Re: phase in quantum optics

From: John Denker (jsd_at_av8n.com)
Date: 10/03/04 

Date: 3 Oct 2004 20:07:57 GMT

Edward Green wrote:
>
> I've noticed recently, for what it's worth, how much of the
> terminology of quantum mechanics is really an exact and appropriate
> application of classical terms, which, in the minds of students and
> students of students, take on an association through unfamiliarity:
> the "wavefunction", as you say, is really an example of a broader
> class of objects called wavefunctions, the "state vector" really is a
> vector in a vector space and the "expectation value" really is an
> expected value or mean of a random variable. But it is almost
> impossible not to receive the initial impression these are special
> terms coined for that mysterious entity called "quantum mechanics".

Not to mention terms like Lagrangian and Hamiltonian! The
story of how two dudes who worked in the late 1700s and early
1800s got their names on two of the most central objects of
QM is quite amusing.
   http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html
   http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html

The answer to the riddle is that wave mechanics is wave mechanics,
and large parts of QM are just wave mechanics. The principles of
mechanics were worked out a long, long time ago. The guys who
invented QM were very well educated in classical mechanics, so
its no surprise that they correctly imported the relevant ideas
and terminology.

Of course there are other parts of QM, including matrix mechanics
(spin and antiparticles) that were a radical departure from
classical ideas.

It is a good exercise and/or test of your understanding of quantum
mechanics to connect it to classical mechanics. For starters,
write out the Lagrangian for an electrical LC oscillator, then
choose canonical coordinates and work out the eqn of motion. For
another, give a thorough description of a classical _field_ such
as the displacement of a piano string, or the signal in a coax.
For fun generalize to waveguides, not just coax. Ken Wilson was
always adept at doing this, to the point where he would mix
metaphors and mix terminology: he might speak of the "rest mass"
of a waveguide mode, when saying "cutoff frequency" would have
been more conventional. It drove some of his colleagues crazy.
They thought it was eccentric, incomprehensible, and possibly
wrong. (You can guess which side of this debate *I* came down
on. My attitude is "the same equations have the same solutions"
and it gave me the creeps to realize that not everybody saw it
this way.)
   http://nobelprize.org/physics/laureates/1982/

> .... about only relative phase being
> physically meaningful. I was going to ask what the limits of this
> idea were: classically the absolute phase often seems to have physical
> meaning -- consider a travelling transverse sine wave on a string...

I don't buy it. Gauge invariance is fundamental. If it appears
to be broken, you should look around and figure out what broke
it.
  ++ If you launch a traveling wave by whacking a string, then
what matters is the *relative* time between whack and observation.
Your innate sense of elapsed time provides a reference, if needed.
  -- OTOH if an adversary launches the wave and keeps the launch-
time secret from you, you'll have a hard time making much sense
of the "absolute" phase.
  ++ If you have a finite string, and the waves are reflecting off
the ends, that will break the symmetry.
  ++ etc.

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