Representation of Waves via Complex Functions

In mathematics, the symbol ${\rm i}$ is conventionally used to represent the square-root of minus one: i.e., one of the solutions of ${\rm i}^2 = -1$. Now, a real number, $x$ (say), can take any value in a continuum of different 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

$$z = x + {\rm i} y,$$ (36)

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 $z^\ast = x-{\rm i} y$.

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., $z\equiv (x, y)$. This idea is illustrated in Fig. 3. The distance, $r=\sqrt{x^2+y^2}$, of the representative point from the origin is termed the modulus of the corresponding complex number, $z$. This is written mathematically as $\vert z\vert=\sqrt{x^2+y^2}$. 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, $z= r \cos\theta+ {\rm i} r\sin\theta$.

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

$${\rm e}^{ {\rm i} \phi} \equiv \cos\phi + {\rm i} \sin\phi,$$ (37)

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 de Moivre's theorem that any complex number, $z$, can be written

$$z = r {\rm e}^{ {\rm i} \theta},$$ (38)

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

Now, a one-dimensional wavefunction takes the general form

$$\psi(x,t) = A \cos(k x-\omega t+\varphi),$$ (39)

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

$$\psi(x,t) = \psi_0 {\rm e}^{ {\rm i} (k x-\omega t)},$$ (40)

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

$$\psi_0 = A {\rm e}^{ {\rm i} \varphi},$$ (41)

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

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

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

$$= A {\rm Re}\left[{\rm e}^{ {\rm i} (k x-\omega t+\varphi)}\right].$$ (42)

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

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

$$\psi(x,t) = \psi_0 {\rm e}^{ {\rm i} (k x-\omega t)}.$$ (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, $A$, and the phase angle, $\varphi$, of the wavefunction into a single complex amplitude, $\psi_0$. Finally, the three dimensional generalization of the above expression is

$$\psi({\bf r},t) = \psi_0 {\rm e}^{ {\rm i} ({\bf k}\cdot{\bf r}-\omega t)},$$ (45)

where ${\bf k}$ is the wavevector.