"QUANTUM UNCERTAINTY"
The famous Heisemberg Principle cames from this subject
In the Heisember uncertainty principle, "delta" is the standard deviation from Statistics
We expand a wave function in terms of an eigenfunction basis
We can evaluate the expectation value for an operator by means of an integral
Now I substitute
All is integrated
The function to integrate is just a summation
We can rearrange
On the left I have the complex conjugated terms
The operator can acts on the right side and then eigenvalues times eigenfunctions can be obtained
However, the eigenvectors form a basis
As a result, cross-terms will be null
Now we rearrange
The expectation value for the operator can be expanded
...in terms of the expansion coefficient
As the same manner, we can evaluate ""
I can write again the integral
I expand again in terms of the eigenfunctions
The "A" operator acts two times on the right side
A similar expression has been obtained
From Statistics we have the standard deviation a well-defined quantity
If we substract, we can obtain the standard deviation
The standard deviation tell us how far away are the available data from its averaged value
If we obtain always the same value (repeated again and again) then "delta A"=0
Quantum Mechanics is a probabilistic (statistical) theory. Remember this!
The Heisemberg Uncertainty principle is an statistical effect, from an statistical theory
