Representation of Waves via Complex Numbers

In mathematics, the symbol ${\rm i}$ is conventionally used to represent the square-root of minus one: i.e., the solution 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

$$z = x + {\rm i}\,y,$$ (775)

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 Figure 58. 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 58: Representation of a complex number as a point in a plane.

Complex numbers are often used to represent waves, and wavefunctions. All such representations depend ultimately on a fundamental mathematical identity, known as de Moivre's theorem (see Exercise 12.1), which takes the form

$${\rm e}^{\,{\rm i}\,\phi} \equiv \cos\phi + {\rm i}\,\sin\phi,$$ (776)

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

$$z = r\,{\rm e}^{\,{\rm i}\,\theta},$$ (777)

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

Now, a one-dimensional wavefunction takes the general form

$$\psi(x,t) = A\,\cos(\phi+k\,x-\omega\,t),$$ (778)

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

$$\psi(x,t) = \psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)},$$ (779)

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

$$\psi_0 = A\,{\rm e}^{\,{\rm i}\,\phi},$$ (780)

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

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

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

$$= A\,{\rm Re}\left[{\rm e}^{\,{\rm i}\,(\phi+k\,x-\omega\,t)}\right].$$ (781)

Thus, it follows from de Moirve's theorem, and Equation (779), that

$${\rm Re}\left[\psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)}\right] =A\,\cos(\phi+k\,x-\omega\,t)=\psi(x,t).$$ (782)

In other words, a general one-dimensional real wavefunction, (779), can be represented as the real part of a complex wavefunction of the form (780). 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

$$\psi(x,t) = \psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)}.$$ (783)

The main advantage of the complex representation, (784), over the more straightforward real representation, (779), is that the former enables us to combine the amplitude, $A$, and the phase angle, $\phi$, of the wavefunction into a single complex amplitude, $\psi_0$.