Re: unweirdness of QM wave equation

[John Denker <jsd@MONMOUTH.COM>]

At 11:39 PM 4/23/00 -0400, I wrote (1)
> There is a principle of pedagogy that says:
>      "learning proceeds from the known to the unknown."

and (2)

> There are some weird things that go on in the
> quantum world, but the wave equation isn't one of them.

Let me be more explicit about statement (2).  Consider the following
coupled differential equations:
         d/dt b + d/dx a = 0            (eq 3)
         d/dt a + d/dx b = 0

In the spirit of statement (1), this should be familiar to all students
_before_ they begin serious study of quantum mechanics.  Simply as abstract
mathematics, they should be able to find wavelike solutions to these
equations.  As physics, they should recognize the option of interpreting
these equations as components of the Maxwell equations (a = E_y, b = B_z).

The students should know how to represent the solutions in terms of real
scalar functions *and* how to represent them in terms of complex numbers.

The students should be able to eliminate one variable in favor of the other
to yield
        (d/dt)^2 a - (d/dx)^2 a = 0     (eq 4)

===

Moving from the known to the almost-as-well-known, consider another set of
coupled differential equations, similar to eq (3) but slightly fancier:
        -d/dt b + (d/dx)^2 a = 0        (eq 5)
         d/dt a + (d/dx)^2 b = 0

Again, before beginning a serious study of quantum mechanics, the students
should be able to find wavelike solutions to these equations.

The students should be able to eliminate one variable in favor of the other
to yield
        (d/dt)^2 a + (d/dx)^4 a = 0     (eq 6)

which can optionally be interpreted as the wave equation for transverse
waves on a stiff rod as discussed on phys-l a couple of months ago.  The
dispersion relation is different from what you'd get for an unstiff string
under tension, but it's nothing to flip out about.

Moving from the known to the possibly unknown:  Students should know that
in the real world, not all objects move at the same speed.  Therefore it
should not be too shocking to find that the foregoing wave equations
describe waves that don't all move at the same speed.  For water waves,
stiffness waves, electromagnetic waveguide waves, etc., speed depends on
wavelength.  It's only in high-school physics that all waves move at the
same speed.

As before, the students should know how to represent the solutions in terms
of real scalar functions *and* how to represent them in terms of complex
numbers.

Finally, moving from the known to the possibly unknown:  Did you notice
that the coupled equations (eq 5) are the Schrödinger equation?  The
variable (a) is the real part of the wavefunction, while the variable (b)
is the imaginary part.

Therefore I ask again: help me out here, folks.  If you can think of any
physical, mathematical, or pedagogical reason for saying that QM wave
equations are qualitatively different from classical wave equations, please
explain.
 -- Having multiple components (a and b) to the dependent variable isn't
special;  that should be familiar from E and M.
 -- Having a nontrivial dispersion relation isn't special, either;  that
should be familiar from the ripple tank.
 -- So what's the fuss about?