determined by the expression (7), and not the value that is used in quantum mechanics today.

7. The equations of general relativity can be applied to all fields that exist in the universe, not just for the gravitational field. Einstein was very close to the creation of a unified field theory. In essence, he created it and because its equation suitable for describing all the interactions that exist in nature. Perhaps, but do not have time during the life of a formulated more precisely.

All of the above conclusions follow only one assumption that elementary particles have a dense core with a diameter equal to the Compton wavelength. This assumption, though one of the solutions of the equations of general relativity, still needs further experimental verification. Such an experiment can be carried out by studying the wave properties of a statistically small charged bodies. In the case of a positive result, many theories of the world structure, including the Big Bang theory, are necessary to revise and refine.

 

The cited literature:

A. Schwarzschild K. On the gravitational field of a point mass in Einstein's theory of / / Albert Einstein and the theory of gravity. 1979. //.pages 199-207.

 

 

 

 

*         *            *

 

Dear Nikolay Saynyuk,

You have many new ideas and have presented them very well. I agree with you on the significance of Compton wavelength and de Broglie wavelength and finite size of particles, although my model is somewhat different from yours. My model does assume a gravitational component of the particle wave, specifically related to gravito-magnetism. I invite you to read my essay, The Nature of the Wave Function. 

Your treatment of planetary orbits as Schrodinger solutions was quite a surprise, and also your harmonic oscillator model of particle collisions converting kinetic and potential energy with subsequent cascading decays.

There is a lot to consider in your essay and some apparently novel ideas. Thanks for submitting your essay and good luck in the contest.

Edwin Eugene Klingman 

*         *            *

 

John A. Macken wrote on Aug. 26, 2012 @ 23:51 GMT

Nikolai,

I agree with many of the points of your essay. I also believe that fundamental particles are not points but waves with quantized angular momentum and a dimension related to the Compton wavelength. I have actually been working on a similar idea for about 10 years. This work still is not complete, but a book manuscript is available online here. When this wave based model of particles is analyzed, it actually generates gravity from first principles (no analogy is made to acceleration). The amazing thing is that this model also makes predictions yielding a previously unknown relationship between the gravitational and electrostatic force. I propose that this is a first step towards unifying these two forces. This relationship between gravity and the electrostatic force is the subject of my essay (Insights into the Unification of Forces). I think that you will find it very interesting since it supports many of the points in your essay. If you are interested, I would like to correspond with you further on this subject. You will find my email address on the link cited above. 

 

 

 

Sergey G Fedosin wrote on Aug. 27, 2012 @ 19:09 GMT

Nicolay

Your equation 1 is the same as equality of rest energy of particle and energy of the wave inside the particle with the energy as for photon: m c² = h c / lambda . Then your deducing of de Broglie wavelength just similar to mine. Wave-particle duality is a consequence of internal standing electromagnetic waves in particles. When we recount with the help of Lorentz transformations these waves in the reference frame, where the particles moving, we find complex wave which amplitude have de Broglie wavelength. See Fedosin S.G. Fizika i filosofiia podobiia ot preonov do metagalaktik. Perm: S.G. Fedosin, 1999, 544 pages. ISBN 5-8131-0012-1. The reason for photon wave-particle duality is other. Every wave quantum (photon) consist of numerous small quanta from the very small charges of substance of electron, if this photon is radiated by the electron in atom. So photon is a particle but the particle is discrete and has wave properties.

About using of Schrodinger equations for planets and your formula (11) see: 

1. M. Oliveira Neto, L.A. Maia, Saulo Carneiro. A DESCRIPTION OF EXTRA-SOLAR PLANETARY ORBITS THROUGH A SCHRODINGER  TYPE DIFFUSION EQUATION. Advances in Space Dynamics 4, H. K. Kuga, Editor, pp 113-121. (2004). 

2. Nottale, L., Schumacher, G., & Lefevre, E. T. Scale-relativity and quantization of exoplanet orbital semi-major axes. Astronomy and Astrophysics, 2000, Vol. 361,P. 379-387. 

3. Nottale, L., Schumacher, G., Gay, J., Scale Relativity and Quantization of the Solar System, 1997, Astron. Astrophys., 322, 1018. 

4. ANTUN RUBCIC and JASNA RUBCIC. THE QUANTIZATION OF THE SOLAR-LIKE GRAVITATIONAL SYSTEMS, FIZIKA B, Vol. 7 (1998) 1, P. 1-14. 

5. ANTUN RUBCIC and JASNA RUBCIC. WHERE THE MOON WAS BORN? FIZIKA A Vol. 18 (2009) 4, P. 185192. 

I quite agree with you in that:

Quarks do not exist. Standard model - this is only an intermediate step in understanding the world.

Wave properties possessed by all objects with the potential field, not just the elementary particles. The substance of nucleons is hold by strong gravitation .

The minimum quantum of action is not a universal constant is suitable for all known interactions in nature. In particular I find quantum of action for the star level of matter and other levels.

Sergey Fedosin