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# 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., (27)

where represents position, represents time, and , , . For instance, if we are considering a sound wave then might correspond to the pressure perturbation associated with the wave at position and time . On the other hand, if we are considering a light wave then might represent the wave's transverse electric field. As is well-known, the cosine function, , is periodic in its argument, , with period : i.e., for all . The function also oscillates between the minimum and maximum values and , respectively, as varies. It follows that the wavefunction (27) is periodic in with period : i.e., for all and . Moreover, the wavefunction is periodic in with period : i.e., for all and . Finally, the wavefunction oscillates between the minimum and maximum values and , respectively, as and vary. The spatial period of the wave, , is known as its wavelength, and the temporal period, , is called its period. Furthermore, the quantity is termed the wave amplitude, the quantity the wavenumber, and the quantity the wave angular frequency. Note that the units of are radians per second. The conventional wave frequency, in cycles per second (otherwise known as hertz), is . Finally, the quantity , appearing in expression (27), 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 , where is an integer. This follows because the maxima of occur at . Note that a given maximum satisfies , where . It follows that the maximum, and, by implication, the whole wave, propagates in the positive -direction at the velocity . Analogous reasoning reveals that (28)

is the wavefunction of a wave of amplitude , wavenumber , angular frequency , and phase angle , which propagates in the negative -direction at the velocity .   Next: Plane Waves Up: Wave-Particle Duality Previous: Introduction
Richard Fitzpatrick 2010-07-20