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Re: [Phys-L] collisions



On 01/23/2014 03:55 PM, Carl Mungan wrote:
Suppose I have two identical baseballs. I mark one with a black dot.
I use two ball launchers to put them on a collision course with each
other. The marked ball is in the left launcher, the unmarked ball in
the right launcher. I tell you everything about the launch speeds,
angles, positions, etc. They collide behind a barrier that I can't
see around. But then they reappear, one to the left and to the right
after the collision behind the barrier. The game is to predict which
baseball has the black dot on it and then go look at the two balls
and see if you're right.

Now repeat this experiment with two identical atoms. Under what
circumstances (if any) can I still win this game? No fair setting the
launchers so that the atoms don't actually collide and just pass each
other by a large distance.

That describes almost perfectly an experiment my buddies
and I did ... using real atoms.

1) Start with identical hydrogen atoms. Cool them off to a
few thousandths of a degree above absolute zero. Using a
seriously powerful magnet, align all the electron spins and
the nuclear spins.

In this regime, the atoms are not like baseballs. They are
not hard little spheres that bounce off each other. Rather,
they are big fluffy clouds that diffract through each other.
If you go by the thermal de Broglie length, the atoms are
the size of bacteria.

In this regime, you cannot distinguish between the two
cases of right-angle scattering. The add-before-you-square
law applies. You see interference between the two cases.
This effectively doubles the 90-degree scattering amplitude.

2) Interestingly enough, at higher temperatures you /can/
distinguish the two cases. The thermal de Broglie length is
small compared to the Bohr radius, so the atoms act like little
baseballs. The two possible trajectories are offset from one
another by about one atomic diameter, and if you look closely
enough you can distinguish the two cases on this basis. Even
though the particles are nominally identical, the trajectories
are distinguishable, and that's all the quantum mechanical laws
care about. In this situation the square-before-you-add law
gives the right answer. You can quantify all this in terms
of wavefunction phase angles et cetera.

3) Now let's ask about the black dot. Let's go back to the
low temperature case, where quantum-mechanical interference
is observed. Atoms the size of bacteria.

Now ... use NMR to flip the nuclear spin of one of the atoms.
This is your black dot.

This changes everything. The the two cases of scattering
are now 100% distinguishable. No more interference. The
scattering amplitude is different in this situation, even
though the physical interaction is the same. The physical
interaction depends on the electron cloud and doesn't care
what the nucleon is doing.

4) But wait, there's more.

Suppose we have a bottle full of identical atoms, with roughly
atmospheric density. (The pressure is very low, on account
of the temperature.)

Now we use NMR to tilt the spin of one atom by a small
amount, and ask how it interacts with the gas. We
can decompose the tilted atom's wavefunction into the
spin-up component and the spin-down component.
-- The spin-up component will scatter according to
the add-before-you-square law, because it is indistinguishable
from the ambient gas atoms.
-- The spin-down component will scatter according to
the square-before-you-add law, because it is distinguishable,
because it bears the black dot.

So, the two components of the wavefunction will undergo
a different scattering phase shift.

So ... what do you think happens next? What can you
think of that causes one part of the wavefunction to
pick up a phase shift relative to the other part?

Drum roll.....

That's σ_z i.e. the generator of rotations around the
z axis.

In other words, the tilted atom will /precess/ around the
axis defined by the ambient gas. Chemists would call this
a molecular-field effect. The ambient gas is essentially
like a magnetic field that causes the tilted atom to
precess.

Note that this molecular field effect is totally dependent
on the fact that the atoms are basically identical. If
you put a tilted deuterium atom in a gas of polarized
hydrogen atoms, there would be no precession. This proves
it's not really a magnetic effect; it's a molecular-field
effect with the same symmetry as a magnetic field.

5) This means that rather than having atoms diffuse around
in the cell, you get /spin waves/.

The spin wave propagates from one end of the cell to the
other, bounces off the wall, comes back, and oscillates
like the sound in an organ pipe. A picture of some spin
wave resonances is here:
http://www.av8n.com/physics/refs/johnson-et-al-hydrogen-1984-p1.pdf

By changing the temperature (and therefore the de Broglie
length) we can sweep back and forth between completely
classical behavior (grossly overdamped diffusion) and
completely wavelike behavior (hi-Q spin wave resonances)
and everything in between (damped spin waves).


6) We were flabbergasted by the spin wave resonances.
We weren't expecting anything of the sort.

We didn't know it at the time, but it turns out that
E.P. Bashkin had /predicted/ spin waves in spin-aligned
hydrogen a couple of years previously, on theoretical
grounds.

We were mightily impressed that anybody understood the
system well enough to make such a prediction. He
published in the Russian literature, which is why we
didn't see it, although that's a pretty lame excuse.