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 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 𝐴𝑈 ( 𝐴𝑈 represents the distance of earth from the sun) and the phase 10 constant ∅ 0 = 𝜋 6 , is the reason for TBL. Moreover, we prove that this huge wave packet belongs 11 to the sun. In the following of the article, based on the solar system wave function, we will enter 12 into the atomic field and arrive to a new atomic model that helps us to describe many phenomena 13 such as the normal Zeeman effect.


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 19 orbits 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 law known as Bode law or 21 Titius-Bode law (TBL). According to this law, the distance of each planet from the sun is equal quantized (or In other words why the orbital angular momentum is quantized: = ħ). Niels Bohr,40 neither in her famous article [2] nor in the years that followed, could not explain the reason for this 41 event. The new atomic model predicts the existence of secondary lines in the hydrogen spectrum 1 . 42 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 54 node, and the position of fifth node (1.3 ) is empty. The seventh, eighth, and ninth nodes are 55 empty, and Ceres is on the tenth node. Jupiter is placed on the eighteenth node and Saturn is on 56 the thirty-third node, and Uranus, Neptune, and Pluto are on the nodes farther from the sun. As 57 you can see, a wave function, with the wavelength = 0.6 , easily predicts the position of the 58 planets and it seems that a huge and standing wave plays a role in determining the position of the 59 planets in solar system. Therefore, we can consider the reason for the TBL to be the existence of 60 a large cosine wave in solar system that oscillates along the axis perpendicular to the plane of solar 61 system. We call this wave ''Solar system wave function''. In this article, we will obtain the equation  ). 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.
The presence of a huge cosine wave in solar system seems strange at first sight, but quantum 64 mechanics eradicates our surprise. Based on quantum mechanics, a wave packet can be assigned to each object which called the ''associated wave'' of that object, and this associated wave is the 66 solution of the Schrodinger equation. In this article, we prove that the above standing and cosine 67 wave function (Solar system wave function) is in the form of a solution of the Schrodinger equation 68 and therefore, based on quantum mechanics, this wave must belong to an object in Solar system; 69 we demonstrate that this object is the sun (We are aware that today Quantum mechanics, the 70 Schrodinger equation, and the de Broglie wavelength relation only use for subatomic scale and 71 subatomic objects. But, in this article, we prove that quantum mechanics is also valid in 72 astronomical scale and we will obtain the shape of Schrodinger equation and de Broglie 73 wavelength relation in astronomical scale). combination of ± is a traveling wave [7]. For example, sin( − + ∅ 0 ) is a traveling 79 wave. Thus, a standing wave is in the form of ( ) ( + ∅ 0 ) or ( ) ( + ∅ 0 ) 80 or ( ) ( + ∅ 0 ) or cos( ) sin( + ∅ 0 ). As mentioned, a cosine standing wave can 81 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 105 number that we will derive its value). Since solar system has a certain size and is not infinitely 106 wide, its wave function must be localized (a wave packet). If we consider an expression in the 107 form − 2 (which is a Gaussian function and plays the role of a wave envelope) in the final 108 function of solar system, in such a case, the final equation is a localized wave or a wave packet 2 .

109
The value of γ, which is a positive number, will be obtained in the following. Thus, the primary 110 form of the wave function of solar system is as follows (equation 1) and the planets are on the 111 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 114 empirical equation which we will obtain it from mathematic methods in the next section.

117
The value of (0,0) equals √3 2 ⁄ . This diagram is drawn by a certain value of , and γ in equation 1, which we 118 will obtain their value in this article. As you can see, the planets are on the nodes of the wave function. Jupiter, Saturn,

119
Uranus, Neptune, and Pluto are on the nodes farther from the sun. The reason why there is no planet in some nodes 120 will explained in the section 7: ''Elliptical orbits and Isotropic Asteroid Belt''. This is due to the unbalanced mass 121 distribution in the protoplanetary disk of solar system.

122
In figure 1, the wave oscillates along the axis over time. But the nodes and the anti-nodes do not 123 move relative to each other along the x-axis. This does not mean that the wave packet is stationary 124 in the space; it is just like passengers sitting on a train who do not move relative to each other but 125 the train is moving relative to the rails. In the same way, solar system wave packet (equation 1) is 126 a standing wave that rotates, along with solar system, around the center of the galaxy.

127
As you observed, function 1 could easily predicts the position of planets. In the continuation of the 128 article, we will prove that this function is in the form of the real part of a solution of the Schrodinger    136 We obtained the equation of a single-frequency wave packet in the previous section, empirically 137 (equation 1). But, a thought experiment [8] arrives us to this conclusion that a single-frequency wave packet cannot be existed. This thought experiment investigates the beat between two single-139 frequency waves. In this thought experiment, it is proved that only a wave with infinite spreading 140 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

Single-Frequency Wave Packet
The above equation is the momentary image of the net wave. Multiply equation 3 by 0 − 0 .

192
We have: Considering ́= − 0 , we have: 195 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 [12]. 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.
Using the variable transformation Lets check the normalization Given a normalized ( ), we get the normalized ( ).

203
Now, how is the time variation of equation 6? Let's go back to equation 2: 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 211 that propagates in the positive direction of the -axis [7]. This means that the location of the nodes 227 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 245 (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  this is the existence of the inverse_square gravitational force of the sun 7 . As you know, the sun 320 formed earlier than the planets [17][18] [19]. 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) [17] and both phases occurred before the formation of the planets. In both states, we have no idea how or why this wave function formed. and then compressed due to collisions with each other and formed larger grains, Planetesimals,

328
Protoplanets and finally planets 9 . At the same time, the inverse_square force of the sun was at 329 work, and changed the circular orbits to elliptical orbits. Don't you see resemblance between the 330 powder ring in Figure 3 and the asteroid belt? In fact in asteroid belt, Planet formation has stopped 331 at the Planetesimals stage. And for some reason the Protoplanet and the planet was not formed.

332
The mass distribution in the asteroid belt is uniform and isotropic, just like the powder ring in 333 Figure 3, and this is another confirmation of our wave theory. Can a theory other than our wave 334 theory explain the uniform and isotropic distribution of Planetesimals in the asteroid belt? 335 If solar system wave function did not exist; The planets had been orbited around the sun in elliptical 336 orbits but the distance of planets from the sun was random and irregular. Without solar system 337 wave packet, it is not possible to reach to the TBL from the protoplanetary disk. Thus, it seems 338 that the existence of a standing wave in the solar system is undeniable.   has been advanced that has found general acceptance.'' [26] 380

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

452
Regarding Figure 5, we said that the associated wave packet of a particle is dark matter. Dark matter has 453 mass. The mass we measure for elementary particles, such as electron and proton, with different ways is 454 actually the mass of the particle itself and its associated wave packet. Therefore, because the proton is 455 heavier than the electron, its wave packet probably is denser and heavier than the electron wave packet. 456 In the next section, first we will investigate the motion of electron around the nucleus in a hydrogen atom 457 ( 1 1 ) and then, based on the diagram of the first ionization energy of the elements, present our general 458 atomic model and we achieve interesting results. 459 460 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 479 and therefore the orbital magnetic dipole moment for hydrogen atom is equal to: is different from Bohr-Somerfield method [29] in old quantum theory. In their model, the normal 488 Zeeman effect is justified by difference in the space orientation of the subshells (Fig. 4a). But in 489 our model, all of the subshells are in the same plane and the difference is in the areas.

491
The diagram of the first ionization energy is a very valuable diagram that arrived scientists to many 492 results (Fig. 7). It was by investigation of this diagram that they realized the existence of subshells, 493 the order of subshells filling, and many other results. For example, this diagram shows that in the 494 first shell is not subshell, or subshell does not exist in the first and second shells, or in another 495 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 521 just speculation. Our theory considers a large part of these lines to be related to hydrogen atoms 522 and theoretically predicts their existence and gives us their wavelengths. Today, these lines are 523 known as the molecular spectrum of hydrogen [30] [31], which this is based on Merton's article 524 [3]. Merton in his article and in the section "Experimental Results" proved in a very vague way 525 that two groups of the secondary spectrum lines are related to hydrogen molecules and finally 526 concluded that: "it is probable that the whole of the secondary spectrum is due to the hydrogen 527 molecule". But we show that this is wrong, and only a part of these lines are related to hydrogen 528 molecules.

529
In the Bohr atomic model, the lines of the emission and absorption spectrum of hydrogen atom are 530 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 532 example, the second Bohr orbit is the fourth orbit in the new atomic model (Fig. 6). Based on this, in the ultraviolet region. The wavelengths displayed in the deuterium spectrum are exactly the 555 same as the wavelengths of the hydrogen spectrum with a very slightly shift toward shorter wavelengths 10 [33] (this is due to the very small difference between Rydberg constant for 557 deuterium and hydrogen [33]). In Fig. 8, and are consecutive wavelengths of the Ballmer 558 series. The number of the secondary lines is very large both at short wavelengths and at long 559 wavelengths.

560
As mentioned, our atomic model (Fig. 5) is based on equation 16. In this equation, the exponential 561 factor is never zero, and therefore the infinite number of orbits can be considered around the 562 nucleus of each atom. By moving away from the nucleus according to equation 16, the effects of 563 the Coulomb force of the nucleus decreases; Therefore, in distant orbits, the electron is practically 564 not bound to the nucleus.

565
In addition to the new atomic model, some secondary lines are definitely related to hydrogen 566 molecules or impurities inside the lamp.  In the following, based on this point, we will explain NZE, in the usual way of proving the NZE 580 in the old quantum mechanics [29]. The difference is that now the cause of the difference in the 581 amount of is the area of the subshells, not their space orientation.

582
Consider the third orbit of an atom and its nine subshells and assume that the area of these subshells 583 are as shown in table 2.  We want to investigate the transition between and states in the presence of a magnetic field.

595
When the magnetic field is zero, the energy of the state is (for all five subshells) and the 596 energy of the state is (for all three P subshells) and because of transition between and a 597 photon will be emitted by energy: ℎ 0 = − . When the field is turned on, the state splits 598 into five equally spaced magnetic sublevels, and the state splits into three equally spaced 599 magnetic sublevels. According to the   Δ = ±1 means that the transition from to is allowed but from to is forbidden. And we 607 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 .

610
For changes of (principal quantum number) any value is allowed. Authorized transitions are 611 shown in Fig. 9. Thus, each line in the emission spectrum splits into three lines by an external magnetic field. As 620 we said, we used the same common method of proving the NZE [29] here. The difference is that 621 here, instead of space orientation of the orbits, the difference in the area of the orbits causes the 622 lines to split. This description of the NZE is easier than the Sommerfeld model.