Page 15 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6 The wave function
2π
k
4π/∆k
Re Ψ(x, t)

x

(a)

Re Ψ(x, t)

x

(b)
Figure 1.2 (a) The real part of the superposition of two plane waves is shown by the
solid curve. The pro®le of the amplitude is shown by the dashed curve. (b) The
positions of the curves in Figure 1.2(a) after a short time interval.

4ð=Äk. At the points of maximum amplitude, the two original plane waves
interfere constructively. At the nodes in Figure 1.2(a), the two original plane
waves interfere destructively and cancel each other out.
As time increases, the plane wave exp[i(kx ÿ ùt)] moves with velocity ù=k.
If we consider a ®xed point x 1 and watch the plane wave as it passes that point,
we observe not only the periodic rise and fall of the amplitude of the
unmodi®ed plane wave exp[i(kx ÿ ùt)], but also the overlapping rise and fall
of the amplitude due to the modulating function 2 cos[(Äkx ÿ Äùt)=2]. With-
out the modulating function, the plane wave would reach the same maximum