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

╧И(x) = sin(2╧А g(x,k)) (5a)
= sin(2╧А g(x,P)/h) (5b)


where P is the particle's 4-momentum, pтИВx + EтИВt. Observers with different velocities
must agree on the value of g for two spacetime vectors, so they'll find nothing to argue
about in Equations (5), despite measuring different individual x and t coordinates for all
the vectors involved. And having defined the propagation vector like this, the
relationship between wave and particle can be summed up in a single equation, a
тАЬspacetime versionтАЭ of Planck's Equation (1):


P = hk (6)


The propagation vector, k, is perpendicular in the spacetime sense to the peaks and
troughs of the waves, the wavefronts for which the phase remains constant. For light
waves, since the 4-momentum P and the propagation vector k are null vectors,
тАЬperpendicular to themselvesтАЭ in the sense that g(k,k)=g(P,P)=0, they're actually both
parallel and perpendicular to the wavefronts. Null vectors are like that.
For matter waves, since P and k are timelike vectors, the wavefronts
perpendicular to them must be spacelike тАФ which means the peaks and troughs of these
waves will seem to тАЬtravelтАЭ faster than light. If the 4-momentum P is that of a particle
with a speed of v, the phase of the wave will have an apparent тАЬspeedтАЭ of 1/v. For
example, a particle moving at 50% of lightspeed will be described by a wave with peaks
that тАЬmoveтАЭ at twice the speed of light.
At first glance this might seem like either a disastrous mistake in the theory, or an
opportunity for sending signals faster than light, but in fact it's neither. Long before
quantum mechanics, the study of waves revealed a crucial distinction between the phase
velocity, which describes how the peaks and troughs of a wave seem to move, and the
group velocity, which describes how disturbances in air, water, and other media
actually propagate from one place to another. For light in a vacuum these two velocities
are identical, but that situation is really quite rare.
How can the peak of a wave merely тАЬseem to moveтАЭ? Imagine setting up a long
row of suspended weights bouncing on the ends of springs, all of them bouncing with
exactly the same frequency, but with each weight reaching its highest point a fraction of a
second later than its neighbour on the left. These weights will form a travelling sine
wave just like the one in Figure 4, and in principle there's nothing to stop you arranging
the time lags so that the peaks тАЬtravelтАЭ as fast as you like from left to right, even faster
 Egan: "Foundations 4"/p.9


than light. But nothing whatsoever is passing from one spring to the next as this
happens. Of course, real waves do spread by transmitting their тАЬbounceтАЭ from place to
place, but the speed at which that happens need not be the same as the apparent тАЬspeedтАЭ
of their peaks and troughs, which simply measures the fact that different parts of the
wave are out of synch.
To make the idea of group velocity more concrete, let's construct a new de