Richard,
I believe you are confused!
***** Richard:
I also expressed concern about the use of the exponential that seemingly results from the relativity fo that origin of the data values. What I did not understand is how the data could be exponential and the fundamental equation not be a wave equation, for exponentials are solutions of wave equations and not the Schroedinger equation unless the potential term V(x) is identically zero.
*****
You need to let me know which particular exponentials you are referring to here!
***** Richard:
Well in my mind I believe this issue has been resolved. The key step in your mathematical derivation is from eq(2.4) to eq(2.5). In eq(2.4) the summantion over the beta terms equals zero because the sum is over indices where the delta function equals zero, and the delta function is a multiplier in thise terms.
*****
The step from eq(2.4) to eq(2.5) does not at all force the beta terms to zero as there is nothing to guarantee that the wave functions do not overlap (as symmetric wave functions are allowed for the "unknowable" data).
***** Richard:
Thus the function called f in equation 2.5 must also be equal to zero.
*****
That statement is simply false!
***** Richard:
It follows that the function V(x) in eq(2.17) is also zero.
*****
Another false statement!
***** Richard:
So your have just derived the wave equation.
*****
I do not know what you are talking about!
***** Richard:
That is expected since you cast the data in the form of exponentials.
*****
Exactly where did I cast my data in the form of exponentials?
***** Richard:
In my mind that is a snooker.
*****
In my mind, you are totally confused as to what I am doing! I am sorry Richard, but I don't think you are following the mathematics.
***** Richard:
I wonder if you snookered yourself as well as the rest of us. But the snooker does not stop there. In deriving the fundamental equation, you take the derivative of the values of the data points called xi and xj. In no place do you define what space is.
*****
I do not anywhere take the derivative of "data points". That is a meaningless concept! I take a derivative of the probability function with respect to the data (as the probability is a function of the data!)
***** Richard:
You just define the values of data. Now we could assume that you meant that the values were defined at points in space given by the indices i and j.
*****
Clearly you have no understanding of what I am doing at all. The indices, i and j, refer to the data of interest and have noting to do with "space". The data is plotted on the x axis for visual convenience only!
***** Richard:
But you go on to claim that the values can be taken in any order. Now obviously you cannot take values of space coordinates in any order.
*****
There are only two coordinates of interest here x and tau. The data is being plotted in that plane!
***** Richard:
One cannot define spatial gradients in such cases.
*****
It is so clear that you do not understand the mathematics of my presentation that I will stop commenting here!
***** Richard:
However, I will grant you that most of your hueristic thinking was in the right direction. I base my opinion on the recent work of Michael Hall with Marcel Reginatto published in the Journal of Physics A and discussed in the 27 April 2002 issue of New Scientist. They took probability theory that contained both known data and unknown data and derived both the Schroedingers equation and the Heisenberg Uncertainty Principle. As far as I can tell they did not make any mathematical errors or confuse any variables.
*****
Well, I have not read their work. If they have comprehended the idea of including an infinite quantity of unknowable data and have deduced both the Schrodinger's equation and the Heisenberg Uncertainty Principle from that idea, I applaud them. It is nice to see someone getting somewhere after 40 years.
Have fun -- Dick |