Page 50 - Electrical Properties of Materials

Two analogies 33

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and velocity, but to a number of other related pairs of physical quantities. It ∗ You might find it interesting to learn
may also help to explain qualitatively some complicated phenomena. We may, that electric and magnetic intensities are
also subject to this law. They cannot
for example, ask the question why there is such a thing as a hydrogen atom con-
be simultaneously measured to arbitrary
sisting of a negatively charged electron and a positively charged proton. Why accuracy.
doesn’t the electron eventually fall into the proton? Armed with our knowledge
of the uncertainty relationship, we can now say that this event is energetically
unfavourable. If the electron is too near to the proton then the uncertainty in
its velocity is high; so it may have quite a high velocity, which means high
kinetic energy. Thus the electron’s search for low potential energy (by moving
near to the proton) is frustrated by the uncertainty principle, which assigns a
large kinetic energy to it. The electron must compromise and stay at a certain
distance from the proton (see Exercise 4.4).

2.5 Two analogies

The uncertainty relationship is characteristic of quantum physics. We would
search in vain for anything similar in classical physics. The derivation is, how-
ever, based on certain mathematical formulae that also appear in some other
problems. Thus, even if the phenomena are entirely different, the common
mathematical formulation permits us to draw analogies.
Analogies may or may not be helpful. It depends to a certain extent on the
person’s imagination or lack of imagination and, of course, on familiarity or
lack of familiarity with the analogue.
We believe in the use of analogies. We think they can help, both in mem-
orizing a certain train of thought, and in arriving at new conclusions and new
combinations. Even such a high-powered mathematician as Archimedes resor-
ted to mechanical analogies when he wanted to convince himself of the truth
of certain mathematical theorems. So this is quite a respectable method, and as
we happen to know two closely related analogies, we shall describe them.
Notice first of all that u(z) and a(k) are related to each other by a Fourier
integral in eqn (2.14). In deriving eqn (2.20), we made the sweeping assump-
tion that a(k) was constant within a certain interval, but this is not necessary.
We would get the same sort of final formula, with slightly different numer-
ical constants, for any reasonable a(k). The uncertainty relationship, as derived
from the wave concept, is a consequence of the Fourier transform connection
between a(k) and u(z). Thus, whenever two functions are related in the same
way, they can readily serve as analogues.
Do such functions appear in engineering practice? They do. The time
variation of a signal and its frequency spectrum are connected by Fourier
transform. A pulse of length τ has a spectrum (Fig. 2.6) exactly like the en-
velope we encountered before. The width of the frequency spectrum, referred
to as bandwidth in common language, is related to the length of the pulse.
All communication engineers know that the shorter the pulse the larger is the
bandwidth to be transmitted. For television, for example, we need to trans-
mit lots of pulses (the light intensity for some several hundred thousand spots
twenty-five times per second), so the pulses must be short and the bandwidth
large. This is why television works at much higher frequencies than radio
broadcasting.