Re: [Phys-l] Quantum of action

[John Denker <jsd@av8n.com>]

On 03/04/2012 04:51 PM, David Bowman wrote:

> I agree with JD that h happens to be the mean area occupied per basis
> state in phase space.  This shows up quite nicely when those states
> are given their Wigner function representation as real-valued (but
> not necessarily positive definite) phase space functions.

Point taken.  I should have added a caveat that the Wigner
function is not a complete description of the state.  Such
things are great for making gee-whiz pictures, but you wouldn't
want to calculate anything serious with them.
 
> But I've always considered the actual deeper *meaning* of Planck's
> constant as it actually being a unit *conversion factor* not all that
> unlike the causal speed limit c being a conversion factor between
> intervals of spacetime denominated in temporal units and those
> denominated in spatial units.  In the particular case of Planck's
> Constant h-bar is the conversion factor between an elapsed action for
> some classical path in phase space and the corresponding phase shift
> in radians of the superposed quantum amplitude of that path between
> the initial and final states.  Whereas h is the conversion factor
> between elapsed action along a path and the number of full cycles of
> phase shift of superposed quantum amplitude.  The factor of 2*[pi]
> between h-bar and h reflects the fact that there are 2*[pi] radians
> of phase in a full cycle of phase.  This identification of h/h-bar as
> conversion factors between elapsed action and complex phase angle
> shows up most strikingly in the Feynman path integral formulation of
> quantum mechanics.

That's true and important and well said.  That is indeed more
fundamental than what I said.

I would just add that it is entirely /consistent/ with what I said.

Indeed, this goes a long way toward answering the last part of the
original question, where it asks "is there a way to derive" this
stuff.  The answer is that you can derive it using the path integrals.
This is not for the faint of heart, but it can be done.

In particular, thermodynamics can be considered the analytic continuation
of quantum mechanics (and vice versa).  The classical limit is obtained
in QM by stationary phase, and in thermodynamics by steepest descent.
It's the same idea, with or without a factor of i in there.  The
Heisenberg uncertainty principle and the second law of thermodynamics
are seen to be intimately related.

States that differ enough in phase will be linearly independent.  The
value of Planck's constant tells you how much action it takes to accomplish
that.  This is crucial when you are trying to construct a basis set.

The wikipedia page on path integrals looks like it was written by somebody
who actually knows what he's talking about, which is quite remarkable.
  http://en.wikipedia.org/wiki/Path_integral_formulation

It includes an annotated "Suggested Reading" section.