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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

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

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

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

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

A one-dimensional wavefunction takes the general form

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

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

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

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

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

$\displaystyle {\rm Re}\left[\psi_0 {\rm e}^{-{\rm i} (\omega t-k x)}\right]...
...\left[A {\rm e}^{ {\rm i} \phi} {\rm e}^{-{\rm i} (\omega t-k x)}\right]$ $\displaystyle ={\rm Re}\left[A {\rm e}^{-{\rm i} (\omega t-k x-\phi)}\right]$    
  $\displaystyle =A {\rm Re}\left[{\rm e}^{-{\rm i} (\omega t-k x-\phi)}\right].$ (C.12)

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

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

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

next up previous
Next: Schrödinger's Equation Up: Wave Mechanics Previous: Electron Diffraction
Richard Fitzpatrick 2016-01-25