(C.6) |

where and are real numbers. In fact, is termed the real part of , and the imaginary part of . This is written mathematically as and . Finally, the

Just as we
can visualize a real number as a point on an infinite straight-line, we can visualize a complex number as
a point in an infinite plane. The coordinates of the point in question are the real and imaginary
parts of the number: that is,
.
The distance,
, of the representative point from the origin is termed the *modulus*
of the corresponding complex number,
. This is written mathematically as
. Incidentally, it follows that
.
The angle,
, that the straight-line joining the representative point to the origin subtends with the
real axis is termed the *argument* of the corresponding complex number,
. This is written mathematically
as
. It follows from standard trigonometry that
, and
.
Hence,
.

Complex numbers are often used to represent waves and wavefunctions. All such representations ultimately depend on a fundamental mathematical identity, known as
*Euler's theorem*, which takes the form

(C.7) |

where is a real number. Incidentally, given that , where is a general complex number, its modulus, and its argument, it follows from Euler's theorem that any complex number, , can be written

(C.8) |

where and are real numbers.

A one-dimensional wavefunction takes the general form

where is the wave amplitude, the phase angle, the wavenumber, and the angular frequency. Consider the complex wavefunction

where is a complex constant. We can write

(C.11) |

where is the modulus, and the argument, of . Hence, we deduce that

(C.12) |

Thus, it follows from Euler's theorem, and Equation (C.9), that

(C.13) |

In other words, a general one-dimensional real wavefunction, (C.9), can be represented as the real part of a complex wavefunction of the form (C.10). For ease of notation, the ``take the real part'' aspect of the previous expression is usually omitted, and our general one-dimension wavefunction is simply written

The main advantage of the complex representation, (C.14), over the more straightforward real representation, (C.9), is that the former enables us to combine the amplitude, , and the phase angle, , of the wavefunction into a single complex amplitude, .