Re: teaching orbitals in the back yard

["John S. Denker" <jsd@MONMOUTH.COM>]

At 05:28 PM 12/18/99 -0500, I wrote:

> 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.

I should have been more explicit.  I recognize that not everybody has 100%
reliable intuitions about this stuff.  I could write out the mathematical
description of the P+ and P- orbitals, but it is more fun to think about
the qualitative physics.  So here goes:

When I was a little kid, we had a pool in the back yard.  It was circular,
about 25 feet across and 4 feet deep.  That's ideal for larval physicists.

I have vivid memories of creating wave patterns in that pool.
 * Of course the simplest is the 1S state:  you leave the water alone and
it just sits there.
 * Next comes the 2S state:  you stand in the middle of the pool and jump
up and down.  If you do it right, you make a wave that has circular
symmetry.  When the middle goes up, the edge goes down, and there is one
circular node about halfway out.
 * Next comes the 2Px state:  You stand at the +X edge of the pool and
jump up and down.  If you do it right, one edge goes up while the other
edge goes down.  There is one node along the X=0 diameter of the
pool.  These modes are rather lightly damped.  A small kid, hitting the
mode on resonance, can stir up quite a big wave, enough to slosh quite a
bit of water clear out of the pool. Guess how I know.
 * Similarly, you can make the 2Py state:  Go over to the +Y edge of the
pool and jump up and down.
 * (There are no Pz states in this system, because it is a 2D system not a
3D system).
 * You can make 3S states et cetera.
---------------

For present purposes, let's focus attention on the P+ orbital.  It is a
running wave, running counterclockwise around the edge of the pool.  There
is a fixed point in the middle, and that's all.  Suppose the Px and Py
orbitals can be written as
  Amplitude = PxShape(x,y) cos(omega t)
  Amplitude = PyShape(x,y) cos(omega t)

Then the P+ orbital can be written as
  Amplitude = PxShape(x,y) cos(omega t) + PyShape(x,y) sin(omega t)
              -----------------------------------------------------
                             sqrt(2)

where I have rewritten it without complex numbers, changing cos to sin
where required.  This is a running wave, but it is stationary in the sense
that its energy is independent of time.

If you want to see the complex-number version, the phasor expression is:
        P+ := sqrt(1/2)(Px + i Py)
        P- := sqrt(1/2)(Px - i Py)

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

The point about all this talk about swimming pools is that the symmetry of
orbitals has nothing to do with chemistry, or even with the physics of
electrons.  It has to do with waves in general, and the geometry of the world.