for example, arbitrarily say that electrons with positive energies
have wavefunctions whose phases rotate counterclockwise, and as
long as we follow that rule consistently within a given calculation,
everything will work. Note that it is not possible to define anything
like a right-hand rule here, because the complex plane shown in the
right-hand side of t/2 doesn’t represent two dimensions of physical
space; unlike a screw going into a piece of wood, an electron doesn’t
have a direction of rotation that depends on its direction of travel.
Superposition of complex wavefunctions example 22. The right side of figure t/3 is a cartoonish representation of
double-slit interference; it depicts the situation at the center, where
symmetry guarantees that the interference is constructive. Sup-
pose that at some off-center point, the two wavefunctions being
superposed areΨ 1 =bandΨ 2 =bi, wherebis a real number
with units. Compare the probability of finding the electron at this
position with what it would have been if the superposition had
been purely constructive,b+b= 2b.
.The probability per unit volume is proportional to the square of
the magnitude of the total wavefunction, so we have
Poff center
Pcenter
=
|b+bi|^2
|b+b|^2=
12 + 1^2
22 + 0^2
=
1
2
.
Figure u shows a method for visualizing complex wavefunctions.
The idea is to use colors to represent complex numbers, accord-
ing to the arbitrary convention defined in figure u/1. Brightness
indicates magnitude, and the rainbow hue shows the argument. Be-
cause this representation can’t be understood in a black and white
printed book, the figure is also reproduced on the back cover of
printed copies. To avoid any confusion, note that the use of rain-
bow colors does not mean that we are representing actual visible
light. In fact, we will be using these visual conventions to represent
the wavefunctions of a material particle such as an electron. It is ar-
bitrary that we use red for positive real numbers and blue-green for
negative numbers, and that we pick a handedness for the diagram
such that going from red toward yellow means going counterclock-
wise. Although physically the rainbow is a linear spectrum, we are
not representing physical colors here, and we are exploiting the fact
that the human brain tends to perceive color as a circle rather than
a line, with violet and red being perceptually similar. One of the
limitations of this representation is that brightness is limited, so we
can’t represent complex numbers with arbitrarily large magnitudes.
Figure u/2 shows a traveling wave as it propagates to the right.
The standard convention in physics is that for a wave moving in a
certain direction, the phase in the forward direction is farther coun-
terclockwise in the complex plane, and you can verify for yourself
Section 13.3 Matter as a wave 913