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Re: Imaginary reality



At 10:29 AM 4/21/00 -0400, TIMOTHY WAGNER wrote:
As a retired teacher I cannot remember how the complex wave equation
y=Aexp(i[kx-wt])

Call that equation 1. That's not what people typically refer to as "the
wave equation". Typically the ordinary wave equation is
(d/dt)^2 y - (d/dx)^2 y = 0

and the simplest "complex" wave equation is the free-space Schrödinger equation
i (d/dt) y - (d/dx)^2 y = 0

Equation 1 (for suitable values of k and w) represents a _solution_ to such
a wave equation, but it isn't the wave equation itself.

was first encountered in my own education. Recently I
began to wonder how this would be introduced, anyway. Accepting the
necessity of its use in QM, I wanted to ask how teachers today related
this complex expression to mind's-eye reality early in the educational
process, or whether they would simply show the concept as a math tool
to have practical applications later.

The answer depends on details of the question....

1) Suppose we emphasize the part of the question that says "early in the
educational process". Then by and large, the answer is "early on, they
don't mention it all". In particular, if the postings to phys-l are
representative, it seems that in the "early" courses the (d/dt^2 - d/dx^2)
wave equation is characterized as "the" wave equation, and the (i d/dt -
d/dx^2) wave equation is firmly pushed over the horizon. This eliminates
any questions as to how the latter equation might be interpreted.

2) Suppose "early" doesn't mean quite so early. Typically by the time the
Schrödinger equation is introduced, the students already know enough about
wave mechanics, imaginary numbers, and probabilities that the
interpretation of the Schrödinger equation poses negligible additional
burdens.

3) Suppose we focus on the subject line rather than the body of the
message. That is, we suppose that the point of the question concerns the
interpretation of probability amplitudes, which is a question that can be
asked independently of any wave equation. As an example, the superposition
laws governing the silver atoms in a Stern-Gerlach machine can be explored
without ever writing down the wave equation that propagates silver atoms
from place to place. An introductory yet thorough example can be found in
_The Feynman Lectures on Physics_, volume III.

4) If it's complex numbers themselves that are causing the problem,
remember that you can _always_ replace complex numbers such as (p + iq) by
vectors in the (p,q) plane. I have done this on a few occasions where I
suspected people were engaging in "cargo cult physics", i.e. applying the
formalism without understanding the interpretation or respecting the
limitations of the formalism. On one such occasion I caught a genuine
error; on the other occasions it turned out that the "fast and loose"
derivation was producing the right answer.

As applied to equation 1, this means that you can write y(x,t) in terms of
sines and cosines (or sines and phase angles) without complex numbers at
all. In particular, I recommend doing so if there are any nonlinearities
around; the theorem (Parseval's theorem) that says |y|^2 = |A|^2 only
applies to _linear_ systems.

In many situations in electrical engineering and elsewhere, equation 1 is
sloppy shorthand for
y = RealPart[Aexp(i[kx-wt])]
Keep this in mind if there is any nonlinearity (even something as humble as
the power dissipated in a resistor).

================
The following pedagogical elements have a partial order:
a) Propagation of a classical wave, using sines and cosines
b) Interference phenomena involving classical waves
c) Complex numbers in general
d) Complex exponentials
e) Application of complex exponentials to waves
f) Probability
g) Complex probability amplitudes
h) Wavelike propagation of probability amplitudes

and the partial orderings include
b>a
d>c
e>d, e>b
g>f, g>d
h>g, h>e