In mathematics, the symbol
is conventionally used to represent the square-root of minus one; that is, the
solution of
(Riley 1974). A real number,
(say), can take any value in a continuum of 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
 |
(11.6) |
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 complex conjugate of
is defined
.
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,
. This idea is illustrated in Figure 11.2.
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,
.
Figure 11.2:
Representation of a complex number as a point in a plane.
|
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 (see Exercise 2), which takes the form
 |
(11.7) |
where
is a real number (Riley 1974). 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
 |
(11.8) |
where
and
are real numbers.
A one-dimensional wavefunction takes the general form
 |
(11.9) |
where
is the wave amplitude,
the phase angle,
the wavenumber, and
the angular
frequency. Consider the complex wavefunction
 |
(11.10) |
where
is a complex constant. We can write
 |
(11.11) |
where
is the modulus, and
the argument, of
.
Hence, we deduce that
![$\displaystyle {\rm Re}\left[\psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-k\,x)}\right]...
...phi)}\right]=A\,{\rm Re}\left[{\rm e}^{-{\rm i}\,(\omega\,t-k\,x-\phi)}\right].$](img3824.png) |
(11.12) |
Thus, it follows from Euler's theorem, and Equation (11.9), that
![$\displaystyle {\rm Re}\left[\psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-k\,x)}\right] =A\,\cos(\omega\,t-k\,x-\phi)=\psi(x,t).$](img3825.png) |
(11.13) |
In other words, a general one-dimensional real wavefunction, (11.9), can be
represented as the real part of a complex wavefunction of the form (11.10).
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
 |
(11.14) |
The
main advantage of the complex representation, (11.14), over the more straightforward
real representation, (11.9), is that the former enables us to combine the amplitude,
, and the
phase angle,
, of the wavefunction into a single complex amplitude,
.