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 section "Elliptical orbits", 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 atomic model inside. Our atomic model explains why the Bohr atomic orbits are quantized (or In other words why the orbital angular momentum is quantized: = ħ). Niels Bohr,39 neither in her famous article [2] nor in the years that followed, could not explain the reason for this 40 event. The new atomic model predicts the existence of secondary lines in the hydrogen spectrum 1 . 41 These are lines that neither the Bohr   is on the second node, Venus is on the third node, earth is on the fourth node, Mars is on the sixth 53 node, and the position of fifth node (1.3 ) is empty. The seventh, eighth, and ninth nodes are 54 empty, and Ceres is on the tenth node. Jupiter is placed on the eighteenth node and Saturn is on 55 the thirty-third node, and Uranus, Neptune, and Pluto are on the nodes farther from the sun. As 56 you can see, a wave function, with the wavelength = 0.6 , easily predicts the position of the 57 planets and it seems that a huge and standing wave plays a role in determining the position of the 58 planets in solar system. Therefore, we can consider the reason for the Bode rule to be the existence 59 of a large cosine wave in solar system that oscillates along the axis perpendicular to the plane of 60 solar system. we call this wave ''Solar system wave function''. In this article, we will obtain the ). 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. quantum mechanics eradicates our surprise. Based on quantum mechanics, a wave packet can be 64 assigned to each object which called the ''associated wave'' of that object, and this associated wave is the solution of the Schrodinger equation. In this article, we prove that the above standing and 66 cosine wave function (Solar system wave function) is in the form of a solution of the Schrodinger 67 equation and therefore, based on quantum mechanics, this wave must belong to an object in Solar 68 system; we demonstrate that this object is the sun (We are aware that today Quantum mechanics, 69 the Schrodinger equation, and the de Broglie wavelength relation only use for subatomic scale and 70 subatomic objects. But, in this article, we prove that quantum mechanics is also valid in 71 astronomical scale and we will obtain the shape of Schrodinger equation and de Broglie 72 wavelength relation in astronomical scale). 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

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, 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 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  This example fully proves that a thought experiment is meaningless because you cannot rely on its 150 results and its results may be wrong. Obviously, this rule (Unreliability of the results of a thought 151 experiment) also is true for the mentioned beat thought experiment. And therefore we cannot 152 comment on the results of the beat experiment until it is performed. Which it means that we cannot 153 comment on the existence or nonexistence of single-frequency wave packet. In addition, attention 154 to the empirical equation 1 we have a single-frequency wave packet and therefore the beat thought 155 experiment is contrary to observation and so is wrong.

156
In the next section, we will obtain equation 1 mathematically and we will show that the equation 192 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.

Lets check the normalization
Given a normalized ( ), we get the normalized ( ).

195
Now, how is the time variation of equation 6? Let's go back to equation 2: Substituting 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 203 that propagates in the positive direction of the -axis [7]. This means that the location of the nodes 219 There is not the structure of ± in equation 12 so the is a standing wave. As you Which is the same as equation 1 for ≤ 0. Therefore, the final form of solar system wave function 237 (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

247
Here we demonstrated that solar system wave function (equation 1) is the real part of a solution of 248 the Schrodinger equation. So, based on quantum mechanics, we can assign it to an object in Solar 249 system. The closest star to solar system is at a distance of 4.8 light-years, which is so far. And the 250 biggest and heavyset object in solar system is sun. Therefore, the wave function of solar system 251 can only belong to the sun. In the section '' Elliptical orbits '', we will discuss more about the 252 formation of the solar system wave packet. De Broglie considered the wave nature for subatomic 253 particles, and here we attributed the wave nature to celestial objects. Neither of these two actions 254 is strange. Rather, they are truths that we must become accustomed to.

264
If we follow the path of proving the Schrodinger equation [11] and put the value ħ instead of ħ, As you can see in Fig. 1 and Fig. 2, because of symmetry of equation 1, the orbits, created by the 297 solar system wave function, are circle. But the real orbits of the planets are elliptic. The reason for this is the existence of the inverse_square gravitational force of the sun 7 . As you know, the sun 299 formed earlier than the planets [15][16] [17]. Simultaneously with the formation of the sun, about   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 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. 9 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.

321
In this section, we present a new atomic model based on the model of the solar system wave 322 function namely Fig. 2). Our atomic model explains why the Bohr atomic orbits are quantized.

392
Regarding Figure 5, we said that the associated wave packet of a particle is dark matter. Dark matter has 393 mass. The mass we measure for elementary particles, such as electron and proton, with different ways is 394 actually the mass of the particle itself and its associated wave packet. Therefore, because the proton is 395 heavier than the electron, its wave packet probably is denser and heavier than the electron wave packet. 396 In the next section, first we will investigate the motion of electron around the nucleus in a hydrogen atom 397 ( 1 1 ) and then, based on the diagram of the first ionization energy of the elements, present our general 398 atomic model and we achieve interesting results. 399 400 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 419 and therefore the orbital magnetic dipole moment for hydrogen atom is equal to:

8-2. The Atomic Hydrogen Model
Where is electric current [23]. In the new atomic model, the direction of the orbital magnetic 422 dipole moment is perpendicular to the plane of the orbit. The dependence of the orbital magnetic 423 dipole moment to the area of the orbit of electron is a very important issue that helps us to justify 424 the normal Zeeman effect. In the following sections, we consider a few subshells for each main 425 orbit, which the areas of these subshells are different from each other. This difference is the cause 426 of difference in the value of , which helps us to justify the normal Zeeman effect. This method 427 is different from Bohr-Somerfield method [24] in old quantum theory. In their model, the normal 428 Zeeman effect is justified by difference in the space orientation of the subshells (Fig. 4a). But in 429 our model, all of the subshells are in the same plane and the difference is in the areas.

431
The diagram of the first ionization energy is a very valuable diagram that arrived scientists to many 432 results (Fig. 7). It was by investigation of this diagram that they realized the existence of subshells, 433 the order of subshells filling, and many other results. For example, this diagram shows that in the 434 first shell is not subshell, or subshell does not exist in the first and second shells, or in another 435 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 461 just speculation. Our theory considers a large part of these lines to be related to hydrogen atoms 462 and theoretically predicts their existence and gives us their wavelengths. Today, these lines are 463 known as the molecular spectrum of hydrogen [25] [26], which this is based on Merton's article 464 [3]. Merton in his article and in the section "Experimental Results" proved in a very vague way 465 that two groups of the secondary spectrum lines are related to hydrogen molecules and finally 466 concluded that: "it is probable that the whole of the secondary spectrum is due to the hydrogen 467 molecule". But we show that this is wrong, and only a part of these lines are related to hydrogen 468 molecules.

469
In the Bohr atomic model, the lines of the emission and absorption spectrum of hydrogen atom are 470 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 472 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 512 effect (NZE), was discovered by Zeeman in 1896. In the years that followed, Sommerfeld, who 513 justified the fine structure by considering elliptical subshells for each orbit [29], assumed that each 514 of these subshells has a different space orientation and all are not in the one plane [30]. For 515 example, the three subshells in the second orbit have three orientations (Fig. 4a). As mentioned, 516 in the new atomic model we considered all of the subshells in one plane and assume that these 517 subshells have different areas. Because of difference between these areas the orbital magnetic 518 dipole moments of subshells ( ) will be different from each other, based on equation 22.

519
in the old quantum mechanics [24]. The difference is that now the cause of the difference in the 521 amount of is the area of the subshells, not their space orientation.

522
Consider the third orbit of an atom and its nine subshells and assume that the area of these subshells 523 are as shown in table 2.

527
In such a case, the difference in the energy levels in the presence of the magnetic field is obtained We want to investigate the transition between and states in the presence of a magnetic field.

535
When the magnetic field is zero, the energy of the state is (for all five subshells) and the 536 energy of the state is (for all three P subshells) and because of transition between and a 537 photon will be emitted by energy: ℎ 0 = − . When the field is turned on, the state splits 538 into five equally spaced magnetic sublevels, and the state splits into three equally spaced 539 magnetic sublevels. According to the

544
and its connection with Schrodinger mechanics and angular momentum is by no means intended in this section. We 545 can use any other sign. Δ = ±1 means that the transition from to is allowed but from to is forbidden. And we 547 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 .

550
For changes of (principal quantum number) any value is allowed. Authorized transitions are 551 shown in Fig. 9. Thus, each line in the emission spectrum splits into three lines by an external magnetic field. As 560 we said, we used the same common method of proving the NZE [24] here. The difference is that 561 here, instead of space orientation of the orbits, the difference in the area of the orbits causes the 562 lines to split. This description of the NZE is easier than the Sommerfeld model.

563
The above relations were obtained in the situation that was in the direction of . If and have 564 a angle with each other, then in the two equations of above relations we will have : 565 = 2 0 : ℎ = ℎ 0 + 2 0 ⟹ ν = 0 + 2 0 ℎ 566 = 0: ℎ = ℎ 0 ⟹ = 0 567 = −2 0 : ℎ = ℎ 0 − 2 0 ⟹ = 0 − 2 0 ℎ 568 8-6. Spin as a Classical Property 569 We know that if we consider the electron as the spinning sphere of charge, in such a case, given collisions [31]. In these experiments, the diameter of the associated wave packet is not measured.

585
The wave packet in collisions is transparent and low effect. The diameter of the wave packet 586 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 593 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 597 and the final atomic model. We have noticed the existence of a wave along with an object since 598 De Broglie. But we never thought it would be in the form of Figures 5 and 2.

599
In our opinion, prediction of the secondary lines of the hydrogen spectrum is a complete success 600 for the new atomic model. Based on the new atomic model, normal Zeeman effect was more easily 601 justified. Anomalous Zeeman effect, which is caused by the spin of electron, is also easily 602 described by the new atomic model, which we have not described due to the length of the article.

603
Based on this article, the associated wave packet of an object (such as electron, proton or sun) is