Gabor's proof of the uncertainty principle

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, $\Lambda T$ is defined by  

$$(\Lambda T)^2 \eq { \int \ t^2 \, b(t)^2\, dt \over \int b(t)^2\, dt }$$ (2)

The spectral bandwidth $\Lambda F$ 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 $\Lambda T$ defined with (2) is infinity, which is unlike the more human measure of width, the distance to the first axis crossing.