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

Throughout these articles we've been using units where c=1, but it's worth leaving the c
in here for a moment, and stating the fact that the momentum, p, of a photon with energy
E is always p=E/c. (This must be true in order for the 4-momentum of the photon to be a
null vector, a spacetime vector with an overall length of zero, as discussed in the
previous article. The relationship is obvious when c=1, but it holds regardless of the
units used.) So the wavelength of light is related to each photon's momentum by:

L = h/p (3)


Equations (2) and (3) are the formulas de Broglie proposed for the period and wavelength
 Egan: "Foundations 4"/p.7


of matter waves. Let's see what such a wave might look like on a spacetime diagram.




Figure 4 shows a travelling sine wave with period T and wavelength L. We don't
actually know that a matter wave will ever take the form of a sine wave, but we might as
well start with a simple possibility like this and see where it leads us. The third axis on
the diagram represents the тАЬstrengthтАЭ of the wave, or amplitude, traditionally labelled ╧И
(the Greek letter psi). Exactly what ╧И means, physically, is something we've yet to
determine. The equation for ╧И in terms of x and t, the wave function, is:


╧И(x,t) = sin(2╧А(x/L тАУ t/T)) (4a)
= sin(2╧А(px тАУ Et)/h) (4b)


It's not hard to see that the wave defined by Equation (4a) will go through a complete
cycle whenever x increases by one wavelength, L, or time increases by one period, T.
The expression 2╧А(x/L тАУ t/T) is known as the phase of the wave: each individual peak
(or trough) in Figure 4 has a certain constant phase, and successive peaks (or troughs)
have a phase of 2╧А more than the last one. The minus sign here, rather than a plus sign,
guarantees that a peak of the wave will move in the positive x direction: to keep
2╧А(x/L тАУ t/T) constant, x must increase as t increases.
If we define the propagation vector for the wave, k, as:


k = (1/L)тИВ x + (1/T)тИВ t


and we write x = xтИВx + tтИВt for the spacetime vector that points from the origin to any
 Egan: "Foundations 4"/p.8


event in flat spacetime, then using the Minkowskian metric, g, we can rewrite Equations
(4) as: