Re: [Phys-L] phase of a classical wavefunction

[John Denker <jsd@av8n.com>]

On 01/21/2015 10:38 AM, Carl Mungan wrote:
> Few intro texts that I am aware say we should have flipped the sign 
> of the kx term instead of the wt term.

Agreed.  Even Feynman drops the ball on this one:
  http://www.feynmanlectures.caltech.edu/I_49.html

=================

One way to stay out of trouble when dealing with "phase"
issues is to switch to the /phasor/ representation:

  ψR(x,t) = a1 sin( kx - ωt) + a2 cos( kx - ωt)    [1]

  ψL(x,t) = b1 sin(-kx - ωt) + b2 cos(-kx - ωt)    [2]

The phase angle does not appear directly in these equations
... although you can compute it in terms of atan2(a1, a2)
if desired.

The advantage here is that ψ is a linear function of
a1 and a2.  In the phase-and-amplitude representation,
ψ would be a linear function of the amplitude, but a 
nonlinear (indeed transcendental) function of the phase.

> Of course, one might also argue that the reflection coefficient
> really should have an absolute value in it. If it's really a ratio of
> amplitudes, it can never be negative. But I think we know what's
> intended.

It's hard to know whether such things should be defined
as always positive.  In particular, it's not obvious to
me that the amplitude should be defined as always positive.
If you write ψ in terms of |something| it becomes a
nonlinear function of that thing, which is sometimes
very unpleasant.  Sometimes it is better to accept an
/overcomplete/ representation, where the sign of the
amplitude can be flipped (whereupon the phase of ψ 
is offset by 180 degrees).  The phase is already open
to fudging by any multiple of 360 degrees, so this
doesn't make anything worse.

Again, switching to the phasor representation makes
all these conundrums go away.  Even though the phase
and amplitude are not uniquely determined (except
maybe by convention), the values of a1 a2 b1 b2
are unique and well behaved.