Wanted to share something which I noticed during my quest for Truth. Some of it you already know but not sure how much deep you have gone on this. The formula for sum of first N Numbers is N (N+1) /2, and sum of first of N^2 numbers is N (N+1) (2N+1) / 6, and sum of first N ^3 is [ N (N+1) / 2]^2, which is basically squaring the Sum of First N Numbers. You all know this.
The Sum of First N Numbers can be thought as an Area of a Triangle dotted with a series of dots approximating the triangle, starting with N dots, N-1, and so on. Since we need to go 0 Dots to make a perfect Triangle, we get the factor of N+1 and factor of 1/2 from the area of Triangle - as it is half of Rectangle. This can also be thought as number of Neighbors one has among N Neighbors.
The second formula can be thought as Volume of a "Double" Pyramid extended by the above dotted triangle to give itself a symmeterical 3-D figure, givng 2N + 1 Layers of N-Dotted Triangles, where they vary from 0 to N. The height has to vary +N and -N on "each side" with 0 as common base, hence the term 2N+1. Now the term 1/3 comes from the Volume of Pyramid Formula.
The story does not stop here. The Quantum Numbers come from background of 3D Harmonics, which are nothing but similar to 1D Fourier Series, which says any wave can be carved out by ad mixing a series of Sin and Cos Waves of Harmonics, e.g., multiples of a common frequency 'f', with different proportions of Amplitude. 3D Harmonics is also similar, which says how to take a Sphere, and carve out different blocks of 3D, 2D and 1D patterns which would add up to sum a 3D Sphere! Think of stacking different layers of thin shells of layers to make an Sphere, e.g, like an Onion. Or take different slices of Oranges cut vertically or horizontally, and stacked up to make a Sphere. Or think of drawing different patterns of quadrants on the surface of the Sphere. Now make this into different combinations, and essentially this is the 3D Harmonic Series equivalent.
Now lets come to the Quantum Numbers. The first quantum number is "n", which show the shell number, and varies from 1 to N, giving (N) States. This can be thought as the layer or the story at which an electron resides or orbits. There was nothing magic about Quantum Jumps, as stated by Planck, as electrons are not dumb to not take a "lane approach to driving" vs driving at any analog levels, Hence they take quantum jumps, otherwise they would collide and interfere with each while "driving", and this also fits with the 3D-Harmonics. It is the radial 1D Expansion of enumeration of first N "points" in Space. Plus there is a deeper mystery on the nature of quantum of jumps which tied with the De Broglies hypothesis of the orbits being integral number of electron wave lenghts. But what is missed is that they are also tied with the wave lenght of the photon absorbed or emitted, yet, admitted in the Energy formula of E = hv. No one talks about the physical process explaning how the photon injection or emmision ties to Electron changing its orbit.
The second quantum number is "l", which varies from 0 to N, giving (N+1) States. This is the Angular Momentum Quantum number, which gives a "circular" sweep to the N points "radius" to make it a circle.
The third quantum number is 'm" and varies from -N to +N, giving (2N+1) States. This is the Magnetic Quantum Number, and it basically takes the mechanical motion of the electron in circular orbit to give a 3D Spherical effect to account for North and South Poles of Magnetic Moment which accounts for spinning - like the Earth's motion
And the fourth quantum number is the Spin 1/2, which is of a Quantum Mechnical nature and there is no classical explaination given. But actually, in my quest, I have found this to be a double covered Sphere which is spun by Photon, giving two N-S pole combinations, where the electron is N-N or S-S artifact of the weaving of the Electron by Photon. The Positron has the opposite artifact because of the same weaving of positron by photon - but in opposite manner.
What does the Sum of N^3 tell about the 4rth Quantum Number? Basically it is the square of the Sum of N, which is like a Cross Product of 2 Dimensions divided by 2 or the Area of a Triangle of sides with N and N+1 Dimensions (the second sweep is 1 unit more). It tells about a 4D effect, where the Cross Vector is either "Up" or "Down", and the 4D Reality is the product of the two Cross Products, e.g. Cross of Cross and like the Triple Product of 3D, which is Volume. This gels with the notion of +h/2 Up and h/2 Down Vector. So Planck Constant is a notion of "minimal" 1D or 2D (as for next dimension it is 1 unit more) unit of some "space".
This is also validated if one takes the Relativity explaination given by the QM for accounting the g=2 for gyromagnetic ratio coupling, but not linking it with the spin of 1/2. The gyromagentic ratio is basically the coupling between Angular Momentum and the Magnetic Moment, and experimentally they found it to be double than what was predicted by Classical Physics. They at one point explained this by Relativistic Effects and Chirality. But my explaination is very Geometrical, and is also blessed by Geometric Algebra, Clifford Algebra, Lie's Algebra, Grassman's Algebra, and SO(3) Symmetries.
Now if you see all these factors, explaination of 3D Harmonics, and the Formulas for Sum N, N^2 and N^3 numbers, you would see there is a lot of similarities and geometrical parallels.
So where is the factor of 1/3 and it is missing in QM? It's pointing to 3 times play of fields, in 3 x 3 x 3 Dimensions.
In another posting, I would explain how the Fine Structure Constant Factor of 1/137.5 is related to different Constants and some famous constants of Science, including Fibonicci Numbers and Golden Ratio.
Oh, yes it is connected with Gold Ratio Angle of 137.5!
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