X4 theory of elastic space
X4 Theory of Elastic Space
When talking about Hydrogen atom in Quantum, we get the concept of electronic cloud (the probability distribution of electron) in some lectures and see a strange phenomenon: The shape of the electronic cloud (the probability distribution of electron) is symmetric about the Z axis, and so, when the Z axis changes its orientation, the shape of the electronic cloud (the probability distribution of electron) will follow suit.
Think about it: the shape of the electronic cloud (the probability distribution of electron) should be the own characteristic of the Hydrogen atom, while the Z axis (the coordinate system) is artificial. If the electronic cloud is the true natural characteristic of the Hydrogen atom, its shape should not follow suit the change of the artificial coordinate system. That is to say the wave functionψ（r, θ，φ）= R(r)Y(θ，φ) might not be the natural solution but the artificial solution. The concept of electronic cloud might not be natural but artificial.
What’s the mistake?
We analyze it carefully and find that people might use 3D dead mathematics of our world to represent 4D alive real cosmos, just as using plane geometry to explain solid，and get its appearance in our world. The ordinary spherical or rectangular coordinate system is established in our world, and is dead (rigid). The coordinate r, θ，φ or X,Y, Z is dead (rigid). And they represent dead (rigid) 3D space in our world.
Even if people find a way to solve the Schrodinger equation in rectangular coordinate system, the probability distribution of electron which is represented with the wave functionψ（X, Y，Z）will also follow suit the coordinate system, because it’s artificial solution.
Turn to X4 Theory, the homogenous coordinate of a space point is denoted with (X1, X2, X3, X4), where X1, X2, X3 are the 3D space elements and X4 is the state (alive) element. It might represent the alive real 4D space. If we want to understand it, we must go into its world. This is philosophy.
Next, we only talk about it in one dimension for simplicity.
Such an alive one dimension space could be denoted with the equation:
X1/ X4 = X or X1 = X X4
It tell us that a particle which is in the position X in the space of our world is in the position X1 = X X4 in the space of its own world. The position X1 might vary if the particle changes its X4 state. So, in fact, the space of the particle’s own world might be alive (elastic).
We know that the Schrodinger equation was induced in the dead (rigid) space of our world. And it is:
Eψ= [( ¯h²/2μ)(d²/dX²) + V(X)] ψ
Next, let’s try to transform it into the particle’s own world and see what the situation will be.
dψ/ dX = (dψ/ dX1)( dX1/ dX) = (dψ/ dX1) [X4+X( dX4/ dX) ]
= X4(dψ/ dX1) + X(dψ/ dX1) ( dX4/ dX)
d²ψ/ dX²=( dX4/ dX) (dψ/ dX1)+ X4(d²ψ/ dX1²)( dX1/ dX)+ (dψ/ dX1) ( dX4/ dX)
+X( dX4/ dX1) (d²ψ/ dX1²)( dX1/ dX)+X(dψ/ dX1) ( d²X4/ dX²)
=( dX4/ dX) (dψ/ dX1) + X4(d²ψ/ dX1²)[X4+X( dX4/ dX) ] + (dψ/ dX1) ( dX4/ dX)
+X( dX4/ dX) (d²ψ/ dX1²)[X4+X( dX4/ dX) ] +X(dψ/ dX1) ( d²X4/ dX²)
=( dX4/ dX) (dψ/ dX1) + X4²(d²ψ/ dX1²)+X X4(d²ψ/ dX1²)( dX4/ dX)
+ (dψ/ dX1) ( dX4/ dX)+ X X4(d²ψ/ dX1²)( dX4/ dX)
+X²( dX4/ dX)² (d²ψ/ dX1²)+X (dψ/ dX1) ( d²X4/ dX²)
=[ X4²+2 X1( dX4/ dX) +X²( dX4/ dX)²] (d²ψ/ dX1²)
+[2( dX4/ dX)+X( d²X4/ dX²)](dψ/ dX1)
Then: Eψ=﹛( ¯h²/2μ) [ (X4²+2 X1( dX4/ dX) +X²( dX4/ dX)²)(d²/ dX1²)
+(2( dX4/ dX)+X( d²X4/ dX²))(d/ dX1) ]+V (X1/ X4)﹜ψ
It’s the God Schrodinger equation. Let’s analyze it:
1. In case of free particle or V does not affect the X4 state of the particle (space point), X4 value is an invariant. The space of the particle’s own world is not elastic.
dX4/ dX≡0, d²X4/ dX²≡0, then:
Eψ= [( ¯h²/2μ) X4²(d²/dX²) + V(X1/ X4)] ψ
When X4 =1, X1 =X, then:
Eψ= [( ¯h²/2μ)(d²/dX²) + V(X)] ψ
It’s the Schrodinger equation we used to be familiar with, in the dead (rigid) space of our world. The solutionψ(X) means the probability distribution in the particle’s own world is the same as that in our world.
2. In case of V affects the X4 state of the particle (space point, and we guess the central Coulomb field is such a case), the traditional Schrodinger equation is no longer applicable to describe the exact situation. It’s the GS equation does.
We don’t know whether the complicated GS equation can has a solution by means of math of mankind. And we also have to determine the gradient of the X4 state field, dX4/ dX, of the electron of the Hydrogen atom elsewhere first.
I would rather to try to guess its meaning with philosophy.
If the GS equation has a solutionψ(X, X1,X4), it should be the natural solution ,and should have the same characteristic in any direction. So the analysis in one dimension is enough to see the characteristic. It might be a series of wave functions. It even might be a concept of 4D field：wave function distribution field. We know that the shape of field often does not follow suit the change of artificial coordinate system and only have something to do with the own structure of the matter concerned.
Chen Li Qiang
May, 2018
