Wave equation

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

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On the previous page, we learned that the plane wave is the wave whose wavefronts are parallel to each other and examined the real plane wave equation made up of real numbers and 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+k_{\scriptsize y} y+k_{\scriptsize z} z-\omega t+\delta\right).\tag{1}\end{align*}

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Wave equation

The equation satisfied not only by plane waves but also by any wave (such as a spherical wave) with speed \(u\) is called the wave equation and can be derived from the real plane wave equation (1).

Derivation of the wave equation

First, if we take the second-order partial differentiation of equation (1) with respect to space, we find that the wave number \(k\) comes out of the plane wave \(\varPsi\). That is,

\begin{align*}\Delta \varPsi&=-(k_{\scriptsize x}^2+k_{\scriptsize y}^2+k_{\scriptsize z}^2)\varPsi\\&=-k^2\varPsi,\tag{2}\end{align*}

where \(\Delta\) is the Laplacian

\begin{align*}\Delta=\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}+\frac{\partial^2 }{\partial z^2}.\tag{3}\end{align*}

The Laplacian can be expressed in the form of an inner product as

\begin{align*}\Delta=\nabla\cdot\nabla=\nabla^2\tag{4}\end{align*}

using nabla \(\nabla\)

\begin{align*}\nabla=\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right).\tag{5}\end{align*}

Next, if we take the second-order partial differentiation of equation (1) with respect to time, we find that the angular frequency \(\omega\) comes out of the plane wave \(\varPsi\). That is,

\begin{align*}\frac{\partial^2 \varPsi}{\partial t^2}=-\omega^2\varPsi.\tag{6}\end{align*}

When we transforme equation (2) into

\begin{align*}\varPsi=-\frac{1}{k^2}\Delta\varPsi\tag{7}\end{align*}

and substitute it into equation (6), the equation

\begin{align*}\frac{\partial^2 \varPsi}{\partial t^2}&=\frac{\omega^2}{k^2}\Delta\varPsi\\&=u^2\Delta\varPsi\tag{8}\end{align*}

is obtained.

※※※In equation (8), we used the equation

\begin{align*}\omega=ku.\tag{9}\end{align*}

Equation (9) can be obtained by transforming the wave speed equation

\begin{align*}\nu\lambda=u\tag{10}\end{align*}

into

\begin{align*}2\pi\nu=\frac{2\pi}{\lambda}u\tag{11}\end{align*}

※※※

This equation (8) is the wave equation, and was derived from the real plane wave, but not only plane waves but any wave (such as a spherical wave) with speed \(u\) satisfies this equation.

On the next page…

On the next page, we will derive a complex plane wave equation that satisfies the wave equation (8) from the real plane wave equation. (1)


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