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Re: orbitals, dumbbell and otherwise



At 02:56 PM 12/18/99 -0600, Jack Uretsky wrote:

Quantum mechanics is not just an annoying correction to classical
mechanics. QM rules the universe; classical mechanics is an approximation
that works in certain extreme circumstances.

Amen, brother!

--------------

There has been a lot of discussion of orbitals, and whether they have
circular symmetry, cylindrical symmetry, or otherwise.

Here's my $.02:

A classical circular orbit has rotational symmetry. A *particle* in that
orbit breaks the symmetry; we need to take some sort of average or
probability to recover the symmetry.

The quantum mechanical Pz orbital is invariant under rotations around the Z
axis. A circle in the XY plane also has this symmetry. However, the
circle lies in the plane where Z=0, while the Pz orbital has exactly zero
amplitude when Z=0. Neither the orbital nor the circle has cylindrical
symmetry; the Z-coordinate is not negligible. The Pz orbital is well
described as dumbbell shaped. It is not well represented by a circular
orbit, and when we pass to the classical limit it does not contribute to
the construction of the classical circular orbit.

Note the Pz orbital has additional symmetries, such as parity, which need
not concern us here.

Meanwhile, back at the ranch, we have the Px and Py orbitals. For present
purposes, it is advantageous to take linear combinations thereof, to
construct the P+ and P- orbitals. The latter are the critters that
correspond most closely to the classical circular orbit. If we square them
we find that the probability has its maximum in the XY plane and is
invariant under rotations around the Z axis. They are presumably what
Ludwik had in mind when he spoke of something orbiting something else.