It is, we are often told, quite the miracle that our boffins can
perform quantum mechanical calculations in exquisitely fine detail
while having a completely indefinite conception of what the equations
of quantum mechanics refer to. But this is very false. A classically
trained Martian, upon leafing through those books of chemistry and
atomic, molecular, and solid state physics from which the boffins
as a matter of fact learn how to solve the universe (rather than
the dustier scriptures of pure quantum mechanics), would surely
conclude that quantum mechanics was about electrons in 'orbitals'
of various shapes and sizes which made up benzene molecules, sodium
chloride crystals, pork chops and the rest. Our Martian friend would
find quite definite pictures of these orbitals, accompanied by
explanations of how the intricate features of their shape explained
the gross properties of the matter at hand and the fine structure
of every kind of spectrum obtained in probing it, and further
accompanied by calculations which use these orbitals as an essential
basis.
So as aspiring boffins the question is not how we can calculate
with no conception at all of the world, but how our everyday belief
in orbitals can be so disconnected from what we tell people we
believe (which is roughly that the world doesn't exist but if it
did it would hardly matter). The reason is apparently simple. The
equations concern a function on a high-dimensional space; our
ordinary space, home of the orbitals, is nowhere to be seen. It
would seem obvious to try to derive our orbitals from this exotic
function on an exotic space; indeed it was obvious to Schroedinger
as he developed wave mechanics. It was understood by both the friends
and foes of wave mechanics that even when it used only a high-dimensional
space it was more intuitive (anschaulich) than the radical matrix
mechanics (which proudly discarded space-time description). But we
tend to hear that Schroedinger's intuitive wave description could
not be maintained (since it was on the high-dimensional space) and
a supposedly equally intuitive particle interpretation of matrix
mechanics simply must be taken up - this we hear, perhaps not by
coincidence, from Schroedinger's enemies and their descendants. In
reality Schroedinger straightaway looked at how to derive a system
of waves in our familiar space from the one wave on a high dimensional
space - these individual waves in everyday space would of course
be our hard-working orbitals.
Schroedinger's first scheme seems unappealing since for an antisymmetric
electron wavefunction (which is just to say for an electron
wavefunction) it gives all electrons the same orbital, which is
hardly what we see. Embracing the antisymmetry Schroedinger further
looked to draw the orbitals from a second-quantized description,
but here we have the problem that a generic wavefunction cannot be
created as just one combination of orbitals (not every antisymmetric
function is a single wedge/outer product, i.e. a single Slater
determinant). But there is a fairly simple way to derive a family
of single-particle wavefunctions, of orbitals, from any many-particle
wavefunction - these are the so-called natural orbitals (which are
'just' the eigenfunctions of the one-particle reduced density
matrix). I suggest that these natural orbitals are exactly the
orbitals our Martian friend saw in our books, and that if we take
them as seriously in our philosophy as our boffins take them in
their calculations, we are led to a conception of the world which
is not just consistent but helpful - a conception which was early
on acknowledged as the ideal intuitive one if only we could actually
tie it to the equations (as we already jump to in those cases where
it is very clear how to).
