But, what about the frequency of the matter wave? de Broglie used the now familiar relation E=h nu to get
.
What is the phase velocity of de Broglie waves?
For particles moving with v<c (which is always true), then the phase
velocity is c (c/v) > c !!! How can this be? The de Broglie waves are moving
faster than the particle they refer to?! Wouldn't they "run away"? The point is
that the particle is not directly associated with the basic wave, but
rather with the modulation of the wave. The same as radio or TV signals, where
it is not the carrier wave that really transmits any information, but the
modulation does. Or yet another way to see it, the probability of finding a
particle is
.
The carrier waves can all have the same amplitude,
is flat (i.e. constant, it carries no information about where the particle
really is). But, the modulation of the wave gives rise to variations in
,
and tells you the spots where the particle is likely to be. It is the motion of
THAT we care about, since that is "where the particle is". The velocity of the
modulation is the group velocity,
.
But, we just found that
which means v_group = V. Excellent! This is as it should be!
You can sometimes see this phenomenon in water, where small fast moving wavelets cruise by, building up into a big crest at some point. You can see the wavelets move through/past the crest, and the crest moves along more slowly...
The bottom line is, there is (generally) a difference between the speed of waves of some well define frequency, and the speed of a localized pulse (such as de Broglie proposed to associate with particles). To build up a localized pulse, you must use a collection of waves of various different frequencies and wave numbers (remember Fourier transforms?) It is this collection we have just been discussing, and the pulse moves along with the group velocity.
Before we look at the experimental evidence that de Broglies waves are correct, let's review our formulae:
.
Non-rel:
.
Rel:
.
We will use the last one, combine it with de Broglie's relation, and get
.
That very same combination of constants appearing out front,
,
also appeared when we looked at Compton scattering,
(We called it the "Compton wavelength" when we used the rest mass of the electron) Rewriting:
Roughly speaking, if the first term under the square root "wins", the particle is non relativistic. If the second term wins, the particle is relativistic. F+T demonstrate this nicely by plotting (p. 64) the log of the left hand side versus the log of (K.E./mc^2). If the world was non-relativistic, then this would give a straight line with slope (-1/2) (because only the first term would appear under the square root)
If the world was purely relativistic, the line would have slope (-1). You can see the smooth transition from non-relativistic to relativistic in the plot...
Also, roughly speaking, if the de Broglie wavelength of a particle is small compared to the characteristic size of a system it is to interact with, then quantum mechanics is not likely to be very important. But if the de Broglie wavelength is large, then you really need to start worrying about quantum effects.
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