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In mathematics, the symbol is conventionally used to represent the squareroot of minus one: i.e., one of the
solutions of
. Now, a real number, (say), can take any value in a continuum of different values lying between and .
On the other hand, an imaginary number takes the general form , where is a real number. It follows that the square of
a real number is a positive real number, whereas the square of an imaginary number is a negative real number. In addition, a general complex number is written

(36) 
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 complex conjugate of is defined
.
Now, just as we
can visualize a real number as a point on an infinite straightline, 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: i.e.,
. This idea is illustrated in Fig. 3.
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 straightline 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,
.
Figure 3:
Representation of a complex number as a point in a plane.

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

(37) 
where is a real number. Incidentally, given that
, where is a general
complex number, its modulus, and
its argument, it follows from de Moivre's theorem that any
complex number, , can be written

(38) 
where and
are real numbers.
Now, a onedimensional wavefunction takes the general form

(39) 
where is the wave amplitude, the wavenumber, the angular
frequency, and the phase angle. Consider the complex wavefunction

(40) 
where is a complex constant. We can write

(41) 
where is the modulus, and the argument, of .
Hence, we deduce that
Thus, it follows from de Moirve's theorem, and Eq. (39), that

(43) 
In other words, a general onedimensional real wavefunction, (39), can be
represented as the real part of a complex wavefunction of the form (40).
For ease
of notation, the ``take the real part'' aspect of the above expression is usually omitted, and our general onedimension wavefunction
is simply written

(44) 
The
main advantage of the complex representation, (44), over the more straightforward
real representation, (39), is that the former enables us to combine the amplitude, , and the
phase angle, , of the wavefunction into a single complex amplitude, .
Finally, the three dimensional generalization of the above expression is

(45) 
where is the wavevector.
Next: Classical Light Waves
Up: WaveParticle Duality
Previous: Plane Waves
Richard Fitzpatrick
20100720