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Physics LibreTexts

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, ψ(x,t)=Acos(kxωt+φ), where x is a position coordinate, t represents time, and A, k, ω>0. For example, if we are considering a sound wave then ψ(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 ψ(x,t) might represent the wave’s transverse electric field. As is well known, the cosine function, cosθ, is periodic in its argument, θ, with period 2π: in other words, cos(θ+2π)=cosθ for all θ. The function also oscillates between the minimum and maximum values 1 and +1, respectively, as θ varies. It follows that the wavefunction (2.1.1) is periodic in x with period λ=2π/k. In other words, ψ(x+λ,t)=ψ(x,t) for all x and t. Moreover, the wavefunction is periodic in t with period T=2π/ω. In other words, ψ(x,t+T)=ψ(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, λ, 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 ω the wave angular frequency. Note that the units of ω are radians per second. The conventional wave frequency, in cycles per second (otherwise known as hertz), is ν=1/T=ω/2π. Finally, the quantity φ, 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 kxωt+φ=j2π, where j is an integer. This follows because the maxima of cosθ occur at θ=j2π. Note that a given maximum satisfies x=(jφ/2π)λ+vt, where v=ω/k. It follows that the maximum, and, by implication, the whole wave, propagates in the positive x-direction at the velocity ω/k. Analogous reasoning reveals that ψ(x,t)=Acos(kxωt+φ)=Acos(kx+ωtφ), is the wavefunction of a wave of amplitude A, wavenumber k, angular frequency ω, and phase angle φ, that propagates in the negative x-direction at the velocity ω/k.


This page titled 2.1: Wavefunctions is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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