3.2: Wavefunctions

A wave is defined as a disturbance in some physical system which is periodic in both space and time. In one dimension, a wave is generally represented in terms of a wavefunction: e.g., 

\begin{equation}\psi(x, t)=A \cos (k x-\omega t+\varphi)\end{equation}

where \(\begin{equation}x\end{equation}\) represents position, \(\begin{equation}t\end{equation}\) represents time, and \(\begin{equation}A, k, \omega>0\end{equation}\).  For instance, if we are considering a sound wave then \(\begin{equation}\psi(x, t)\end{equation}\) might correspond to the pressure perturbation associated with the wave at position \(\begin{equation}x\end{equation}\) and time \(\begin{equation}t\end{equation}\). On the other hand, if we are considering a light wave then \(\begin{equation}\psi(x, t)\end{equation}\) might represent the wave's transverse electric field. As is well-known, the cosine function, \(\begin{equation}\cos (\theta)\end{equation}\), is periodic in its argument, \(\begin{equation}\theta\end{equation}\), with period \(\begin{equation}2 \pi\end{equation}\): i.e., \(\begin{equation}\cos (\theta+2 \pi)=\cos \theta \text { for all } \theta\end{equation}\). The function also oscillates between the minimum and maximum values  and , respectively, as \(\begin{equation}\theta\end{equation}\) varies. It follows that the wavefunction (3.2.1) is periodic in \(\begin{equation}x\end{equation}\) with period \(\begin{equation}\lambda=2 \pi / k\end{equation}\): i.e., \(\begin{equation}\psi(x+\lambda, t)=\psi(x, t) \text { for all } x \text { and } t\end{equation}\). Moreover, the wavefunction is periodic in \(\begin{equation}t\end{equation}\) with period \(\begin{equation}T=2 \pi / \omega: \text { i.e.}, \psi(x, t+T)=\psi(x, t)\end{equation}\) for all \(\begin{equation}x \text { and } t\end{equation}\). Finally, the wavefunction oscillates between the minimum and maximum values \(\begin{equation}-A \text { and }+A\end{equation}\), respectively, as \(\begin{equation}x \text { and } t\end{equation}\) vary. The spatial period of the wave, \(\begin{equation}\lambda\end{equation}\) , is known as its wavelength, and the temporal period, \(\begin{equation}T\end{equation}\), is called its period. Furthermore, the quantity \(\begin{equation}A\end{equation}\) is termed the wave amplitude, the quantity \(\begin{equation}k\end{equation}\) the wavenumber, and the quantity \(\begin{equation}\omega\end{equation}\) the wave angular frequency. Note that the units of \(\begin{equation}\omega\end{equation}\) are radians per second. The conventional wave frequency, in cycles per second (otherwise known as hertz), is \(\begin{equation}\nu=1 / T=\omega / 2 \pi\end{equation}\). Finally, the quantity \(\begin{equation}\varphi\end{equation}\), appearing in expression (3.2.1), is termed the phase angle, and determines the exact positions of the wave maxima and minima at a given time. In fact, the maxima are located at \(\begin{equation}k x-\omega t+\varphi=j 2 \pi\end{equation}\), where \(\begin{equation}j\end{equation}\) is an integer. This follows because the maxima of \(\begin{equation}\cos (\theta) \text { occur at } \theta=j 2 \pi\end{equation}\) . Note that a given maximum satisfies \(\begin{equation}x=(j-\varphi / 2 \pi) \lambda+v t, \text { where } v=\omega / k\end{equation}\). It follows that the maximum, and, by implication, the whole wave, propagates in the positive \(\begin{equation}x\end{equation}\)-doirection at the velocity \(\begin{equation}\omega / k\end{equation}\). Analogous reasoning reveals that

\begin{equation}\psi(x, t)=A \cos (-k x-\omega t+\varphi)=A \cos (k x+\omega t-\varphi)\end{equation}

is the wavefunction of a wave of amplitude \(\begin{equation}A, \text { wavenumber } k, \text { angular frequency } \omega\end{equation}\), and phase angle \(\begin{equation}\varphi\end{equation}\), which propagates in the negative \(\begin{equation}x \text { -direction at the velocity } \omega / k\end{equation}\).