Solar System Wave Function and its Achievements

5 Pluto, Ceres and all planets of solar system except Neptune, with a high approximation, follow a 6 rule called Titius-Bode rule or Bode rule, which can by no means be considered as a stochastic 7 event. This rule shows that the distance of the planets from the sun in Solar system is regulated. 8 Here, we prove that the existence of a standing and cosine wave packet in solar system, with the 9 wavelength λ = 0.6 𝐴𝑈 ( 𝐴𝑈 represents the distance of earth from the sun) and the phase constant 10 ∅ 0 = 𝜋 6 , is the reason for Bode rule. Moreover, we prove that this huge wave packet belongs to 11 the sun. In the following of the article, based on the solar system wave function, we will enter into 12 the atomic field and arrive to a new atomic model that helps us to describe many phenomena such 13 as the normal Zeeman effect. 14


Introduction
The planets of solar system move around the sun in elliptical orbits such that the sun is in one of 18 the focal points of these ellipses. These ellipses are very close to the circle, and in fact the orbits 19 of the planets of solar system are concentric circles. Pluto, Ceres and all planets of Solar 20 system except Neptune, with a high approximation follow a rule known as Bode rule or 21 Titius-Bode rule. According to this rule, the distance of each planet from the sun is equal 22 to = 0.4 + 0.3 × 2 , where 0.4 is the distance of Mercury from the sun (or more 23 precisely the length of the semi-major axis of Mercury's orbit) and = 0,1,2,3, …. [1]. Table. 1 24 shows the high accuracy of the Bode rule. If this rule was only true for three or four planets, then 25 we could call it a coincidence, but when it is true for seven planets, plus Ceres and Pluto, there is 26 definitely a reason for it. It was historically based on this rule that Ceres was discovered in 1801 27 [1]. In this article, we will find the reason for the existence of the Bode rule. In fact, we will prove 28 that the presence of a cosine and standing wave packet in solar system is the reason for existence 29 of Bode rule; and moreover, we will prove that this wave packet belongs to the sun. In this article 30 we prove that the Schrodinger equation is valid at the scale of the solar system and is not limited 31 to subatomic particles. Bode rule does not predict the distance of Neptune from the sun but, this 32 article is able to give us the distance of Neptune. In this paper and in the "Elliptical orbits" section, 33 we prove that each model for explain the formation of solar system without considering the solar 34 system wave function is incomplete and wrong. 35 Next, using the model of solar system standing wave, we enter into the fundamental topics of 36 quantum mechanics and arrive at an atomic model that has both the Schrodinger wave function 37 and the Bohr    combination of ± is a traveling wave [7]. For example, sin( − + ∅ 0 ) is a traveling 78 wave. Thus, a standing wave is in the form of ( ) ( + ∅ 0 ) or ( ) ( + ∅ 0 ) 79 or ( ) ( + ∅ 0 ) or cos( ) sin( + ∅ 0 ). As mentioned, a cosine standing wave can 80 predict the positions of planets. Therefore, the form of the standing wave of solar system must be ) and then we will show that our choice is correct (δ is a constant 104 number that we will derive its value). Since solar system has a certain size and is not infinitely 105 wide, its wave function must be localized (a wave packet). If we consider an expression in the 106 form − 2 (which is a Gaussian function and plays the role of a wave envelope) in the final 107 function of solar system, in such a case, the final equation is a localized wave or a wave packet 2 .

108
The value of γ, which is a positive number, will be obtained in the following. Thus, the primary 109 form of the wave function of solar system is as follows (equation 1) and the planets are on the 110 nodes of this wave function ( Fig. 1): In equation 1, γ, C and δ are constant values and we obtain their values in this article. This is an 113 empirical equation which we will obtain it from mathematic methods in the next section.

116
The value of (0,0) equals √3 2 ⁄ . This diagram is drawn by a certain value of , and γ in equation 1, which we 117 will obtain their value in this article. As you can see, the planets are on the nodes of the wave function. Jupiter, Saturn,

118
Uranus, Neptune, and Pluto are on the nodes farther from the sun. The reason why there is no planet in some nodes 119 will explained in the section 7: ''elliptical orbits''. This is due to the unbalanced mass distribution in the protoplanetary 120 disk of solar system.

121
In figure 1, the wave oscillates along the axis over time. But the nodes and the anti-nodes do not 122 move relative to each other along the x-axis. This does not mean that the wave packet is stationary 123 in the space; it is just like passengers sitting on a train who do not move relative to each other but 124 the train is moving relative to the rails. In the same way, solar system wave packet (equation 1) is 125 a standing wave that rotates, along with solar system, around the center of the galaxy.

126
As you observed, function 1 could easily predicts the position of planets. In the continuation of the 127 article, we will prove that this function is in the form of the real part of a solution of the Schrodinger    135 We obtained the equation of a single-frequency wave packet in the previous section, empirically 136 (equation 1). But, a thought experiment [20] arrives us to this conclusion that a single-frequency 137 wave packet cannot be existed. This thought experiment investigates the beat between two single-138 frequency waves. In this thought experiment, it is proved that only a wave with infinite spreading 139 can be single-frequency. This means that a wave packet, which is localized, cannot be single  In the next section, we will obtain equation 1 mathematically and we will show that the equation 1 is the real part of a solution of the Schrodinger equation.   3 In fact, all experiments are initially a thought experiment. Some of them remain a thought experiment due to the human inability to perform the experiment in the real world, and some become real physical experiment. 4 In the Electromagnetic (EM) waves we cannot consider one 0 for two or many waves in which their is different from each other, because for all of the EM waves we have: = where is the velocity of light. But for matter waves the issue is different. In the matter waves we have = ħ 2 2 [10]. As you can see is the function of and . Therefore, it is possible to choose one value of 0 for the waves in which their is different from each other. is obtained by normalize of ( ). Since equation 2 is derived from the superposition principle, it 172 is the solution of the Schrodinger equation [10].

173
To obtain ( , ) from equation 2, we calculate the superposition of all of the waves in one 174 moment, which we consider to be the origin of time ( = 0), and then we can obtain the net wave 175 at any other time. We have: The above equation is the momentary image of the net wave. Multiply equation 3 by 0 − 0 .

178
We have: Considering ́= − 0 , we have: Using the variable transformation Lets check the normalization Given a normalized ( ), we get the normalized ( ).

189
Now, how is the time variation of equation 6? Let's go back to equation 2: We put 0 − 0 in the equation: This integral is similar to integral 5, which led to ( , 0) (Equation 6). Therefore, we have: Due to the presence of the factor 0 − 0 , equations 7 and 8 represent a traveling wave packet 197 that propagates in the positive direction of the -axis [7]. This means that the location of the nodes 212 There is not the structure of ± in equation 12 so the is a standing wave. As you following the path we have taken from equation 2 to equation 13, we reach the following relation; Which is the same as equation 1 for ≤ 0. Therefore, the final form of solar system wave function

230
(equation 1) is as follows: In this equation, the larger the α is, the more the width of wave packet, along the x-axis. We drew system. The closest star to solar system is at a distance of 4.8 light-years, which is so far. And the 237 biggest and heavyset object in solar system is sun. Therefore, the wave function of solar system 238 can only belong to the sun. In the '' Elliptical orbits '' section, we will discuss more about the 239 formation of the solar system wave packet. De Broglie considered the wave nature of light for 240 subatomic particles, and here we attributed the wave nature to celestial objects. Neither of these 241 two actions is strange. Rather, they are truths that we must become accustomed to.  [14]. But in this section we will show that the real 262 part of the solution of the Schrodinger equation is the oscillation of Dark matter.

263
In this article, we showed that a standing wave oscillation had been effective in the formation of   inverse_square force of the sun was at work, and changed the circular orbits to elliptical orbits. 296 6 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 7 The wave function of the solar system probably was formed either when the sun was a protostar or when the newborn sun was on 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. 8 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.
If solar system wave function did not exist; The planets had been orbited around the sun in elliptical 297 orbits but the distance of planets from the sun was random and irregular. Without solar system 298 wave packet, it is not possible to reach to the Titius-Bode rule from the molecular nebula and the 299 protoplanetary disk. Thus, the existence of a standing wave in the solar system is undeniable. But, 300 we do not know if solar system wave packet is a frozen standing wave or oscillating. Coulomb do not affect the associated wave packet of an object and only affect the particle. An 318 example can help to understand the issue; consider a person who has an umbrella over his head 319 and you pull him towards yourself. This work has no effect on the opening and closing of the 320 umbrella (the wave function of the solar system is also an umbrella above the sun). The change in 321 the wavelength of the associated wave packet of an object is probably related to another factor 322 such as the change in the mass of matter or the other factors that we do not yet know. Thus the 323 observations show that the potential has no effect on the associated wave packet of matter and we must eliminate potential from the Schrodinger equation. This is a shocking result, but there is no 325 escaping it.

326
Here we need to say an important point first. Consider Fig. 5. This is the shape of our atomic 327 model, which is based on the diagram of wave function of the solar system (Fig. 2). This model is 328 a unifying model that includes both the Schrodinger wave function and the Bohr atomic orbits.

430
Regarding Figure 5, we said that the associated wave packet of a particle is dark matter. Dark matter has 431 mass. The mass we measure for elementary particles, such as electron and proton, with different ways is 432 actually the mass of the particle itself and its associated wave packet. Therefore, because the proton is 433 heavier than the electron, its wave packet probably is denser and heavier than the electron wave packet. 434 In the next section, first we will investigate the motion of electron around the nucleus in a hydrogen atom 435 ( 1 1 ) and then, based on the diagram of the first ionization energy of the elements, present our general 436 atomic model and we achieve interesting results. 437 438 We know from mechanics that a mass or an electric charge that is affected by a central force moves In such a case, the area of orbit equals : 1 = 1 1 . The orbit of electron is a current loop 457 and therefore the orbital magnetic dipole moment for hydrogen atom is equal to:

9-2. The Atomic Hydrogen Model
Where is electric current [23]. In the new atomic model, the direction of the orbital magnetic 460 dipole moment is perpendicular to the plane of the orbit. The dependence of the orbital magnetic 461 dipole moment to the area of the orbit of electron is a very important issue that helps us to justify 462 the normal Zeeman effect. In the following sections, we consider a few subshells for each main 463 orbit, which the areas of these subshells are different from each other. This difference is the cause 464 of difference in the value of , which helps us to justify the normal Zeeman effect. This method 465 is different from Bohr-Somerfield method [24] in old quantum theory. In their model, the normal 466 Zeeman effect is justified by difference in the space orientation of the subshells (Fig. 4a). But in 467 our model, all of the subshells are in the same plane and the difference is in the areas.

469
The diagram of the first ionization energy is a very valuable diagram that arrived scientists to many 470 results (Fig. 7). It was by investigation of this diagram that they realized the existence of subshells, 471 the order of subshells filling, and many other results. For example, this diagram shows that in the 472 first shell is not subshell, or subshell does not exist in the first and second shells, or in another 473 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 499 just speculation. Our theory considers a large part of these lines to be related to hydrogen atoms 500 and theoretically predicts their existence and gives us their wavelengths. Today, these lines are 501 known as the molecular spectrum of hydrogen [25] [26], which this is based on Merton's article 502 [3]. Merton in his article and in the section "Experimental Results" proved in a very vague way 503 that two groups of the secondary spectrum lines are related to hydrogen molecules and finally 504 concluded that: "it is probable that the whole of the secondary spectrum is due to the hydrogen 505 molecule". But we show that this is wrong, and only a part of these lines are related to hydrogen 506 molecules.

507
In the Bohr atomic model, the lines of the emission and absorption spectrum of hydrogen atom are 508 the result of quantum jumps. We use the same assumption in the new atomic theory. Consider Figure 6. As we said, if the Bohr model has orbits, the new atomic model has 2 orbits. For 510 example, the second Bohr orbit is the fourth orbit in the new atomic model (Fig. 6). Based on this, The splitting of spectrum lines in the presence of an external magnetic field, or the normal Zeeman 549 effect (NZE), was discovered by Zeeman in 1896. In the years that followed, Sommerfeld, who 550 justified the fine structure by considering elliptical subshells for each orbit [29], assumed that each 551 of these subshells has a different space orientation and all are not in the one plane [30]. For 552 example, the three subshells in the second orbit have three orientations (Fig. 4a). As mentioned, 553 in the new atomic model we considered all of the subshells in one plane and assume that these 554 subshells have different areas. Because of difference between these areas the orbital magnetic 555 dipole moments of subshells ( ) will be different from each other, based on equation 22.

556
In the following, based on this point, we will explain NZE, in the usual way of proving the NZE 557 in the old quantum mechanics [24]. The difference is that now the cause of the difference in the 558 amount of is the area of the subshells, not their space orientation.

559
Consider the third orbit of an atom and its nine subshells and assume that the area of these subshells 560 are as shown in table 2.  We want to investigate the transition between and states in the presence of a magnetic field.

572
When the magnetic field is zero, the energy of the state is (for all five subshells) and the 573 energy of the state is (for all three P subshells) and because of transition between and a 574 photon will be emitted by energy: ℎ 0 = − . When the field is turned on, the state splits 575 into five equally spaced magnetic sublevels, and the state splits into three equally spaced 576 magnetic sublevels. According to the

581
and its connection with Schrodinger mechanics and angular momentum is by no means intended in this section. We 582 can use any other sign. Δ = ±1 means that the transition from to is allowed but from to is forbidden. And we 584 also should consider a selection rule for changing area of the subshells. we have: That is, only those transitions are allowed in which either does not change or changes by 2 0 .

587
For changes of (principal quantum number) any value is allowed. Authorized transitions are 588 shown in Fig. 9. Thus, each line in the emission spectrum splits into three lines by an external magnetic field. As 597 we said, we used the same common method of proving the NZE [24] here. The difference is that 598 here, instead of space orientation of the orbits, the difference in the area of the orbits causes the 599 lines to split. This description of the NZE is easier than the Sommerfeld model.

600
The above relations were obtained in the situation that was in the direction of . If and have collisions [31]. In these experiments, the diameter of the associated wave packet is not measured.

622
The wave packet in collisions is transparent and low effect. The diameter of the wave packet 623 usually is defined as one-half of maximum value of the function at the center. for subatomic particles, and here we attributed the wave nature to celestial objects. Neither of these 630 two actions is strange. Rather, they are truths that we must become accustomed to. and Schrodinger each were achieved to a piece of reality, and Figure 5 is probably total of reality 634 and the final atomic model.

635
In our opinion, prediction of the secondary lines of the hydrogen spectrum is a complete success 636 for the new atomic model. Based on the new atomic model, normal Zeeman effect was more easily 637 justified. Anomalous Zeeman effect, which is caused by the spin of electron, is also easily 638 described by the new atomic model, which we have not described due to the length of the article.

639
Based on this article, the associated wave packet of a matter (such as electron, proton or sun) is Here, we proved that the potential must be removed from the Schrodinger equation 645 and we proved that the word of '' thought experiment '' is meaningless and an experiment must be 646 done in the real world so that we can talk about its results. In this article, we turned quantum 647 mechanics into a simpler and more beautiful theory.