.(continued)
Now consider adding two waves with nearly the same k and frequency, but not quite. Use the mathematical identity
.
Proof: use
Which gives
What's the picture? We're adding 2 nearly identical waves, and get back just
what we started with, times
(which is a "modulation" of the amplitude)
The carrier wave is
,
which moves at velocity V =
(also called the phase velocity above). But the modulation is itself also waving, and it moves with a velocity (called the "group velocity") given by
For light in a vacuum, c is a constant,
,
there's no difference! Phase velocity and group velocity is the same! But matter waves will be different:
.
This means that the velocity of a matter wave is not constant, it varies with the wavelength (this is quite different from light!), and the phase and group velocities will differ.
(Quiz today...)
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