"HEISEMBERG UNCERTAINTY PRINCIPLE"
PROBLEM: Obtain "delta x" (uncertainty) for the wave function "psi"
The wave function is symmetric and it corresponds to a particle in a quantum well
It is normalized
Remember that "delta x" is the standard deviation
We write the standard deviation formula
Firstly, we write ""
We calculate the expectation value for "X"
We rearrange the equation
We use a common factor
We expand the braket
We simplify
We write again the common factor
We have a polynomial integral
Easy to perform
We substitute for 0 and "a"
We simplify
we write again everything
We write the common factor
We rearrange the solution
Algebra is important
And the solution is shown
"a/2" has been obtained
It is logic
The wave function is symmetric, then "=a/2"
The expectation value is the center of the quantum well
Now we evaluate the second part of the formula
We write the integral
We expand the braket
We integrate
It is easy
We rearrange
We write a common factor
We substitute by "a" and 0
The solution is shown
It is a detailed calculation
Now we write the complete formula
Now we obtain the standard deviation
It s a simple result
The Heisemberg uncertainty for "X" operator is shown
It is an statistical uncertainty
If a measure "X" in the quantum well, I know the expectation value
And I also know the statistical uncertainty
This is the Heisemberg uncertainty in a measurement process
The wave function IS NOT in a stationary state for the "X" operator. So we have a finite "delta X"
