Dear Basudeba,
You wrote: "We want clarity of our thought by asking questions that we feel important."
Good, from your response it is evident to me that unfortunately you misunderstand the meaning of the spatial derivative and the Schroedinger equation. I will try to explain, and hope that you will receive my explanations in the spirit of the words you wrote.
First, on the spatial derivative: It refers to a rate of change of some quantity (in the denominator) as you change position along the specified coordinate. Let us suppose we want to know how a field, symbolized by, say, F, varies as we consider positions along the x-axis. We would then write
[math]\frac{dF}{dx}[/math]
to denote the rate of change of F with respect to a change of position along the x-axis.
If dF/dx=0 then that means the F is constant along the x-axis. Note that the constant need not be 0, because all the spatial derivative tells us is how F changes as x changes. It does not tell us anything about the value of F itself. if F is greater than zero, then F has the same constant value everywhere along the interval of x that you are considering, and if F is smaller than zero then it has the same negative constant value everywhere along that interveal. if F=0, then it is zero everywhere along the interval of x that you are considering.
If dF/dx is smaller than 0, then the Field Strength decreases as you go farther along x (i.e. it becomes weaker). Again, this tells us nothing about the value of F itself, at any point it could be positive, negative or zero but in the direction of positive x it will be less than at that point, and in the direction of negative x it will be greater.
If dF/dx is larger than zero, then the Field strength increases as you go farther along x, this is just the opposite of the previous case, again with no implication for the value of F itself.
The quantity
[math]\frac{d^2F}{dx^2}[/math]
tells us the "rate of change of the rate of change" of F as we go along x. Notice that we are still considering the rate of change along one direction, not two, as you seem to think.
If d^2F/dx^2=0, then this means that the rate of change of F (i.e. dF/dx) is constant. Note that the constant need not be zero. If the constant dF/dx is greater than zero, then this means that the rate at which F changes as x changes is positive, so F becomes larger at a constant rate as x increases. If the constant dF/dx is negative, it means that F becomes smaller at a constant rate as x increases. If it the constant dF/dx is zero, it means that that F remains the same constant value as you go along x, but again, this does not mean that F itself is zero (see above).
If d^2F/dx^2 is smaller than zero, then this means that the rate of change of the rate of change is decreasing as you along x. Note that it dF/dx can be either positive, negative, or zero. If dF/dx is positive, then d^F/dx^2 smaller than zero means that the rate of change of F in the positive direction becomes smaller as you go along x. For every constant interval of x, the amount by which F becomes larger (more positive) decreases. if dF/dx is negative, then this means that the rate of change of F in the negative direction becomes larger as you go along x. For every constant interval of x, the amount by which F becomes more negative increases. If dF/dx=0 it means that F has reached a local maximum, it has gone as positive as it could, and now it is going to decrease in value further along x. Again, all of this has no bearing on the value of F itself, in any of these cases it can be positive, negative or zero (for example you can have local maximum that is negative if all the surrounding values for F are more negative).
I will let you figure out what happens if d^2F/dx^2 is larger than zero, you have enough information to be able to do it. Please do it, for only then will you know whether you really have understood these concepts.
Once you have done this exercise, can you guess what a third order derivative (d^3F/dx^3) means? Hopefully you understand now that it does not mean what you wrote above.
All this is covered in any introductory calculus course or text. If you had calculus before, then I would suggest that you review it, and if not, then I strongly advise you to learn it before you make further attempts to clarify your thoughts about how the world works. I really think that without a minimum knowledge of calculus, it is not really possible to do understand how the world works.
Now, on to the Schroedinger equation. Let me just mention upfront that you have written the time-independent Schroedinger equation. This is fine, but keep in mind that it only applies to states of constant Energy. If you want to describe the time evolution of any arbitrary quantum state, then you must use the time-dependent Schroedinger equation. It is the latter, incidentally, which has the imaginary numer i in it.
The equation you wrote is the one dimensional Schroedinger equation which is used most often as a toy example when the dimensionality of the system does not matter for a particular problem. The three dimensional Schroedinger equation uses the three dimensional generalization of the spatial derivative, which is called the gradient. It is very straightforward to generalize the concepts mentioned above to three dimensions: You set up a coordinate system with three perpendicular directions, say i,j, k, and then do the same thing as above for each direction and sum.
If what you said
"A three dimensional equation is a third order equation implying volume. Addition of three areas does not generate volume [x+y+z ≠ (x.y.z)] and [x2+y2+z2 ≠ (x.y.z)]. Thus, there is no wonder that it has failed to explain spectra other than hydrogen. "
were correct, then the Schroedinger equation would not even give the solution to the Hydrogen atom. The reason that it can't give an analytical solution for larger atoms (numerically, one can find solutions to arbitrary precision) is something much more subtle. But let me stop here and ask you to sincerely reflect on the following:
You made some arguments based on elementary mathematical mistakes that any math/physics freshman could instantly recognize. Should that not give you pause to consider whether your other arguments might not be based on similar misunderstandings? I am not necessarily saying that all of your arguments are wrong, but I think that the above illustrates that it behooves you to check the basis of your arguments (especially the mathematics upon which it is based) before you use strong words like "proof" and "hoax" to make highly controversial claims. It is really in your best interest, because I doubt that many physicists will take the time to interact with you otherwise.
Finally, let me mention that on one point I stand corrected: I said that "extra dimensions came later" (i.e. after the development of QM in the mid 1920's" and you wrote " Extra dimensions came before QM". The earliest use of extra dimensions in physics (as opposed to science fiction) that I know of is when Theodor Kaluza wrote to Einstein that he had developed a 5-dimensional theory that unifies EM with GR. I checked, that was in 1919, so you were correct.
All the best,
Armin
