"Greg Egan - Foundations 4 - Quantum Mechanics" - читать интересную книгу автора (Egan Greg)

Broglie wave by adding together several waves, all of the form given by Equations (4),
but with a range of different frequencies. In the region where all these waves are more or
less in phase with each other, they'll produce a kind of mound, or wave packet.




Figure 5 shows the result of adding waves of both higher and lower frequencies
to the original wave of Figure 4. The overall height of the wave packet, ignoring the
individual dips and rises and just looking at an тАЬenvelopeтАЭ stretching from peak to peak,
is greatest at the point where all the waves are perfectly in phase with each other тАФ but
it's clear that this point doesn't move at the same speed as the individual peaks. So how
fast does it move?
Start with a simple fact that we established in the previous article: the length of a
particle's 4-momentum vector P is just its rest mass, m, and hence m2=тАУg(P,P)=E2тАУp2.
Two waves with slightly different energies and momenta тАФ say E1 and E2, p1 and p2 тАФ
that happen to be in phase will only stay in phase where (p1x тАУ E1t) remains equal to
(p2x тАУ E2t), since apart from factors of 2╧А/h, these are the respective phases of the two
waves. So as time t increases, x must increase at a rate of (E2 тАУ E1)/(p2 тАУ p1) to keep
the phases equal. Now, since E2тАУp2=m2 for both waves, we have:
 Egan: "Foundations 4"/p.10


(E 2 ) 2 тАУ (p 2 ) 2 (E 1 ) 2 тАУ (p 1 ) 2
=
(E 2 )2 тАУ (E 1 )2 (p 2 ) 2 тАУ (p 1 ) 2
=
(E2тАУE1)(E2+E1) = (p2тАУp1)(p2+p1)
(E2тАУE1)/(p2тАУp1) = (p2+p1)/(E2+E1)


The right hand side of the last line here is just the value of p/E for an тАЬaverageтАЭ wave.
Using the formulas we derived in the previous article, p=mv/тИЪ(1тАУv2) and E=m/тИЪ(1тАУv2),
p/E is simply equal to the particle's velocity v. So the velocity of a wave packet, the
group velocity for de Broglie's matter waves, matches the particle's velocity.
In most experiments, particles like electrons can be localised to some degree:
even when you can't pin them down to the nearest nanometre, you know that they're
inside your apparatus and not on the other side of the planet. This suggests that they
should generally be described by wave packets, which involve a localised тАЬbumpтАЭ in ╧И,
rather than a sine wave that goes on forever. But we've seen that the process of creating
that bump means adding together waves that have a range of different momenta. To
localise the particle, to give it anything like a definite position, we've had to give up the
idea that it has a single, precise momentum.
This is just one manifestation of a famous aspect of quantum mechanics known as
the uncertainty principle. It's a simple mathematical fact about wave packets that the
more sharply defined they are, the greater the range of wavelengths needed to build them
тАФ and that's just as true for sound waves and water waves as it is for waves in quantum
mechanics. Since wavelength equates to momentum for a de Broglie matter wave, the
more sharply defined a particle's position, the less well-defined its momentum will be.
The particle is localised where there's a bump in ╧И, but what about all the peaks