Re: Imaginary reality

["John S. Denker" <jsd@MONMOUTH.COM>]

At 11:11 AM 4/21/00 -0700, Daniel Schroeder wrote:
> Why does the wavefunction have to be a two-component object?  Because
> a single component could contain all the needed information about
> the position of a particle but not also the momentum (or vice-versa).
> If the wavefunction is to encode a *complete* description of the
> state of a particle, then one component simply isn't enough.
> For example, try to find a way to use a single-component function
> to describe a particle with a well-defined momentum (including
> direction) and completely undefined [position] (equal chance of
> finding it at any x).

That's all true as far as it goes, but it's not the only possible answer to
the original question.  Note that the wavefunction that started this thread,
        y=Aexp(i[kx-wt])                Eq. 1
could indeed be the solution to a quantum wave equation, but it could
equally well be a solution to a completely classical wave equation
(compression waves on a slinky, transverse waves on a string, EM waves in a
coax, et cetera).

A mathematician would say that there are two linearly independent solutions
"because" we have a second-order linear differential equation.  (This begs
the question of why so many things in nature are well described by
second-order linear differential equations.  For homework, list three
interesting physical processes that are _not_ well described by
second-order linear diffeqs.)

Anyway, given two real solutions to the wave equation, equation 1 is a
convenient way to bundle both of them into one complex equation.

> Given that the wavefunction has to be a two-component object, why
> is it so convenient to use complex numbers to encode the two
> components?  Beats me.  Always seemed a little spooky, to tell
> the truth.

Why spooky?  There are lots of cases where we do similar things, e.g.
bundling 3 or 4 scalar equations into one vector equation.  Whenever there
are N solutions to the same equation, it is just a matter of time before
somebody contrives a formalism that allows us to bundle them together.