Combining the results from the previous SunQM series papers, a (high resolution) 3D probability density map has been constructed and it is able to describe the whole Solar system with time-dependent orbital movement. It is the Eigen description of our Solar system using Schrodinger equation’s solution. In it, the Eigen n’ values of all planet have been calculated. These Eigen n’ values give both the orbital r and the surface r information for each planet. The result revealed that for all planets, their Eigen n’ values in all three dimensions are equal (n’r = n’θ = n’φ = n’). For example, for a planet at orbit {1,5//6} (in the Solar {N,n//6} QM structure), if it has Eigen n’ = n*q^w = 5*6^11 = 1.81E+9 in each of rθφ-3D dimension, then it will have an orbital r = 1.57E+11 m, and surface r = 7.89E+6 m. This is very close to Earth’s orbital r = 1.49E+11 m and surface r = 6.38E+6 m. For Asteroid belt and the cold-KBO, their Eigen n’(s) in the r- and θ-dimension are equal (n’r = n’θ). For example, Asteroid belt’s Eigen n’ = 48 in both r- and θ-dimension. We found that a planet’s wave function in the φ-dimension is composed by a group of wave functions that further forms a (group) wave packet out of the phase wave. Because Schrodinger equation/solution can accurately describe a Solar system as well as a hydrogen atom, it implies that either the whole universe or a single quark can also be described by Schrodinger equation and solution (simultaneously at the resolution levels of either quark, or proton, or star, or galaxy, or even our universe). Several lower resolution 3D probability density maps (also based on Schrodinger equation’s solution) for the whole Solar system have also been successfully built. A summary of the major results from the phase-1 study of the SunQM series has been listed. This work indisputably proved that Solar system is a QM system.
Combining the results from the previous SunQM series papers, a (high resolution) 3D probability density map has been constructed and it is able to describe the whole Solar system with time-dependent orbital movement. It is the Eigen description of our Solar system using Schrodinger equation’s solution. In it, the Eigen n’ values of all planet have been calculated. These Eigen n’ values give both the orbital r and the surface r information for each planet. The result revealed that for all planets, their Eigen n’ values in all three dimensions are equal (n’r = n’θ = n’φ = n’). For example, for a planet at orbit {1,5//6} (in the Solar {N,n//6} QM structure), if it has Eigen n’ = nq^w = 56^11 = 1.81E+9 in each of rθφ-3D dimension, then it will have an orbital r = 1.57E+11 m, and surface r = 7.89E+6 m. This is very close to Earth’s orbital r = 1.49E+11 m and surface r = 6.38E+6 m. For Asteroid belt and the cold-KBO, their Eigen n’(s) in the r- and θ-dimension are equal (n’r = n’θ). For example, Asteroid belt’s Eigen n’ = 48 in both r- and θ-dimension. We found that a planet’s wave function in the φ-dimension is composed by a group of wave functions that further forms a (group) wave packet out of the phase wave. Because Schrodinger equation/solution can accurately describe a Solar system as well as a hydrogen atom, it implies that either the whole universe or a single quark can also be described by Schrodinger equation and solution (simultaneously at the resolution levels of either quark, or proton, or star, or galaxy, or even our universe). Several lower resolution 3D probability density maps (also based on Schrodinger equation’s solution) for the whole Solar system have also been successfully built. A summary of the major results from the phase-1 study of the SunQM series has been listed. This work indisputably proved that Solar system is a QM system.
In SunQM-3 series, we studied Solar {N,n} QM within the boundary of the traditional Schrodinger equation/solution and Born’s rule. In SunQM-4 series, we start to relax that boundary. In the current paper, for a planet in nLL QM state doing circular orbital movement, we deduced out (a full-QM deduced) |Φ(φ)|^2 * |T(t)|^2 for a planet’s time-dependent probability density in φ-dimension. To satisfy the well-known QM rule that a matter wave’s group velocity equals to 2x of its phase velocity, we have to define a non-Born calculation as |T(t)|^2 ∝ [exp(-i * ω(n,ph) * t)]^2 where ω(n,ph) is the phase angular frequency of a planet’s matter wave in φ-dimension. To obtain a physical meaningful |Φ(φ)|^2 * |T(t)|^2, we have to define a non-Born probability (NBP) density calculation as |Φ(φ)|^2 ∝ Φ(φ), or its φ-dimensional probability density function is directly proportional to its matter wave function. Combining with SunQM-3s11’s result, we built a complete Solar system with time-dependent circular orbital movement using the full-QM deduced non-Born probability density 3D map. This 3D probability density described a Solar system not only at planet’s Eigen description level, but also at any level of resolution (down to proton level, or up to the whole universe level). Therefore, we propose that “Simultaneous-Multi-Eigen-Description (SMED)” is one of many nature attributes of QM. We believe that by adding the non-Born calculation to Born calculation, the QM will become more self-consistent and more complete.
{N,n} QM contains two major parts: the Bohr formula-based part and the Schrodinger equation-based part. 1) For the Bohr formula-based part, after combining all results from the previous SunQM series papers, a {N,n//6} QM Structure (Master) Periodic Table (with N = -23..15, n = 1..12) is established. A summarized result of the {N,n//6} QM structure has been given for each region of this (Master) Periodic Table, including the cosmic {N,n//6} QM (at N > 5), the Solar {N,n//6} QM (at N= 5..-5), the chemical bond dominated world (at N= -2..-11), the atomic and nuclear {N,n//6} QM (at N= -12..-15), and the elementary particles and sub-quark {N,n//6} QM (at N ≤ -16). The common properties (that across all regions of the Master Period Table) has been discussed. For example, by adding many short-life intermediate states (n’ = 1..6^6) in between the major quantum state n=1..5, we can transform a quantum description (with n=1..5} into a (continuous) classical physics description (with n’ = 1..6^6). 2) For the Schrodinger equation-based part, we summarized that how to use {N,n//6} QM plus non-Born probability (NBP) plus Simultaneous-Multi-Eigen-Description (SMED) to describe a planet (or an atom in this planet) that orbiting around Sun, and a photon that propagating away (or towards to) a (Bohr) atom (and it can also be used for a 180° scattering). We further discussed the possibility to extend this description for any kind of movement (elliptical/parabolic/hyperbolic orbiting, scattering, etc.), with any inclination/eccentricity/precession. We also discussed how to use Schrodinger equation/solution for the N-body movement description. We pointed out that there is a conceptual difference between the {N,n} QM and the traditional QM in describing the H-atom’s electron orbit: while the probability (i.e., the Born probability) is the foundation of the traditional QM, the single trajectory (i.e., NBP peak’s time-dependent trajectory) is the foundation of the {N,n} QM (although this single trajectory will pick one of the many possible tracks based on the probability). 3) In the cosmic {N,n//6} QM, we explored the possibility that those super large cosmic “great walls” with distances ~ 9E+9 light-years from Earth are the (~ 9E+9 years old) self-images of our own Milky Way galaxy and be viewed at different angles. If this is correct, then our universe may should be a positive curved 3D space, with the circumference of ~ 9E+9 light-years. This means, just like we are practically living in the 2D spherical space on the surface of a 3D ball something (i.e., a planet), we may practically live in the 3D spherical space on the surface of a 4D ball something. 4) We pointed out that the wave mechanics is equally powerful as that of particle mechanics in solving a QM problem, and a particle mechanics based holographic-description and SMED is waiting to be developed. 5) Because the Bohr formula rn = r1 * n^2 correlates to the (free-fall) accelerated distance formula d = (1/2) *g *t^2 and Newton formula F = m * a, we believed that Bohr formula is the Number-One important formula in physics. We believed that all these results make {N,n} QM theory becomes one of the most (roughly) completed and (roughly) self-consistent theories in physics.
In SunQM-4, to build a full-QM deduced Solar system’s 3D probability density map with time-dependent orbital movement, we developed a non-Born probability (NBP) method. In the current paper, we showed that NBP is directly proportional to the wave function (rather than Born probability’s conjugated-squared wave function). For a cos(x) wave function, its NBP is simply to lift-up wave function by one to make its min =0, and then divided by two to make its max =1, so its NBP = [1 + cos(x)]/2. The trigonometric formula [1 + cos(x)]/2 = [cos(x/2)}^2 revealed that for a planet doing orbital movement in Solar system, its NBP ground state is n=1, and its Born probability ground state is n = 1/2. We showed that NBP is valid not only for {N,n} QM’s nLL QM state, but also for the 1D infinity deep square potential well QM, for the circular 1D QM, and for the plane wave QM. NBP demonstrates more direct and intuitive physical meaning than that of Born probability. In contrast to that Born probability is only for the standing wave function, NBP is for a uni-directional traveling wave function (which naturally includes the linearly combined two opposite-directional traveling waves, or a standing wave). Therefore, Born probability is possible merely a special case of (the more generalized) NBP. With NBP, we may explain the flute sound wave (an air mass density vibration) mechanics directly as the 1D quantum mechanics! This may significantly change our view on QM and its application on our daily-life-world’s (Newtonian) physics. The planet formation through accretion is also discussed by using NBP.
In this paper, we tried to use the non-Born probability (NBP, developed in SunQM-4) to describe Earth atmosphere’s phenomenon in θφ-2D dimension. After analyzing Earth atmosphere’s global circulation pattern, we found that it is related only to the θ-1D QM. Because the spinning Earth’s θ-1D dimensional matter wave is in a true standing wave mode, we found that the Born probability analysis gives more sharp and conclusive result than that of the NBP analysis. Earth atmosphere’s elliptical shape is caused by that the |2,1,1> QM state is the most heavily populated (by air mass); its Hadley/Ferrel/Polar global circulation cells are formed by the |2,1,1> QM’s mass-peaking effect at 0° latitude and |2,1,0> QM’s mass-peaking effect at 60° latitude; and its jet streams may can be explained as the residue |3,2,1> and |3,2,0> states embedded in the extremely suppressed |2,0,0> sub-shell. To describe Earth’s local weather, we have to use NBP because it gives φ (besides θ) dimensional information. We demonstrated that a hurricane’s low air pressure center can be described by a NBP density minimum center (or a negative peak of a combined Re[Y(l,m)] wave functions); an hurricane array can be described by that a single wave function mode of Re[Y(l,m)] is (unevenly) amplified; the polar vortex’s low air pressure center can be described by a NBP density minimum center (or a negative peak of a combined Re[Y(l,m)] wave functions) of low temperature; a single day (7/22/2018)’s extreme weather can be described by that a single wave function mode of Re[Y(l=9,m=7)] was (unevenly) amplified; El Nino related warm/cold water pools oscillation can also be explained by NBP in a combined Re[Y(l,m)], or by the 1D infinitely deep potential well QM. This work clearly revealed the true physical meaning of the QM’s wave function: it is the probability density of a physical variable’s value, although in the wave form. Also, the physical meaning of the Schrodinger equation becomes: it directly describes a dynamic distribution of a physical variable’s value (although in its probability’s wave form).
In this paper, we developed several new methods to describe a photon’s emission and propagation process: 1) To explain a photon that emitted from a Bohr atom from the n=3 to n=2 electron orbit transition, we added a virtual electron that doing RF rotation in a virtual 2D orbit with n=2.43, and it can intuitively explain the frequency and the transverse wave of the photon; 2) Using the newly designed {N,n} QM field theory and the non-Born probability (NBP), we described the photon’s emission and propagation process as a series excitation of a {N,n} QM field’s nLL QM state from low n to high n (e.g., from the {N,n} QM field’s ground state n=1, or |1,0,0> to the first excited state n=2, or |2,1,1>, then to the second excited n=3, or |3,2,2>, and then to the third excited state n=4, or |4,3,3>, …). To describe a photon (that propagated to high n in the {N,n} QM field) with a limited size, in the NBP density formula (of the nLL QM state in the {N,n} QM field), we used the multiplier n’ for the exponential index so that we can describe a photon with any appropriate size that we want; 3) We hypothesized that a photon has an onion-like multi-shells physical structure (a superposition of many QM states), and has a “standard” size of about 100x of its wavelength. We explained a photon’s wave-particle duality as that its out-shells have the wave character, while its inner core has the particle character. We explained a photon’s double-slit experiment as that its out-shell wave’s interference guided its inner core’s particle motion; 4) We explored the possibility that whether the redshift (with Hubble’s constant ~ 70 (km/s)/Mpc) is the natural attribute of the photon propagation; 5) Alternatively, we used only two QM states to describe a photon’s emission and propagation: the ground state of {N,n} QM field |1,0,0> and the excited state of {N,n} QM field |2,1,1>, with r1 = c * t, so that the speed of the light c is incorporated into the description. 6) We believe that the {N,n} QM field theory can be used for other particles’ description as well. Finally, we explained the physical meaning of the matter wave’s wave function as the NBP 3D distribution of a particle/celestial body’s movement trajectory (that is described by the Schrodinger equation).
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.