u/1. A representation of complex
numbers using color and bright-
ness. 2. A wave traveling toward
the right. 3. A wave traveling to-
ward the left. 4. A standing wave
formed by superposition of waves
2 and 3. 5. A two-dimensional
standing wave. 6. A double-slit
diffraction pattern.
that this is the case by comparing with the convention defined by
u/1. The function being plotted here is Ψ =eikx, wherek= 2π/λ
is the spatial analog of frequency, with an extra factor of 2π for
convenience. For the use of the complex exponential, see sec. 10.5.6,
p .627; it simply represents a point on the unit circle in the complex
plane. The wavelengthλ is a constant and can be measured, for
example, from one yellow point to the next. The wavelength isnot
different at different points on the figure, because we are using the
colors merely as a visual encoding of the complex numbers — so,
for example, a red point on the figure is not a point where the wave
has a longer wavelength than it does at a blue point.
Figure u/3 represents a wave traveling to the left.
Figure u/4 shows a standing wave created by superimposing the
traveling waves from u/2 and u/3, Ψ 4 = (Ψ 2 + Ψ 3 )/2. (The rea-
son for the factor of 2 is simply that otherwise some portions of Ψ 4
would have magnitudes too great to be represented using the avail-
able range of brightness.) All points on this wave have real values,
represented by red and blue-green. We made the superposition real
by an appropriate choice of the phases of Ψ 2 and Ψ 3. This is always
possible to do when we have a standing wave, but it isonlypossible
for a standing wave, and this is the reason for all of the disclaimers
in the captions of previous figures in which I took the liberty of
representing a traveling wave as a sine-wave graph.
Figure u/5 shows a two-dimensional standing wave of a particle
in a box, and u/6 shows a double-slit interference pattern. (In the
latter, I’ve cheated by making the amplitude of the wave on the914 Chapter 13 Quantum Physics