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DoctorDick:
I think I understand your confusion; it appears that you misunderstand the meaning of the indices. The data consists of a set of numbers. The index refers to which number in the set we are talking about. Thus the statement that the sum is over i not equal to j says that the pair of numbers taken from that data set to be used in the Dirac delta function shall never be the same piece of data. It does not mean that two different members of the data set can not have the same numerical value. That is to say it is certainly possible that xi=xj without requiring i=j.
Yanniru:
You specifically excluded that possibility by using unknowable data the make every number unique. You cannot have it both ways. Maybe it is unnecessary to use unknowable data that way???
DoctorDick:
The xi values are just plotted on the x axis. My data are displayed as a bunch of points on the x axis!
Yanniru:
Nonetheless xi are vales not spatial coordinates
DoctorDick:
a constant may be added to all my data without removing the possibility of modeling the data. This is, in essence, identical to the standard proposition that the origin of the coordinate system used to represent the data cannot have an influence on the answer to the problem
Yanniru:
I do not believe that adding a constant to all the data is identical to shifting the origin of a coordinate system. That happens if the data is exponential, but not always.
DoctorDick:
The data and the points in the mathematical space are exactly the same thing; they don’t go back and forth in interpretation at all.
Yanniru:
Again I have to disagree. The values of the data do not equal the values of the coordinates. What you have, and have never explicitly specified, is values of data that vary from spatial point to spatial point, which is normally expressed as
xi(x).
I believe that the delta function can be dispensed with and still derive the same algorithm. However, I am having a problem with the combination of data values xi and their respective positions in space x. If I use such a large summation of data points that they may be approximated as continuous in space, then the derivation goes through. But that is equivalent to having a data point at every almost infinitisimal point in space. An alternative of using descrete space and math also seems undesirable as we then would require a value of xi at every descrete point in space. Perhaps the nearly infinite sum is not so bad as that is just what quantum measurements effectively do.
However, note that with an infinite sum of data points, the spatial dependence of psi is given by the data directly, ie. psi->f(xn(x)), as is expected from quantum theory. I here have to admit that I have not written down the derivation. It's all in my head right now, and as you well know, subject to error.
Regards,
Richard
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