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

size. This means i itself, being equal to i times 1, will lie at 90┬░ from 1, and all the
imaginary numbers will form a line perpendicular to the real number line. Complex
numbers can then be visualised as points whose x and y coordinates are equal to their real
and imaginary parts.
 Egan: "Foundations 4"/p.12




Figure 6 shows part of the complex plane, with the point representing a complex
number z=2+3i marked on the diagram. To introduce some convenient notation, the real
and imaginary parts of z, in this case 2 and 3i, are usually written Re z and Im z. The
distance of z from 0, which is тИЪ(22+32)=тИЪ(13), is known as the magnitude of z, and is
written |z|. The angle from the real line to z, in this case 56.3┬░, is known as the
argument of z, and is written arg z.
Why should we care about these angles and distances? It turns out that the
metaphor we used to construct the diagram, where we treated multiplication by тАУ1 or i as
a kind of rotation, works seamlessly for all complex numbers, so long as you also take
into account their magnitude. Multiplying any two complex numbers w and z produces a
result, wz, whose magnitude is |w||z|, and whose argument is arg z + arg w. In other
words, multiplying z by w тАЬstretchesтАЭ z by a factor of |w|, and rotates it by an angle of
arg w. For example, in Figure 6, the product of 2i and z has a magnitude of 2тИЪ(13) тАФ
which is |2i| times that of z тАФ and it is rotated arg 2i, or 90┬░, away from z itself.
The number z*, also marked on the diagram, is known as the complex
conjugate of z. It has the same real part as z, but its imaginary part is тАУIm z.
Similarly, it has the same magnitude as z, but its argument is тАУarg z. Because of this,
z*z must be a real number, since the sum of the two arguments comes to zero, and its
magnitude must be |z*||z|=|z|2. If you check, (2тАУ3i)(2+3i) = 4тАУ6i+6iтАУ9i2 = 13, or |z|2.
To describe a cyclic de Broglie wave that never changes size, we could use the
complex number whose argument is equal to the phase of the wave, 2╧А(px тАУ Et)/h:


╧И(x,t) = cos(2╧А(px тАУ Et)/h) + i sin(2╧А(px тАУ Et)/h) (7)
 Egan: "Foundations 4"/p.13


This wave always has a magnitude of 1, but it moves in a circle around the complex
plane, from 1 to i to тАУ1 to тАУi and back to 1 again. Such a wave can exhibit constructive
and destructive interference: if you split it into two beams, then recombine the beams
with their phase unchanged, you'll recover the original wave with a magnitude of 1;
however, if you cause one beam to be precisely half a cycle out of phase with the other,
the two waves will have opposite values when they meet (e.g. if one is i, the other will
be тАУi), and they'll add up to zero. Other phase differences will produce results in
between those two extremes.
There's a more concise way to write Equation (7), but it requires a brief
mathematical detour. Most readers will be familiar with the concept of exponential
growth: there are many systems, from populations of bacteria to bank deposits earning
compound interest, that grow at a rate proportional to their own size. In most real
situations the growth occurs in finite steps, but it's possible to imagine an idealised case
where growth is continuous. For example, a bank deposit earning 10% тАЬnominalтАЭ