Re: IONS in metals

[Maurice Barnhill <mvb@UDel.Edu>]

James Mclean wrote:
> Ludwik Kowalski wrote:
> > Bob Sciamanda wrote:
> >
> > > Forces are human inventions which help us model our
> > > observations. However, quantum mechanical models are
> > > not always simply understandable in terms of this concept.
> >
> > As far as I remember, the central concept of QM is potential, V.
> > We must specify a potential function to write down the
> > Schroedinger equations for specific problems, for example,
> > Coulomb's potential in the case of the single hydrogen atom.
> > What prevents me from using F=-grad(V) and to think that
> > there is a force behind any smooth potential function?
>
> I think what Bob was emphasizing is that in QM, the Pauli exclusion
> principle "pushes things around", but is not describable as a force.
> That's not to say that -grad(V) isn't important or sensable;  it's just
> that it is no longer the whole story.
>
> --
> --James McLean
> jmclean@chem.ucsd.edu
> post doc
> UC San Diego, Chemistry

I think that the situation is a little worse than that.  Ehrenfest's
Theorem states that  d<p>/dt = - <grad V>, which is as close as you get
to F=ma in the Schroedinger Equation.  Since  [x,p]<>0, you can't even
define a p(x) in order to construct a point-by-point analogy to Newton's
Second Law.  So you can define the force as F = - grad V, but you don't
have an equation to put it into.  When we talk about forces in Quantum
Mechanics we are talking in metaphors and/or trying to understand
semiclassical limits.  Of course, talking that way can be very helpful
to your intuition.

You can go to momentum space so that dp/dt is well defined, but then you
lose V(x).  You can't win, except in the semiclassical limit.
-- 
Maurice Barnhill, mvb@udel.edu                
http://www.physics.udel.edu/~barnhill/          
Physics Dept., University of Delaware, Newark, DE 19716