##

The Heisenberg Uncertainty Principle

The wave packets we tried above satisfy an
**uncertainty principle which is a property of waves**.
That is
.

For the ``square'' packet the full width in
is
.
The width in
is a little hard to define, but, lets use the first node in the probability
found at
or
.
So the width is twice this or
.
This gives us

which certainly satisfies the limit above.
Note that if we change the width of
, the width of
changes to keep the uncertainty product constant.
For the Gaussian wave packet, we can rigorously read the
**RMS width of the probability distribution**
as was done at the end of the section on the
Fourier Transform of a Gaussian.

We can again see that as we vary the width in k-space, the width in x-space varies to keep the product constant.

The **Gaussian wave packet gives the minimum uncertainty**.
We will prove this later.
If we translate into momentum
, then

So the **Heisenberg Uncertainty Principle** states.
It says we cannot know the position of a particle and its momentum at the same time
and tells us the limit of how well we can know them.
If we try to localize a particle to a very small region of space, its momentum becomes uncertain.
If we try to make a particle with a definite momentum, its probability distribution spreads out over space.

Jim Branson
2013-04-22