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2.3: Representation of Waves via Complex Functions

In mathematics, the symbol $${\rm i}$$ is conventionally used to represent the square-root of minus one: in other words, 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 $z = x + {\rm i}\,y,$ 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 lying on an infinite straight-line, we can visualize a complex number as a point lying in an infinite plane. The coordinates of the point in question are the real and imaginary parts of the number: that is, $$z\equiv (x,\,y)$$. This idea is illustrated in Figure [f13.2]. 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 $$|z|=(x^{\,2}+y^{\,2})^{1/2}$$. Incidentally, it follows that $$z\,z^\ast = x^{\,2} + y^{\,2}=|z|^{\,2}$$. 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 wavefunctions. All such representations depend ultimately on a fundamental mathematical identity, known as Euler’s theorem , that takes the form ${\rm e}^{\,{\rm i}\,\phi} \equiv \cos\phi + {\rm i}\,\sin\phi,$ 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, $$r=|z|$$ its modulus, and $$\theta={\rm arg}(z)$$ its argument, it follows from Euler’s theorem that any complex number, $$z$$, can be written $z = r\,{\rm e}^{\,{\rm i}\,\theta},$ where $$r=|z|$$ and $$\theta={\rm arg}(z)$$ are real numbers.

A one-dimensional wavefunction takes the general form

$\label{e12.8} \psi(x,t) = A\,\cos(k\,x-\omega\,t+\varphi),$ where $$A$$ is the wave amplitude, $$k$$ the wavenumber, $$\omega$$ the angular frequency, and $$\varphi$$ the phase angle. Consider the complex wavefunction

$\label{e12.10} \psi(x,t) = \psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)},$ where $$\psi_0$$ is a complex constant. We can write $\psi_0 = A\,{\rm e}^{\,{\rm i}\,\varphi},$ where $$A$$ is the modulus, and $$\varphi$$ the argument, of $$\psi_0$$. Hence, we deduce that \begin{aligned} {\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].\end{aligned} Thus, it follows from Euler’s theorem, and Equation (2.3.4), that ${\rm Re}\left[\psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)}\right] =A\,\cos(k\,x-\omega\,t+\varphi)=\psi(x,t).$ In other words, a general one-dimensional real wavefunction, (2.3.4), can be represented as the real part of a complex wavefunction of the form (2.3.5). 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

$\label{e12.13} \psi(x,t) = \psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)}.$ The main advantage of the complex representation, (2.3.8), over the more straightforward real representation, (2.3.4), 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 previous expression is $\psi({\bf r},t) = \psi_0\,{\rm e}^{\,{\rm i}\,({\bf k}\cdot{\bf r}-\omega\,t)},$ where $${\bf k}$$ is the wavevector.

Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
