2.One-Dimensional Wavefunction

One of the important characteristics of a traveling wave is that a traveling wave is actually a self-sustaining disturbance of the medium through which it propagates. Consider a mechanical wave that propagates along a string and a sound wave in the air. Wave on a string is transverse, i.e. the medium is perpendicularly displaced to the motion of the wave. While a sound wave in the air is longitudinal, i.e. the medium is displaced in the direction of wave's motion. In both cases, and also for all cases, the individual participating atoms remain in the vicinity of their equilibrium positions although the energy-carrying disturbance advances through the medium. Thus, a traveling wave does not transport the medium through which it travels, and this property permits waves to propagate at very great speeds.

Figure 2.1. (a) Longitudinal wave. (b) Transverse wave.

Since the disturbance ψ is moving, one dimensionally, it can be written as a function of position x and time t:

                                                               

                                                                  ψ = f(x,t)                                                       (2.1)

For instance, holding time constant, say at t = 0, the shape of the disturbance can be expressed:

ψ(x,t)t=0 = f(x,0) = f(x)                                 (2.2)

Equation (2.2) represents the shape or wave’s profile at a certain time. Consider taking ‘snapshots’ of the wave’s profile over a time interval t, in this case, the unaltered wave’s profile has eventually moved along the x-axis a distance vt. Where v is the speed of coordinate system, say S’, which travels along with the pulse (see Figure 2.2 below). Since ψ is stationary constant profile, as we move along with S’, the functional form of the wave profile is the same as given by Equation (2.2). We introduce further coordinate x’ which corresponds to coordinate system S’: 

ψ = f(x')                                                        (2.3)

And it follows from Figure 2.2 that:

x' = x - vt                                                       (2.4)

So that ψ can be written in terms of the variables associated with the stationary S coordinate system as:

  ψ(x,t) = f(x - vt)                                                 (2.5)

Equation 2.5 represents the most general form of the one-dimensional wavefunction.

Figure 2.2. Moving reference frame

1. Waves

One of the most interesting phenomena in physics is wave phenomenon. Consider the nature of light: "Is light a wave phenomenon or a particle phenomenon?" If light is a particle phenomenon, then the particle shall be well-defined in a 'small' region in space. This is the essential feature of a particle, its localization in a well-defined "small" region in space. However, practically, this particle interacts with its environment; for instance, it has a gravitational field that interacts with the earth, the moon, and the sun, etc. And this field, cannot be separated  from the particle since it spreads out into space. Real particles interacts via fields, and, in a sense, the field is the particle and the particle is the field (E.Hecht. Optics. Addison-Wesley, 1998).  Adequately, if  light is a beam of submicroscopic particles (photons) , they are by no means "ordinary" mini-ball classical particles.

In contrast to the localization feature of a particle, a classical traveling wave is a self-sustaining disturbance of a medium, which moves through space transporting energy and momentum which is non-localized. However, the waves are not continuous entities since the media supporting the waves are atomic. That is, counter-intuitive to the concept of ideal wave as a continuous entity that exists over an extended region. The only possible exception might be the electromagnetic wave,  which was first suggested by Einstein as the statistical manifestation of a fundamentally granular underlying microscopic phenomenon.