# 2.1: Wavefunctions

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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 \(\omega/k\).