Hello everybody, and welcome back to Exploring Quantum Physics.
I'm guest lecturer Ian Appelbaum, and I'll be telling you about Stern-Gerlach
experiment. In the last lecture, we found that
incorporating the correct relativistic and variance into the quantum mechanical
wave equation. Constructing the Dirac equation, not the
Schrodinger equation, left us with the requirement that the electron wave
function has two components. Our hope is that understanding this
degree of freedom will lead to an explanation of the anomalism effect,
where gas discharge spectral lines are split into more than the three components
of a Lorentz triplet. The Stern-Gerlach experiment, the subject
of today's lecture, was instrumental in revealing the underlying physics of this
so-called anomaly. The experimental geometry of the
Stern-Gerlach Experiment was something like this figure.
Where silver was heated in a vacuum so that individual atoms with thermal energy
kBT of several thousand kelvin were emitted from a furnace, and collimated
through two sequential slits. This beam then passed through a region of
high magnetic field gradient perpendicular to its velocity.
These were neutral atoms, so the magnetic field itself did not affect the
trajectory. However, for atoms carrying a magnetic
moment, the gradient of the magnetic field will impart a force and deflect the
beam. We can predict the observed deflection by
first calculating force. This is just the gradient of potential
energy which we know from the dipole interaction energy.
The displacement is given by the solution to the classical equation of motion for a
constant force. Using the time of flight, length over
thermal velocity, we then have a complete expression that we can use to obtain
quantitative values of deflection. For a gradient of about ten Tesla per
centimeter over a distance of three centimeters.
We predicted deflection of 100 microns for a moment of one Bohr magnetron
oriented parallel to the gradient. If the moment is oriented anti parallel,
the deflection is in the opposite direction.
Now if the magnetic moments of the neutral atoms are distributed randomly
upon emission from the furnace, as expected from a classical degree of
freedom. This deflection will vary continuously
and merely result in a broadened beam of width times D.
So what happened? What did Stern and Gerlach actually see?
Here's what Stern and Gerlach saw on the glass slide removed from the apparatus,
after converting the nearly transparent deposited silver into black silver
sulfide. On the left is the deposited pattern from
the silver beam without the perpendicular field gradient, and on the right is the
pattern with the field gradient. Overlaying the pole geometry used to
create the gradient shows that that horizontal deflection is greatest in the
middle, simply because the field gradient is higher there.
The length scale shows that this deflection is in the order of our
calculated 100 microns. But the most remarkable thing about the
pattern with the magnetic gradient is that it is not simply smeared as expected
from randomly oriented magnetic moments, but that they're split into two well
resolved beams. This postcard was sent to Bohr and it
says below something like, congratulations on confirmation of your
atomic theory. Stern and Gerlach thought that they were
measuring the orbital magnetic moment predicted by Bohr's model.
But this was 1922 before the Schrodinger equation, and the realization that the L
equals zero S dates have zero angular momentum.
It doesn't measure the orbital component, they were measuring the magnetic moment
of the intrinsic angular momentum of the unpaired 5S electron.
This is rather unfortunately called spin for historical reasons.
And the twofold splitting of the atomic beam is due to the two possible
Eigenvalues of the Z component, plus or minus h bar over two.
It's then natural to identify an additional quantum number, the spin
magnetic quantum number M sub S, that takes the value of plus or minus one
half. A so-called g factor, nominally equal to
two, due to a relativistic effect called Thomas procession from Lorentz boost,
must be included in the conversion from angular momentum to magnetic moment.
We therefore identify the two components of the electron part of the Dirac wave
function as the amplitudes of spin up and spin down.
For those interested in the serendipity behind this experiment, and the
historical development of the correct interpretation.
I rem, recommend this popular account by Friedrich and Herschbach in Physics
Today, very highly. Incidentally, long after this extremely
important experiment, Stern won the Nobel Prize during World War II.
But Gerlach was snubbed, in part because he stayed in Germany to work on wartime
weapons development for the Nazis. I said it was unfortunate that the word
we use for electronic intrinsic magnetic moment is spin.
And this is because, although it's natural to try to understand how an
individual particle can carry intrinsic angular momentum, there's no classical
analog to it. Still, one can find figures like this
propagating a false notion in text books. In fact, the idea of spin was at first
ridiculed for the very reason that the geometrical interpretation leads to
nonsense such as this. If an electron is a spinning charged
sphere, then what's the velocity at the electron surface?
We can calculate the classical electron radius, by equating the electrostatic
potential energy needed to assemble charge E into the sphere with the
electron rest mass. And then use it to determine the velocity
necessary to generate one Bohr magneton. The result is the speed of light divided
by the fine structure constant. A dimensionalist quantity equal to about
one over 137, far less than unity. The velocity then would have to be over
two orders of magnitude higher than the speed of light.
Clearly nonsense. This issue played an important role in
history. Goudsmit and Uhlenbeck are credited with
the correct interpretation of the Stern-Gerlach experiment in 1925, as
being due to the intrinsic electron angular momentum.
But perhaps only because they were lucky enough to have an open-minded advisor,
Ehrenfest, who said, well, that's a nice idea, though it may be wrong.
But you don't yet have a reputation, so you have nothing to lose.
Whereas the same idea had apparently been suggested several years before by Kronig,
who was thoroughly discouraged by his mentor Pauli, who said, it's indeed very
clever, but of course has nothing to do with reality.
Llewellyn Thomas, who introduced the Thomas G factor, wrote a humorous letter
to Goudsmith after his paper's publication saying the following.
I think you and Uhlenbeck have been very lucky to get your spinning electron
published and talked about before Pauli heard of it.
It appears that more than a year ago Kronig, believed in the spinning electron
and worked out something. The first person he showed it to was
Pauli. Pauli ridiculed the whole thing so much
that the first person became also the last, and no one else heard anything of
it. Which all goes to show that the
infallibility of the Deity does not extend to his self-styled vicar on Earth.