Page 35 - Introduction to Information Optics

20 1. Entropy Information and Optics

H(V)

0 V m

Fig. 1.4. An ideal low-pass channel.

that is, I/At ^ v m, then the output response will quite faithfully reproduce the
input signal. However, if the signal spectrum spreads beyond the passband of
the channel, the output response will be severely distorted, which fails to
reproduce the input signal. We note that this duration-bandwidth product
provides a profound relationship with the Heisenberg's uncertainty principle in
quantum mechanics [1.7], such as

Ax Ap ^ /z, (1.70)

where Ax and Ap are the position and momentum errors, respectively, and h
is Plank's constant. In other words, the uncertainty principle states that the
position variable x and its momentum variable p cannot be observed or
measured simultaneously with arbitrary accuracy.
In fact, the uncertainty relation can also be written in the form of energy
and time variables:

AEAf (1.71)
where A£ and At are the corresponding energy and time errors. Since £ = hv,
hence AE = /iAv, we see that

AEAf = (1.72)
Thus, we see that Av At ^ 1 is the Heisenberg uncertainty relation.