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

and troughs in Figure 5, on which the bump is superimposed? Interference experiments
with electrons can produce results exactly like those with light shown in Figure 1, so the
variation in phase suggested by these peaks and troughs seems undeniable. But it turns
out that it's only the difference in phase between two split halves of an electron beam
that can be detected тАФ no experiment has ever measured peaks and troughs in an
individual beam. Every water wave or sound wave produces a detectable rise and fall in
water height or air pressure, so why should matter waves be different? How can they
have a phase that shows up in interference experiments, but not in the wave itself?
It seems we were wrong to assume that matter waves take the form of sine
waves. This doesn't invalidate any of our results тАФ which have all been based merely
on the cyclic nature of the wave, not its exact value тАФ but somehow, matter waves must
be cyclic without growing weaker and stronger. That sounds paradoxical, but a vector
can change direction cyclically without changing strength, and rotating vectors can
certainly produce interference effects by pointing in different directions. Some matter
waves are in fact vectors, but the simplest possible values for ╧И are numbers that possess
 Egan: "Foundations 4"/p.11


a kind of тАЬinternalтАЭ direction that has nothing to do with directions in spacetime. They're
known as complex numbers.


Complex Numbers

Several times in the history of arithmetic, people have stumbled upon the fact that they'd
left out a useful class of numbers that obeyed all the same rules as the numbers with
which they were already familiar. Negative numbers, fractions, and irrational numbers
can all be manipulated by the same kind of operations as the natural numbers (0, 1, 2,
3тАж). If I tell you that xтАУ6 = yтАУ7, you don't need to stop and wonder what kind of
numbers x and y are, before you conclude that x = yтАУ1. It makes no difference; the
rules of algebra don't discriminate.
The real numbers тАФ which consist of all integers, all fractions, and all irrational
numbers тАФ seem to be about as complete as you could hope for: there are no gaps left to
fill between them. However, the fact remains that if you assume that there's a number,
i, such that i2=тАУ1, you can subject it to all kinds of algebraic manipulation without ever
coming to grief. (Compare this with the assumption that there's a number j such that
0j=1. Multiply by two, and you get 0j=2. Subtract the first equation from the second
and you've proved that 0=1. That's grief.)
The ordinary rules of algebra тАФ if you leave out notions of order, such as always
being able to classify y as less than x, greater than x, or equal to x тАФ don't discriminate
against i any more than they discriminate against ╧А or тИЪ2, and including i both enriches
and simplifies almost every field of mathematics. Real multiples of i, such as 3i or
тАУ6.2i, are known as imaginary numbers. Sums of real and imaginary numbers, such
as 1+4i or 2тАУтИЪ2i, are known as complex numbers.
Just as the number line is a useful way to visualise the set of real numbers, the
complex plane provides the perfect equivalent for complex numbers. If you think of
the real numbers as having a direction тАФ the positive numbers pointing right and the
negative numbers pointing left тАФ then multiplying any number by тАУ1 changes its
direction by 180┬░, without changing its size. This metaphor can be extended by letting
multiplication by i change the direction of any number by 90┬░, again without changing its