```Uncertainty Principle

These are some examples to illustrate the uncertainty principle.
I also include a proof of the uncertainty principle, although
we will be doing a more general proof when we cover the
formalism of quantum mechanics in chapter 3.

There is also an accompanying matlab file which can be used to generate
the plots below as well as others.  Although it is not easy to read
someone else's code, the code is well documented.  For those familiar
with Matlab, I would encourage that you look at it.  I strongly recommend
that all of you learn some Matlab.
uncertainty.m

The wave functions considered are

Gaussian	psi(x) = C exp(ikx) exp(-(x/width))^2)
"quartic"	psi(x) = C exp(ikx) exp(-(x/width))^4)

where C in each case is a constant that gives normalization to 1.
To place this on a computer I have set hbar equal to 1 so the
uncertainty principle is sigmax sigmap greater than or equal to 1/2 .
The Gaussian wave function above is the minimum uncertainty wave
packet, where sigmax sigmap = 1/2 .

Results:
function   width  k   sigmax   sigmap   (sigmax sigmap)
----------------------------------------------------------------
Gaussian     1    6   0.5      1	0.5	       graph
Gaussian     1    6   0.125    4        0.5            graph
"quartic"  0.25   6   0.489    1.195    0.584          graph
"quartic"  0.25   6   0.122    4.77     0.583          graph
----------------------------------------------------------------
The computer code does not give exactly 0.5 for (sigmax sigmap), but something
like 0.499 .  This difference is due to discretizing the integrals for
the expectation values.

```