Subject: Re: CCL:orbital and reality

From: Malcolm E Davis <malcolm.davis _(a)_ bms.com>
Date: Mon, 31 Jan 2005 13:47:03 -0500

Many people have found the QM picture of molecules troubling because of 
various quantum "bizarreness".  However, when push comes to shove, what 
makes it bizarre is that it is just different than our macroscopic, 
billiard ball experience.  Folks have tried to show that there is some 
"normal" reality underlying QM that we just cannot measure.  Such 
"Hidden Variable" theories have been ruled out by "Bell's Inequality" 
and the "Aspect" experiments.  So, wave functions are as real as it 
gets.  This is not to say that QM will never need refinement, but just 
as QM does not negate Classical Mechanics in its region of 
applicability, the refinements are only likely to show QM to be an 
approximation which holds in some limit.  In fact, this has actually 
already happened with QED which "unifies" the electromagnetic force with 
the strong and weak nuclear forces.  So, while it is true that 
scientific theory is always a model of reality, it is of equal validity 
to claim that we may discover that our understanding of gravity is all 
wrong and that it doesn't obey an inverse square law as it is to claim 
that QM will be overturned and that wavefunctions are not "real".

Now about orbitals.  When one solves the time-independent Schroedinger 
equation for a hydrogen atom, one gets a series of "orbital" states that 
the electron can exist in.  These orbitals can be represented as exact 
mathematical constructs, and the hydrogen spectrum indicates that the 
electron does exist only in these discrete states.  For example, the 
electron can be observed to make the transition between the 1s and 2s 
energy states.  The hydrogen atom in its ground state has its electron 
in the 1s orbital.  If you want to debate the reality of this orbital in 
the ground state then you really are off in the realm of philosophy 
where I will now argue that you aren't real, and we will move on without 
you.  As for the "reality" of the unoccupied 2s orbital, I grant that 
there could be some debate about the "reality" of a propensity.  
Nevertheless once the hydrogen atom is excited to the 2s state, the 2s 
orbital gives exactly the wave function of the electron, and I would 
again claim that that is as real as anything gets.

Of course, this is where the physicists stop, and chemists enter.  (You 
know the whole "angels fear to tread" thing.)  The multi-body 
Schroedinger equation for anything more complicated than the hydrogen 
atom precludes exact solution.  But here we take advantage of some other 
mathematical properties of the exact hydrogen states.  They form a 
"complete basis" which just means that any arbitrary distribution can be 
created from the proper linear combination of these states.  In fact it 
is possible to perturb even a hydrogen atom by application of an 
external field such that the stable states are no longer the exact 
hydrogen orbitals, but rather linear combinations of them.  So for these 
more complicated cases, the "exact" solution of the wavefunction is 
approximated in different ways as expansions of these basis functions 
(or approximations thereof).  This does lead to many misrepresentations 
of reality.  For example, one of my favorites is assigning atomic 
"charges" by adding up the coefficients of the basis functions centered 
on an atom and then making claims about partial charge transfer between 
non-bonded atoms.

So in the muddier world of chemistry there are approximations galore 
relative to the full time-dependent multi-body Schroedinger equation of 
QM.  (Maybe there are even ways that QED could be brought into the 
equation as it seems folks are making more and more macroscopic QM 
manifestations everyday, see for example work on Bose-Einstein 
condensates).  Nonetheless the "exact" solutions to the multi-body 
Schroedinger equation exist.  That we can't represent them as simple 
mathematical functions does not preclude their existence.  Moreover, the 
fact that molecular spectra can be measured indicates that they are 
"real." 

All that said, there are certainly times when a simple approximate 
theory is more useful than a much more difficult to calculate more exact 
theory.  For example, one would not bother to do a full QM treatment of 
the orbital mechanics of a satellite.  I see this as where some of the 
arguments of "this" representation vs. "that" representation come in.  
For the satellite Classical Mechanics works.  This does not mean that QM 
does not govern the orbit.  It just means that QM is an impractical 
level of detail for the problem at hand.  It isn't a matter or 
either-or.  In the same vein, for a lot of chemistry one can push dots 
around or apply rules which don't reference "orbitals" in any way.

Cheers,
Malcolm Davis