Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
We should not look upon the uncertainty principle as being merely an unfortunate
limitation on our ability to know nature with in®nite precision. We can use
it to our advantage. For example, when combining the time±energy uncertainty
relation with Einstein's mass±energy relation (E ˆ mc 2 ) we obtain the relation
mt h=c 2 . This result is very useful in our quest to understand the universe, in
particular, the origin of matter.
Wave packets and group velocity
Energy (that is, a signal or information) is transmitted by groups of waves, not a
single wave. Phase velocity may be greater than the speed of light c, `group
velocity' is always less than c. The wave groups with which energy is transmitted
from place to place are called wave packets. Let us ®rst consider a simple case
where we have two waves ' 1 and ' 2 : each has the same amplitude but di€ers
slightly in frequency and wavelength,
' 1 …x; t† ˆA cos…!t kx†;
' 2 …x; t† ˆA cos‰…! ‡ !†t …k ‡ k†xŠ;
where ! ! and k k. Each represents a pure sinusoidal wave extending to
in®nite along the x-axis. Together they give a resultant wave
' ˆ ' 1 ‡ ' 2
ˆ Afcos…!t kx†‡cos‰…! ‡ !†t …k ‡ k†xŠg:
Using the trigonometrical identity
cos A ‡ cos B ˆ 2 cos A ‡ B
2
cos A B ;
2
we can rewrite ' as
2!t 2kx ‡ !t kx !t ‡ kx
' ˆ 2 cos cos
2
2
ˆ 2 cos 1 …!t kx† cos…!t kx†:
2
This represents an oscillation of the original frequency !, but with a modulated
amplitude as shown in Fig. 4.18. A given segment of the wave system, such as AB,
can be regarded as a `wave packet' and moves with a velocity v g (not yet determined).
This segment contains a large number of oscillations of the primary wave
that moves with the velocity v. And the velocity v g with which the modulated
amplitude propagates is called the group velocity and can be determined by the
requirement that the phase of the modulated amplitude be constant. Thus
v g ˆ dx=dt ˆ !=k ! d!=dk:
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