t/1. Oscillations can go back
and forth, but it’s also possible
for them to move along a path
that bites its own tail, like a cir-
cle. Photons act like one, elec-
trons like the other.
- Back-and-forth oscillations can
naturally be described by a seg-
ment taken from the real num-
ber line, and we visualize the cor-
responding type of wave as a
sine wave. Oscillations around a
closed path relate more naturally
to the complex number system.
The complex number system has
rotation built into its structure,
e.g., the sequence 1, i, i^2 , i^3 ,
... rotates around the unit circle in
90-degree increments. - The double slit experiment em-
bodies the one and only mystery
of quantum physics. Either type
of wave can undergo double-slit
interference.
A complete investigation of these issues is beyond the scope of
this book, and this is why we have normally limited ourselves to
standing waves, which can be described with real-valued wavefunc-
tions. Figure t gives a visual depiction of the difference between
real and complex wavefunctions. The following remarks may also
be helpful.
Neither of the graphs in t/2 should be interpreted as a path
traveled by something. This isn’t anything mystical about quantum
physics. It’s just an ordinary fact about waves, which we first en-
countered in subsection 6.1.1, p. 354, where we saw the distinction
between the motion of a wave and the motion of a wave pattern. In
bothexamples in t/2, the wave pattern is moving in a straight line
to the right.
The helical graph in t/2 shows a complex wavefunction whose
value rotates around a circle in the complex plane with a frequencyf
related to its energy byE=hf. As it does so, its squared magnitude
|Ψ|^2 stays the same, so the corresponding probability stays constant.
Which direction does it rotate? This direction is purely a matter
of convention, since the distinction between the symbolsiand−iis
arbitrary — both are equally valid as square roots of−1. We can,912 Chapter 13 Quantum Physics