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Although it is easy to verify the uncertainty principle in many special cases,
it is not easy to deduce it.
The difficulty begins from finding a definition of the width
of a function that leads to a tractable analysis.
One possible definition uses a second moment;
that is, is defined by

| |
(2) |

The spectral bandwidth is defined likewise.
With these definitions,
Dennis **Gabor** prepared a widely reproduced proof.
I will omit his proof here;
it is not an easy proof;
it is widely available;
and the definition
(2)
seems inappropriate
for a function we often use,
the **sinc** function, i.e., the FT of a step function.
Since the sinc function drops off as *t*^{-1},
its width defined with
(2)
is infinity,
which is unlike the more human measure of width,
the distance to the first axis crossing.

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Stanford Exploration Project

10/21/1998