Solar System Wave Function and its Achievements

Pluto, Ceres and all planets of solar system except Neptune, with a high approximation, follow a 6 law called Titius-Bode law (TBL) or Bode law, which can by no means be considered as a 7 stochastic event. This law shows that the distance of the planets from the sun in Solar system is 8 regulated. Here, we prove that the existence of a standing and cosine wave packet in solar system, 9 with the wavelength λ = 0.6 ( represents the distance of earth from the sun) and the phase 10 constant , is the reason for TBL. Moreover, we prove that this huge wave packet belongs 11 to the sun (Certainly the presence of a standing wave in space is not without an origin). We will 12 prove that without solar system wave function, it is not possible to reach to the TBL from the 13 protoplanetary disk of solar system. In the following of the article, based on the solar system wave 14 function, we will enter into the atomic field and arrive to a new atomic model that helps us to 15 describe many phenomena such as the normal Zeeman Effect. 16


Introduction
The planets of solar system move around the sun in elliptical orbits such that the sun is in one of 20 the focal points of these ellipses. These ellipses are very close to the circle, and in fact the 21 orbits of the planets of solar system are concentric circles. Pluto, Ceres and all planets of Solar 22 system except Neptune, with a high approximation follow a law known as Bode law or 23 Titius-Bode law (TBL). According to this law, the distance of each planet from the sun is equal 24 to , where 0.4 is the distance of Mercury from the sun (or more 25 precisely the length of the semi-major axis of Mercury's orbit) and [1]. Table. 1 26 shows the high accuracy of the Bode law. If this law was only true for three or four planets, then 27 we could call it a coincidence, but when it is true for seven planets, plus Ceres and Pluto, there is 28 definitely a reason for it. It was historically based on this law that Ceres was discovered in 1801 29 [1]. In this article, we will find the reason for the existence of the TBL. In fact, we will prove that 30 the presence of a cosine and standing wave packet in solar system is the reason for existence of 31 TBL; and moreover, we will prove that this wave packet belongs to the sun. In this article we prove 32 that the Schrodinger equation is valid at the astronomical scale and is not limited to subatomic 33 particles. TBL does not predict the distance of Neptune from the sun but, this article is able to give 34 us the distance of Neptune. In this paper and in the section "Elliptical orbits and Isotropic Asteroid

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Belt", we prove that each model for explain the formation of solar system without considering the 36 solar system wave function is incomplete. We will prove that without solar system wave packet, 37 it is not possible to reach to the TBL from the protoplanetary disk. The distribution of 38 Planetesimals in the asteroid belt is isotropic. In this Article we prove that this isotropy only can 39 explain by our wave theory. 40 Next, using the model of solar system standing wave, we enter into the fundamental topics of 41 quantum mechanics and arrive at an atomic model that has both the Schrodinger wave function 42 and the Bohr atomic model inside. Our atomic model explains why the Bohr atomic orbits are 43 quantized (or In other words why the orbital angular momentum is quantized: ). Niels Bohr,44 neither in her famous article [2] nor in the years that followed, could not explain the reason for this 45 event. The new atomic model predicts the existence of secondary lines in the hydrogen spectrum 1 . 46 These are lines that neither the Bohr   . Of course, this secondary spectrum is now known as the molecular spectrum, which this issue is based on Merton's article [3]. In this article, we show that this is wrong and only a part of the secondary spectrum lines is relevant to the hydrogen molecule.
planets and it seems that a huge and standing wave plays a role in determining the position of the 63 planets in solar system. Therefore, we can consider the reason for the TBL to be the existence of 64 a large cosine wave in solar system that oscillates along the axis perpendicular to the plane of solar 65 system. We call this wave ''Solar system wave function''. In this article, we will obtain the equation 66 of this wave function. But what does this wave belong to? We answer this question in this article.

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The presence of a huge cosine wave in solar system seems strange at first sight, but quantum 68 mechanics eradicates our surprise. Based on quantum mechanics, a wave packet can be attributed 69 to each object which called the ''associated wave'' of that object, and this associated wave is the 70 solution of the Schrodinger equation. In this article, we prove that the above standing and cosine 71 wave function ( number that we will derive its value). Since solar system has a certain size and is not infinitely 110 wide, its wave function must be localized (a wave packet). If we consider an expression in the 111 form (which is a Gaussian function and plays the role of a wave envelope) in the final 112 function of solar system, in such a case, the final equation is a localized wave or a wave packet 2 .

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The value of γ, which is a positive number, will be obtained in the following. Thus, the primary In equation 1, γ, C and δ are constant values and we obtain their values in this article. This is an 118 empirical equation which we will obtain it from mathematic methods in the next section.

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The value of equals . This diagram is drawn by a certain value of , and γ in equation 1, which we 122 will obtain their value in this article. As you can see, the planets are on the nodes of the wave function. Jupiter, Saturn,

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Uranus, Neptune, and Pluto are on the nodes farther from the sun. The reason why there is no planet in some nodes 124 will explained in the section 7: ''Elliptical orbits and Isotropic Asteroid Belt''. This is due to the unbalanced mass 125 distribution in the protoplanetary disk of solar system.

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In figure 1, the wave oscillates along the axis over time. But the nodes and the anti-nodes do not 127 move relative to each other along the x-axis. This does not mean that the wave packet is stationary 128 in the space; it is just like passengers sitting on a train who do not move relative to each other but 129 the train is moving relative to the rails. In the same way, solar system wave packet (equation 1) is 130 a standing wave that rotates, along with solar system, around the center of the galaxy.

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As you observed, function 1 could easily predicts the position of planets. In the continuation of the 132 article, we will prove that this function is in the form of the real part of a solution of the Schrodinger    138 We obtained the equation of a single-frequency wave packet in the previous section, empirically 139 (equation 1). But, a thought experiment [8] arrives us to this conclusion that a single-frequency 140 wave packet cannot be existed. This thought experiment investigates the beat between two single-141 frequency waves. In this thought experiment, it is proved that only a wave with infinite spreading 142 can be single-frequency. This means that a wave packet, which is localized, cannot be single 143 frequency. This experiment apparently has a logical and unambiguous face. but there is a very 144 important problem about thought experiments. A thought experiment, even if is based on a logical 145 process, may lead to wrong results. We prove this with an example. Here we use 146 "counterexample" method (which is mathematic method) and with a counterexample we disproof 147 of the generalization "All thought experiments lead to the right result". A well-known example is  In the next section, we will obtain equation 1 mathematically and we will show that the equation  The above equation is the momentary image of the net wave. Multiply equation 3 by . 198 We have: 199 5 In the Electromagnetic (EM) waves we cannot consider one for two or many waves in which their is different from each other, because for all of the EM waves we have:

Single-Frequency Wave Packet
where is the velocity of light. But for matter waves the issue is different. In the matter waves we have [12]. As you can see is the function of and . Therefore, it is possible to choose one value of for the waves in which their is different from each other. Given a normalized , we get the normalized . This integral is similar to integral 5, which led to (Equation 6). Therefore, we have: In this equation, the larger the α is, the more the width of wave packet, along the x-axis. We drew Solar system. The closest star to solar system is at a distance of 4.8 light-years, which is so far.

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And the biggest and heavyset object in solar system is sun. Therefore, the wave function of solar 266 system can only belong to the sun. In the section ''Elliptical orbits and Isotropic Asteroid Belt'', we 267 will discuss more about the formation of the solar system wave packet. De Broglie considered the 268 wave nature for subatomic particles, and here we attributed the wave nature to celestial objects.

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Neither of these two actions is strange. Rather, they are truths that we must become accustomed 270 to. If we follow the path of proving the Schrodinger equation [13] and put the value ħ instead of ħ,

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In this section we will show that the nebular theory without our wave function theory is an 287 incomplete theory and our wave function model must be attached to the nebular theory in order to 288 explain the formation of the solar system.

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The equation 14 in cylindrical coordinate is:  Protoplanet and planet was not formed [19]). Can a theory other than our wave theory explain the 316 uniform and isotropic distribution of Planetesimals in the asteroid belt? 317 If solar system wave function did not exist; the planets might have been formed around the sun but 318 the distance of planets from the sun was random and irregular. Without solar system wave packet, 319 it is not possible to reach to the TBL from the protoplanetary disk. Thus, it seems that the existence 320 of a standing wave in the solar system is undeniable. Therefore, the nebular theory becomes more 321 complete by our wave function theory.

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The reason why there is no planet in some nodes in figure 1 and 2 is due to the unbalanced mass 323 distribution in the protoplanetary disk of solar system.  7 We know from classical mechanics that the elliptic orbits of the planets (Kepler's first law) are the result of Newton's law of gravitation, which is an inverse_square relation 8 The wave function of the solar system probably was formed either when the sun was a protostar or when the newborn sun was entered to the Main-sequence. The distance between these two phases is very short (less than 50 million years) [15] and both phases occurred before the formation of the planets. In both states, we have no idea how or why this wave function formed. 9 "Collisions" were more like gentle touches. The particles were far too small to attract each other gravitationally at this point, they were able to stick together through electrostatic forces-the same "static electricity" that makes hair stick to a comb [20]. 10 Since the mass distribution in the Protoplanetary disk had not been uniform. During formation of the planets, in some nodes of the solar system wave function, less dusts were collected and in some nodes, more dusts were placed next to each other. Therefore, in some nodes a planet had been formed and in others it had not been formed.   In this article, we showed that a standing wave oscillation had been effective in the formation of 341 orbits of solar system planets and created the Bode rule. On the other hand, we know that Dark 342 matter interacts with baryonic matters (baryonic in astronomical terms) and affects them 11 . 343 Therefore, it can be concluded that the standing wave oscillation in Fig. 1, with the equation 1,

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probably is the oscillation of Dark matter (Or dark matter is one of the candidate for the nature of   to the growing Planetesimals via radiation pressure) [27]. But in our article is not any simplification 379 or assumption. Our paper, without eliminating the physical parameters of solar system, justifies has been advanced that has found general acceptance.'' [28] 391

Solar System Wave Function and the Other Theories
In addition, some calculations and attempts were made which showed that TBL was a matter of

463
Regarding Figure 5, we said that the associated wave packet of a particle is dark matter. Dark matter has 464 mass. The mass we measure for elementary particles, such as electron and proton, with different ways is 465 actually the mass of the particle itself and its associated wave packet. Therefore, because the proton is 466 heavier than the electron, its wave packet probably is denser and heavier than the electron wave packet. 467 In the next section, first we will investigate the motion of electron around the nucleus in a hydrogen atom 468 ( ) and then, based on the diagram of the first ionization energy of the elements, present our general 469 atomic model and we achieve interesting results. 470

471
We know from mechanics that a mass or an electric charge that is affected by a central force moves is different from Bohr-Somerfield method [31] in old quantum theory. In their model, the normal 499 Zeeman effect is justified by difference in the space orientation of the subshells (Fig. 4a). But in 500 our model, all of the subshells are in the same plane and the difference is in the areas.

502
The diagram of the first ionization energy is a very valuable diagram that arrived scientists to many 503 results (Fig. 7). It was by investigation of this diagram that they realized the existence of subshells, 504 the order of subshells filling, and many other results. For example, this diagram shows that in the 505 first shell is not subshell, or subshell does not exist in the first and second shells, or in another 506 example, this diagram shows that 4 subshell fills earlier than 3 . lamp [3], and some have attributed them to hydrogen molecules in the lamp [3]. But all this was 532 just speculation. Our theory considers a large part of these lines to be related to hydrogen atoms 533 and theoretically predicts their existence and gives us their wavelengths. Today, these lines are 534 known as the molecular spectrum of hydrogen [32] [33], which this is based on Merton's article 535 [3]. Merton in his article and in the section "Experimental Results" proved in a very vague way 536 that two groups of the secondary spectrum lines are related to hydrogen molecules and finally 537 concluded that: "it is probable that the whole of the secondary spectrum is due to the hydrogen 538 molecule". But we show that this is wrong, and only a part of these lines are related to hydrogen 539 molecules.

540
In the Bohr atomic model, the lines of the emission and absorption spectrum of hydrogen atom are 541 the result of quantum jumps. We use the same assumption in the new atomic theory. Consider 542 Figure 6. As we said, if the Bohr model has orbits, the new atomic model has orbits. For 543 example, the second Bohr orbit is the fourth orbit in the new atomic model (Fig. 6). Based on this, These two wavelengths along with the previous wavelength can be easily seen in Fig. 8. This  In the following, based on this point, we will explain NZE, in the usual way of proving the NZE 591 in the old quantum mechanics [31]. The difference is that now the cause of the difference in the 592 amount of is the area of the subshells, not their space orientation.

593
Consider the third orbit of an atom and its nine subshells and assume that the area of these subshells 594 are as shown in table 2.

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Subshell of the third orbit Area of subshell

598
In such a case, the difference in the energy levels in the presence of the magnetic field is obtained 599 from the following equation: We want to investigate the transition between and states in the presence of a magnetic field.

606
When the magnetic field is zero, the energy of the state is (for all five subshells) and the 607 energy of the state is (for all three P subshells) and because of transition between and a 608 photon will be emitted by energy: . When the field is turned on, the state splits 609 into five equally spaced magnetic sublevels, and the state splits into three equally spaced 610 magnetic sublevels. According to the That is, only those transitions are allowed in which either does not change or changes by .

621
For changes of (principal quantum number) any value is allowed. Authorized transitions are 622 shown in Fig. 9. Thus, each line in the emission spectrum splits into three lines by an external magnetic field. As 631 we said, we used the same common method of proving the NZE [31] here. The difference is that 632 here, instead of space orientation of the orbits, the difference in the area of the orbits causes the 633 lines to split. This description of the NZE is easier than the Sommerfeld model.

634
The above relations were obtained in the situation that was in the direction of . If and have 635 a angle with each other, then in the two equations of above relations we will have : sphere. Which in this case is higher than the velocity of light. But in our model, the electron is 644 surrounded by a wave packet (Fig. 5). If we assume associated wave of a particle is dark matter, collisions [38]. In these experiments, the diameter of the associated wave packet is not measured.

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The wave packet in collisions is transparent and low effect. The diameter of the wave packet 659 usually is defined as one-half of maximum value of the function at the center.