right-hand half of the picture much greater than it would actually
be.)
A paradox resolved example 23
Consider the following paradox. Suppose we have an electron
that is traveling wave, and its wavefunction looks like a wave-train
consisting of 5 cycles of a sine wave. Call the distance between
the leading and trailing edges of the wave-trainL, so thatλ =
L/5. By sketching the wave, you can easily check that there are
11 points where its value equals zero. Therefore at a particular
moment in time, there are 11 points where a detector has zero
probability of detecting the electron.
But now consider how this would look in a frame of reference
where the electron is moving more slowly, at one fifth of the speed
we saw in the original frame. In this frame,Lis the same, butλ
is five times greater, becauseλ=h/p. Therefore in this frame
we see only one cycle in the wave-train. Now there are only 3
points where the probability of detection is zero. But how can this
be? All observers, regardless of their frames of reference, should
agree on whether a particular detector detects the electron.
The resolution to this paradox is that it starts from the assumption
that we can depict a traveling wave as a real-valued sine wave,
which is zero in certain places. Actually, we can’t. It has to be
a complex number with a rotating phase angle in the complex
plane, as in figure u/2, and aconstantmagnitude.Linearity of the Schr ̈odinger equation
Some mathematical relationships and operations arelinear, and
some are not. For example, 2√ ×(3+2) is the same as 2×3+2×2, but
1 + 1 6 =
√
1 +
√
- Differentiation is a linear operation, (f+g)′=
f′+g′. The Schr ̈odinger equation is built out of derivatives, so
it is linear as well. That is, if Ψ 1 and Ψ 2 are both solutions of the
Schr ̈odinger equation, then so is Ψ 1 +Ψ 2. Linearity normally implies
linearity with respect both to addition and to multiplication by a
scalar. For example, if Ψ is a solution, then so is Ψ + Ψ + Ψ, which
is the same as 3Ψ.
Linearity guarantees that the phase of a wavefunction makes no
difference as to its validity as a solution to the Schr ̈odinger equa-
tion. If sinkxis a solution, then so is the sine wave−sinkxwith
the opposite phase. This fact is logically interdependent with the
fact that, as discussed on p. 894, the phase of a wavefunction is un-
observable. For measuring devices and humans are material objects
that can be described by wavefunctions. So suppose, for example,
that we flip the phase of all the particles inside the entire labora-
tory. By linearity, the evolution of this measurement process is still
a valid solution of the Schr ̈odinger equation.
The Schr ̈odinger equation is a wave equation, and its linearitySection 13.3 Matter as a wave 915