6.1: Complex Numbers

Before we begin, however, we need briefly to review complex numbers. Complex numbers are intrinsic to quantum mechanics, and indeed the entire theory wouldn’t work if we didn’t use complex numbers as part of it.

A complex number is a number that has both a real and an “imaginary” part. The name “imaginary” is perhaps unfortunate, because it suggests there’s something less tangible about imaginary numbers than there is about real numbers. Remember, however, that even real numbers, when used in science, are abstract mathematical representations of the systems that they are standing in for. Even real numbers are imaginary, in that sense of the word.

All imaginary numbers can be constructed from \(\ i\), sometimes called “the” imaginary number, which is defined as:

\(\ i=\sqrt{-1}\)

you may remember from math that you can’t take a square root of a negative number. In fact, you can, but you don’t get a real number as a result; you get an imaginary number. By the same token, you may remember that the square of any number is positive. That only applies to real numbers; the square of any real number is positive. However, square both sides of the equation above and you can see that:

\(\ i^{2}=-1\)

You can construct any other imaginary number by just multiplying \(\ i\) by a real number. So, \(\ 3i\), \(\ \pi i\), and \(\ -2.9 \times 10^{21} i\) are all imaginary numbers.

You can then write any complex number as the real part plus the imaginary part. So, \(\ 2+3 i\) is a complex number. You can’t simplify it any further than that. Remember that \(\ i\) is not a variable here, but a number, just as concrete as any other number. It’s not a number that you could place on a numberline, because a numberline only has the real numbers on it. But it’s just as. . . well, just as real as a real number. The expression \(\ 2+3 i\) is fundamentally different from the expression \(\ 2+3 \pi\). You can view \(\ 2+3 \pi\) as being completely reduced, as there’s no need to reduce it further (as there would be with the expression \(\ 2+(3)(4)\), which can be reduced to 14). However, you could, if you wished, reduce \(\ 2+3 \pi\) with your calculator, and write down an imperfect single-valued representation of it: 11.424778. No such further reduction may be done with the number \(\ 2+3 i\). The two parts of this number, 2 and \(\ 3i\), are like two components of a vector; they both have an independent identity. However, when we get to using vectors to represent spin states of particles don’t confuse the real and imaginary parts of a complex number with components of those vectors. The value \(\ 2+3 i\) represents a single complex number. You can reduce the real part and the imaginary part of a complex number down so that the first is represented by a single real number, and the second is represented by a second real number multiplied by \(\ i\).

For every complex number, there is a partner number called the complex conjugate. Along with the noun complex conjugate there is a verb complex conjugate. In order to complex conjugate a number, you replace every instance of \(\ i\) with \(\ -i\). So, the complex conjugate of \(\ 2+3 i\) is \(\ 2-3 i\). In algebra, we use the symbol ^∗ to indicate the complex conjugate of a quantity. If you have a complex number \(\ a\) (i.e. a variable in algebra that may not just have a real value, but which may have a fully complex value), the complex conjugate of \(\ a\) is represented as \(\ a^{*}\). Thus, if \(\ a=2+3 i\), then \(\ a^{*}=2-3 i\).