Mathematical Methods for Physicists: A concise introduction - Site Map

WAVE PACKETS AND GROUP VELOCITY
If we further increase the absolute value of z, the function sin (z†=z runs alternately
through maxima and minima, the function values of which are small compared
with the principal maximum at z ˆ 0, and quickly converges to zero.
Therefore, we can conclude that superposition generates a wave packet whose
amplitude is non-zero only in a ®nite region, and is described by sin (z†=z (see Fig.
4.19).
The modulating factor sin (z†=z of the amplitude assumes the maximum value 1
as z ! 0. Recall that z ˆ k…x ! 0 t), thus for z ˆ 0, we have
x ! 0 t ˆ 0;
which means that the maximum of the amplitude is a plane propagating with
velocity
dx
dt ˆ ! 0 ˆ d!
dk ;
0
that is, ! 0 is the group velocity, the velocity of the whole wave packet.
The concept of a wave packet also plays an important role in quantum
mechanics. The idea of associating a wave-like property with the electron and
other material particles was ®rst proposed by Louis Victor de Broglie (1892±1987)
in 1925. His work was motivated by the mystery of the Bohr orbits. After
Rutherford's successful -particle scattering experiments, a planetary-type
nuclear atom, with electrons orbiting around the nucleus, was in favor with
most physicists. But, according to classical electromagnetic theory, a charge
undergoing continuous centripetal acceleration emits electromagnetic radiation
Figure 4.19.
A wave packet.
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