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# Representation of Waves via Complex Functions

In mathematics, the symbol is conventionally used to represent the square-root 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 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: 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 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 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 one-dimensional 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
 (42)

Thus, it follows from de Moirve's theorem, and Eq. (39), that
 (43)

In other words, a general one-dimensional 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 one-dimension 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: Wave-Particle Duality Previous: Plane Waves
Richard Fitzpatrick 2010-07-20