A misinterpretation of the uncertainty principle

It is easy to misunderstand the uncertainty principle. An oversimplification of it is to say that it is ``impossible to know the frequency at any particular time.'' This oversimplification leads us to think about a truncated sinusoid, such as in Figure 1. We know the frequency exactly, so $\Lambda F$ seems zero, whereas $\Lambda T$ is finite, and this seems to violate (1). But what the figure shows is that the truncation of the sinusoid has broadened the frequency band. More particularly, the impulse function in the frequency domain has been convolved by the sinc function that is the Fourier transform of the truncating rectangle function.

 

windcos
Figure 1 Windowed sinusoid and its Fourier transform. 1