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 \(\omega/k\).