Complex plane wave

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On this page, we will derive a complex plane wave equation that satisfies the wave equation from the real plane wave equation.

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On the previous page…

On the previous page, we derived the wave equation

\begin{align*}\frac{\partial^2 \varPsi’}{\partial t^2}&=u^2\Delta\varPsi’\tag{1}\end{align*}

satisfied not only by plane waves but also by any wave (such as a spherical wave) with speed \(u\) from the real plane wave equation with speed \(u\)

\begin{align*}\varPsi'(q,t)&=a\cos\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right)\\&=a\cos\left(k_{\scriptsize x} x+k_{\scriptsize y} y+k_{\scriptsize z} z-\omega t+\delta\right).\tag{2}

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Complex plane wave

A complex plane wave is a plane wave that consists of a complex exponential function and satisfies the wave equation like a real plane wave. The equation for the complex plane wave can be obtained from the equation for the real plane wave.

Derivation of complex plane wave

Consider the plane wave that has the same wavenumber vector \(boldsymbol k\), angular frequency \(\omega), and phase shift \(\delta\) as the real plane wave equation (1), and whose amplitude is an imaginary number \(ai\). This plane wave can be written as

\begin{align*}\varPsi'{}'(q,t)&=ai\sin\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right).\tag{3}\end{align*}

The speed of the complex plane wave (3) is the same as the speed of the real plane wave (1), so equation (3) also satisfies the wave equation (2).

Here, by adding the complex plane wave equation (3) to the real plane wave equation (2), we get the new complex plane wave

\(\varPsi\)

\begin{align*}\varPsi&=\varPsi’+\varPsi'{}’\\&=a\cos\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right)+ai\sin\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right)\\&=ae^{i\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right)}.\tag{4}\end{align*}

In equation (4), Euler’s formula

\begin{align*}e^{i\theta}=\cos\theta+i\sin\theta\tag{5}\end{align*}

is used, and the plane wave (4)can be expressed not by trigonometric functions but by complex exponential functions. This plane wave is also called a complex plane wave, and since the speed of the complex plane wave (4) is the same as the speed of the real plane wave (1) and the complex plane wave (3), the equation (4) also satisfies the wave equation (2).

Why consider complex plane waves?

Both real waves and complex waves are solutions of the wave equation (2). However, the solutions of the wave equation (2) used in quantum mechanics are not real waves, but complex waves because there are phenomena that cannot be explained by real waves. In fields other than quantum mechanics, it is correct that “the imaginary part of complex waves is ignored because waves in the real world are the real part”, but it is wrong in quantum mechanics.

In future pages, we will see examples where complex waves must be used.

On the next page…

On the next page, by substituting Einstein’s equation and de Broglie’s equation, which have wave-particle duality, into the complex plane wave equation (4) that expresses only wave properties, we derive the complex plane wave equation that has wave-particle duality.


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