Representation of Waves via Complex Numbers

In mathematics, the symbol ${\rm i}$ is conventionally used to represent the square-root of minus one: that is, the solution of ${\rm i}^{ 2} = -1$ . A real number, $x$ (say), can take any value in a continuum of values lying between $-\infty$ and $+\infty$ . On the other hand, an imaginary number takes the general form ${\rm i} y$ , where $y$ 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

$$z = x + {\rm i} y,$$ (C.6)

where $x$ and $y$ are real numbers. In fact, $x$ is termed the real part of $z$ , and $y$ the imaginary part of $z$ . This is written mathematically as $x={\rm Re}(z)$ and $y={\rm Im}(z)$ . Finally, the complex conjugate of $z$ is defined $z^\ast = x-{\rm i} y$ .

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, $z\equiv (x, y)$ . The distance, $r=(x^{ 2}+y^{ 2})^{1/2}$ , of the representative point from the origin is termed the modulus of the corresponding complex number, $z$ . This is written mathematically as $\vert z\vert=(x^{ 2}+y^{ 2})^{1/2}$ . Incidentally, it follows that $z z^\ast = x^{ 2} + y^{ 2}=\vert z\vert^{ 2}$ . The angle, $\theta=\tan^{-1}(y/x)$ , 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, $z$ . This is written mathematically as ${\rm arg}(z)=\tan^{-1}(y/x)$ . It follows from standard trigonometry that $x=r \cos\theta$ , and $y=r \sin\theta$ . Hence, $z= r \cos\theta+ {\rm i} r\sin\theta$ .

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

$${\rm e}^{ {\rm i} \phi} \equiv \cos\phi + {\rm i} \sin\phi,$$ (C.7)

where $\phi$ is a real number. Incidentally, given that $z=r \cos\theta + {\rm i} r \sin\theta= r (\cos\theta+{\rm i} \sin\theta)$ , where $z$ is a general complex number, $r=\vert z\vert$ its modulus, and $\theta={\rm arg}(z)$ its argument, it follows from Euler's theorem that any complex number, $z$ , can be written

$$z = r {\rm e}^{ {\rm i} \theta},$$ (C.8)

where $r=\vert z\vert$ and $\theta={\rm arg}(z)$ are real numbers.

A one-dimensional wavefunction takes the general form

$$\psi(x,t) = A \cos(\omega t-k x-\phi),$$ (C.9)

where $A>0$ is the wave amplitude, $\phi$ the phase angle, $k$ the wavenumber, and $\omega$ the angular frequency. Consider the complex wavefunction

$$\psi(x,t) = \psi_0 {\rm e}^{-{\rm i} (\omega t-k x)},$$ (C.10)

where $\psi_0$ is a complex constant. We can write

$$\psi_0 = A {\rm e}^{ {\rm i} \phi},$$ (C.11)

where $A$ is the modulus, and $\phi$ the argument, of $\psi_0$ . Hence, we deduce that

$${\rm Re}\left[\psi_0 {\rm e}^{-{\rm i} (\omega t-k x)}\right]... ...\left[A {\rm e}^{ {\rm i} \phi} {\rm e}^{-{\rm i} (\omega t-k x)}\right] ={\rm Re}\left[A {\rm e}^{-{\rm i} (\omega t-k x-\phi)}\right]$$

$$=A {\rm Re}\left[{\rm e}^{-{\rm i} (\omega t-k x-\phi)}\right].$$ (C.12)

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

$${\rm Re}\left[\psi_0 {\rm e}^{-{\rm i} (\omega t-k x)}\right] =A \cos(\omega t-k x-\phi)=\psi(x,t).$$ (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

$$\psi(x,t) = \psi_0 {\rm e}^{-{\rm i} (\omega t-k x)}.$$ (C.14)

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, $A$ , and the phase angle, $\phi$ , of the wavefunction into a single complex amplitude, $\psi_0$ .