2.1: Wavefunctions

A wave is defined as a disturbance in some physical system that is periodic in both space and time. In one dimension, a wave is generally represented in terms of a wavefunction: for instance, $$\label{ew} \psi(x,t) = A\,\cos(k\,x-\omega\,t+\varphi),$$ where $x$ is a position coordinate, $t$ represents time, and $A$, $k$, $\omega >0$. For example, if we are considering a sound wave then $\psi(x,t)$ might correspond to the pressure perturbation associated with the wave at position $x$ and time $t$. On the other hand, if we are considering a light-wave then $\psi(x,t)$ might represent the wave’s transverse electric field. As is well known, the cosine function, $\cos \theta$, is periodic in its argument, $\theta$, with period $2\pi$: in other words, $\cos(\theta+2\pi)=\cos\theta$ for all $\theta$. The function also oscillates between the minimum and maximum values $-1$ and $+1$, respectively, as $\theta$ varies. It follows that the wavefunction (2.1.1) is periodic in $x$ with period $\lambda=2\pi/k$. In other words, $\psi(x+\lambda,t)=\psi(x,t)$ for all $x$ and $t$. Moreover, the wavefunction is periodic in $t$ with period $T=2\pi/\omega$. In other words, $\psi(x,t+T)=\psi(x,t)$ for all $x$ and $t$. Finally, the wavefunction oscillates between the minimum and maximum values $-A$ and $+A$, respectively, as $x$ and $t$ vary. The spatial period of the wave, $\lambda$, is known as its wavelength, and the temporal period, $T$, is called its period. Furthermore, the quantity $A$ is termed the wave amplitude, the quantity $k$ the wavenumber, and the quantity $\omega$ the wave angular frequency. Note that the units of $\omega$ are radians per second. The conventional wave frequency, in cycles per second (otherwise known as hertz), is $\nu=1/T=\omega/2\pi$. Finally, the quantity $\varphi$, appearing in expression (2.1.1), 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 $k\,x-\omega\,t+\varphi = j\,2\pi$, where $j$ is an integer. This follows because the maxima of $\cos\theta$ occur at $\theta=j\,2\pi$. Note that a given maximum satisfies $x=(j-\varphi/2\pi)\,\lambda+ v\,t$, where $v=\omega/k$. It follows that the maximum, and, by implication, the whole wave, propagates in the positive $x$-direction at the velocity $\omega/k$. Analogous reasoning reveals that $$\psi(x,t) = A\,\cos(-k\,x-\omega\,t+\varphi)=A\,\cos(k\,x+\omega\,t-\varphi),$$ is the wavefunction of a wave of amplitude $A$, wavenumber $k$, angular frequency $\omega$, and phase angle $\varphi$, that propagates in the negative $x$-direction at the velocity $\omega/k$.