【Easy Quantum Mechanics】Real plane wave

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On this page, we will learn that a plane wave is a wave whose wavefronts are parallel to each other and examine a real plane wave equation made up of real numbers.

Real plane wave

A plane wave is a wave whose wavefronts are parallel to each other, which differs from a spherical wave whose wavefronts are concentrically spherical. There are two types of plane waves: real plane waves consisting of real numbers and complex plane waves consisting of complex numbers. On this page, we will focus on real plane waves.

1D real plane wave

First, let us consider a one-dimensional real plane wave, such as string vibration. It can be expressed as

\begin{align*}\varPsi(q,t)&=a\cos\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right)\\&=a\cos\left(k_{\scriptsize x} x-\omega t+\delta\right),\tag{1}\end{align*}

where \(a\) is an amplitude, \(\boldsymbol k\) is a wavevector whose magnitude is equal to a wavenumber \(k\) and whose direction is along the direction of wave propagation, \(\omega\) is an angular frequency, \(\delta\) is a phase shift, \(\boldsymbol q\) is a one-dimensional position vector, \(k_x\) is an x-component of the wavevector \(\boldsymbol k\).

Real plane wave equation

Let us check that equation (1) is correct. Since the period of a cosine function is \( 2 \pi \), equation (1) repeats its values every \(\frac{2\pi}{k_x}\) along the spatial direction. Also, since the wavenumber \(k_x\) is defined as the number of waves per length \(2\pi\), \(\frac{k_x} { 2\pi}\) represents the number of waves per unit length, and \(\frac {2 \pi} {k_x} \), the reciprocal of \(\frac{k_x}{2\pi}\), corresponds to the length per wave, which is the wavelength \(\lambda\)

\begin{align*}\lambda=\frac{2\pi}{k_x}.\tag{2}\end{align*}

The fact that \(\varPsi\) repeats its value for each wavelength \(\lambda\) accurately describes the nature of the wave.

Next, since the period of a cosine function is \( 2\pi\), equation (1) repeats its values every \(\frac{2\pi}{\omega}\) in the time direction.Also, since angular frequency \(\omega\) is defined as the angle per unit time, \(\frac{\omega}{2\pi}\) represents a frequency \(\nu\), which is the number of waves per unit time, and can be expressed as

\begin{align*}\nu=\frac{\omega}{2\pi}.\tag{3}\end{align*}

\(\frac{2\pi}{\omega}\), the reciprocal of the frequency \(\nu\), corresponds to the time taken to complete one cycle of an oscillation, which is a period \(T\)

\begin{align*}T=\frac{1}{\nu}.\tag{4}\end{align*}

The fact that \(\varPsi\) repeats its value for each period \(T\) accurately describes the nature of the wave.

In equation (1), the plane wave speed \(u\) is expressed as the product of the frequency \(\nu\), which represents the number of waves per unit time, and the wavelength \(\lambda\), which represents the length per wave. The plane wave speed can be written as

\begin{align*}u=\nu\lambda.\tag{5}\end{align*}

2D/3D real plane wave

Finally, let us consider a two-dimensional and three-dimensional real plane waves. Membrane vibration is an example of 2D real plane waves, and field vibration is an example of 3D real plane waves. Using the position vector \(\boldsymbol q\), the equation for the two-dimensional real plane wave can be written as

\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-\omega t+\delta\right),\tag{6}\end{align*}

and the equation for the three-dimensional real plane wave can be written as (7)

\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{7}\end{align*}

where \(k_x\), \(k_y\), and \(k_z\) are the \(x\)-, \(y\)-, and \(z\)-components of the wave vector \(\boldsymbol k\), respectively.

On the next page…

On the next page, starting from equation (7) for the real plane wave with speed \(u\), we derive the wave equation satisfied not only by plane waves but also by any wave (such as a spherical wave) with speed \(u\).

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