Page 11 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2 The wave function

1.1 Wave motion
Plane wave
A simple stationary harmonic wave can be represented by the equation
2ðx
ø(x) ˆ cos
ë
and is illustrated by the solid curve in Figure 1.1. The distance ë between peaks
(or between troughs) is called the wavelength of the harmonic wave. The value
of ø(x) for any given value of x is called the amplitude of the wave at that
point. In this case the amplitude ranges from ‡1to ÿ1. If the harmonic wave is
A cos(2ðx=ë), where A is a constant, then the amplitude ranges from ‡A to
ÿA. The values of x where the wave crosses the x-axis, i.e., where ø(x) equals
zero, are the nodes of ø(x).
If the wave moves without distortion in the positive x-direction by an amount
x 0 , it becomes the dashed curve in Figure 1.1. Since the value of ø(x)atany
point x on the new (dashed) curve corresponds to the value of ø(x) at point
x ÿ x 0 on the original (solid) curve, the equation for the new curve is
2ð
ø(x) ˆ cos (x ÿ x 0 )
ë
If the harmonic wave moves in time at a constant velocity v, then we have the
relation x 0 ˆ vt, where t is the elapsed time (in seconds), and ø(x) becomes
2ð
ø(x, t) ˆ cos (x ÿ vt)
ë
Suppose that in one second, í cycles of the harmonic wave pass a ®xed point
on the x-axis. The quantity í is called the frequency of the wave. The velocity

ψ(x)
x 0
λ

x
λ/2 λ 3λ/2 λ2

Figure 1.1 A stationary harmonic wave. The dashed curve shows the displacement of
the harmonic wave by x 0 .