> We've talked about why the amplitudes should be complex numbers, and why the rule for converting amplitudes to probabilities should be a squaring rule.
The squaring rule is actually a special case of multiplying a number by its complex conjugate, which the article doesn't mention, unfortunately.
That is to say, if we have a number z = x + iy, we can obtain its norm from sqrt(xx + yy). But another way to express this is simply sqrt(z z). The product z z is just (x + iy)(x - iy). That of course is just x^2 - (iy)^2 which goes to x^2 - (-1y^2) -> x^2 + y^2.
Geometrically, the conjugate of a complex number has the opposite angle. If z is 20 degrees from the real axis, z* is -20 degrees. Since multiplication of complex numbers is additions of their arguments (i.e. angle components), the two cancel out and the result is on the real number line.
The squaring rule is actually a special case of multiplying a number by its complex conjugate, which the article doesn't mention, unfortunately.
That is to say, if we have a number z = x + iy, we can obtain its norm from sqrt(xx + yy). But another way to express this is simply sqrt(z z). The product z z is just (x + iy)(x - iy). That of course is just x^2 - (iy)^2 which goes to x^2 - (-1y^2) -> x^2 + y^2.
Geometrically, the conjugate of a complex number has the opposite angle. If z is 20 degrees from the real axis, z* is -20 degrees. Since multiplication of complex numbers is additions of their arguments (i.e. angle components), the two cancel out and the result is on the real number line.