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Hello everybody, and welcome back to 
Exploring Quantum Physics. 

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I'm guest lecturer Ian Appelbaum, and 
I'll be telling you about Stern-Gerlach 

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experiment. 
In the last lecture, we found that 

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incorporating the correct relativistic 
and variance into the quantum mechanical 

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wave equation. 
Constructing the Dirac equation, not the 

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Schrodinger equation, left us with the 
requirement that the electron wave 

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function has two components. 
Our hope is that understanding this 

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degree of freedom will lead to an 
explanation of the anomalism effect, 

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where gas discharge spectral lines are 
split into more than the three components 

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of a Lorentz triplet. 
The Stern-Gerlach experiment, the subject 

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of today's lecture, was instrumental in 
revealing the underlying physics of this 

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so-called anomaly. 
The experimental geometry of the 

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Stern-Gerlach Experiment was something 
like this figure. 

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Where silver was heated in a vacuum so 
that individual atoms with thermal energy 

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kBT of several thousand kelvin were 
emitted from a furnace, and collimated 

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through two sequential slits. 
This beam then passed through a region of 

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high magnetic field gradient 
perpendicular to its velocity. 

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These were neutral atoms, so the magnetic 
field itself did not affect the 

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trajectory. 
However, for atoms carrying a magnetic 

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moment, the gradient of the magnetic 
field will impart a force and deflect the 

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beam. 
We can predict the observed deflection by 

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first calculating force. 
This is just the gradient of potential 

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energy which we know from the dipole 
interaction energy. 

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The displacement is given by the solution 
to the classical equation of motion for a 

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constant force. 
Using the time of flight, length over 

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thermal velocity, we then have a complete 
expression that we can use to obtain 

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quantitative values of deflection. 
For a gradient of about ten Tesla per 

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centimeter over a distance of three 
centimeters. 

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We predicted deflection of 100 microns 
for a moment of one Bohr magnetron 

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oriented parallel to the gradient. 
If the moment is oriented anti parallel, 

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the deflection is in the opposite 
direction. 

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Now if the magnetic moments of the 
neutral atoms are distributed randomly 

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upon emission from the furnace, as 
expected from a classical degree of 

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freedom. 
This deflection will vary continuously 

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and merely result in a broadened beam of 
width times D. 

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So what happened? 
What did Stern and Gerlach actually see? 

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Here's what Stern and Gerlach saw on the 
glass slide removed from the apparatus, 

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after converting the nearly transparent 
deposited silver into black silver 

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sulfide. 
On the left is the deposited pattern from 

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the silver beam without the perpendicular 
field gradient, and on the right is the 

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pattern with the field gradient. 
Overlaying the pole geometry used to 

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create the gradient shows that that 
horizontal deflection is greatest in the 

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middle, simply because the field gradient 
is higher there. 

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The length scale shows that this 
deflection is in the order of our 

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calculated 100 microns. 
But the most remarkable thing about the 

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pattern with the magnetic gradient is 
that it is not simply smeared as expected 

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from randomly oriented magnetic moments, 
but that they're split into two well 

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resolved beams. 
This postcard was sent to Bohr and it 

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says below something like, 
congratulations on confirmation of your 

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atomic theory. 
Stern and Gerlach thought that they were 

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measuring the orbital magnetic moment 
predicted by Bohr's model. 

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But this was 1922 before the Schrodinger 
equation, and the realization that the L 

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equals zero S dates have zero angular 
momentum. 

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It doesn't measure the orbital component, 
they were measuring the magnetic moment 

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of the intrinsic angular momentum of the 
unpaired 5S electron. 

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This is rather unfortunately called spin 
for historical reasons. 

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And the twofold splitting of the atomic 
beam is due to the two possible 

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Eigenvalues of the Z component, plus or 
minus h bar over two. 

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It's then natural to identify an 
additional quantum number, the spin 

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magnetic quantum number M sub S, that 
takes the value of plus or minus one 

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half. 
A so-called g factor, nominally equal to 

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two, due to a relativistic effect called 
Thomas procession from Lorentz boost, 

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must be included in the conversion from 
angular momentum to magnetic moment. 

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We therefore identify the two components 
of the electron part of the Dirac wave 

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function as the amplitudes of spin up and 
spin down. 

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For those interested in the serendipity 
behind this experiment, and the 

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historical development of the correct 
interpretation. 

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I rem, recommend this popular account by 
Friedrich and Herschbach in Physics 

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Today, very highly. 
Incidentally, long after this extremely 

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important experiment, Stern won the Nobel 
Prize during World War II. 

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But Gerlach was snubbed, in part because 
he stayed in Germany to work on wartime 

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weapons development for the Nazis. 
I said it was unfortunate that the word 

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we use for electronic intrinsic magnetic 
moment is spin. 

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And this is because, although it's 
natural to try to understand how an 

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individual particle can carry intrinsic 
angular momentum, there's no classical 

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analog to it. 
Still, one can find figures like this 

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propagating a false notion in text books. 
In fact, the idea of spin was at first 

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ridiculed for the very reason that the 
geometrical interpretation leads to 

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nonsense such as this. 
If an electron is a spinning charged 

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sphere, then what's the velocity at the 
electron surface? 

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We can calculate the classical electron 
radius, by equating the electrostatic 

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potential energy needed to assemble 
charge E into the sphere with the 

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electron rest mass. 
And then use it to determine the velocity 

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necessary to generate one Bohr magneton. 
The result is the speed of light divided 

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by the fine structure constant. 
A dimensionalist quantity equal to about 

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one over 137, far less than unity. 
The velocity then would have to be over 

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two orders of magnitude higher than the 
speed of light. 

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Clearly nonsense. 
This issue played an important role in 

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history. 
Goudsmit and Uhlenbeck are credited with 

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the correct interpretation of the 
Stern-Gerlach experiment in 1925, as 

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being due to the intrinsic electron 
angular momentum. 

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But perhaps only because they were lucky 
enough to have an open-minded advisor, 

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Ehrenfest, who said, well, that's a nice 
idea, though it may be wrong. 

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But you don't yet have a reputation, so 
you have nothing to lose. 

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Whereas the same idea had apparently been 
suggested several years before by Kronig, 

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who was thoroughly discouraged by his 
mentor Pauli, who said, it's indeed very 

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clever, but of course has nothing to do 
with reality. 

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Llewellyn Thomas, who introduced the 
Thomas G factor, wrote a humorous letter 

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to Goudsmith after his paper's 
publication saying the following. 

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I think you and Uhlenbeck have been very 
lucky to get your spinning electron 

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published and talked about before Pauli 
heard of it. 

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It appears that more than a year ago 
Kronig, believed in the spinning electron 

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and worked out something. 
The first person he showed it to was 

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Pauli. 
Pauli ridiculed the whole thing so much 

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that the first person became also the 
last, and no one else heard anything of 

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it. 
Which all goes to show that the 

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infallibility of the Deity does not 
extend to his self-styled vicar on Earth.