Original web page can be found at
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/SternGerlach/SternGerlach.html
This is just a copy as allowed provided no commercial use is made of the material and the author is acknowledged.
This
page summarises the classic SternGerlach experiment on "spin" and
extends the treatment to a discussion of correlation experiments. As is often
the case, I build up maximum complexity as I examine the experimental details,
and then hide them in a 'box'. This time the box will turn out to be literal.
Here
we concentrate on electrons, which have only two spinstates. We also mention
photons, which also have two spinstates. The approach is largely based on one
by Feynman which he used for objects with three spin states: see R.P. Feynman,
R.B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol III,
Chapter 5 for this discussion.
We begin by considering a macroscopic
charged ball that is thrown between the poles of a magnet. If the ball is not
spinning, a "knuckleball" to a baseball fan, it will not be
deflected. However, if it is spinning it will be deflected as shown: We ignore:


For
the case shown above, a close up view of the ball shows that it is spinning
as shown to the right: We
shall call this orientation "spin up" since it is deflected up by
the magnets. The
total amount of deflection is a function of



By
contrast a "spin down" electron would have its spin oriented as
shown to the left: 

Such
an object is deflected down by the magnets. 
All
of the above is just classical 19th century electricity and magnetism.
An "electron gun" produces a
beam of electrons. Further information may be found here. If the beam from the electron gun is
directed to the magnets, as shown to the right, the beam is split into two
parts. One half of the electrons in the beam are deflected up, the other half
were deflected down. The amount of deflection up or down is exactly the same
magnitude. Whether an individual electron is deflected up or down appears to
be random. Stern and Gerlach did a version of this
experiment in 1922. 

This
is very mysterious. It seems that the "spin" of electrons comes in
only two states. If we assume, correctly, that the rate of spin, total charge,
and charge distribution of all electrons is the same, then evidently the magnitude
of the angle the spin axis makes with the horizontal is the same for all
electrons. For some electrons, the spin axis is what we are calling "spin
up", for others "spin down".
You
should beware of the term "spin." If one uses the "classical
radius of the electron" and the known total angular momentum of the
electron, it is easy to calculate that a point on the equator of the electron
is moving at about 137 times the speed of light! Thus, although we will
continue to use the word "spin" it is really a shorthand for
"intrinsic angular momentum."
As promised at the beginning, we now make
the situation a bit more complex. Consider the arrangement shown to the
right: Note that the polarity of the middle
longer magnet is reversed from the other two. We have also drawn the path of
a "spin up" object. When the object emerges from the magnets it is
going the same direction as before it entered them with the same speed. 

The path of a "spin down" object
is: 

For a beam of electrons, onehalf will go
follow the upper path while and other half will follow the lower path: 

Finally, we imagine putting a small block
of lead in the path of the "spin down" electrons. Here, onehalf of the incident beam, the
spindown electrons, will be stopped inside the apparatus, while all the
spinup electrons will emerge in the same direction as before they entered
the magnets and at the same speed. Thus this is a "filter" that
selects spinup electrons. 

Now, again as promised, we simplify by
taking all three magnets and the beam stopper and put it in a box. In the
figure we also have included an electron gun firing a beam of electrons at
the box. So onehalf of the incident beam of
electrons will emerge. It will be important to notice that we
have painted an arrow on the front side of the box to indicate what direction
is "up." You can't see it yet, but there is also an arrow pointing
in the same direction on the back of the box. 

Note
that onehalf of the incident beam of electrons on the filter emerge from the
box, while the other half do not. This is independent of the orientation of the
filter; in both of the orientations shown below onehalf of the incident
electrons emerge, while the other half do not.
Evidently
the direction of "up" is defined by the orientation of the filter
doing the measurement. This is sometimes called spatial quantisation, a
term I do not like.
We now put a second filter behind the
first with the same orientation. The second filter has no effect. Half of the
electrons from the electron gun emerge from the first box, and all of
those electrons pass through the second filter. So, once "up" is
defined by the first filter, it is the same as the "up" defined by
the second. 

Now we put the second filter behind the
first and upside down relative to the first. As always, half of the beam of
electrons from the electron gun emerge from the first filter, and none
of those electrons emerge from the second filter. So, evidently once the
first filter defines "up" that definition is the second filter's
definition of "down." 

Here is another orientation for the second
filter, this time oriented at 90° relative to the first one. To repeat once again, half of the beam of
electrons from the electron gun emerge from the first filter. It turns out
that onehalf of those electrons pass through the second filter. So if we
have two definitions of "up" from two filters at right angles to
each other, one half of the electrons will satisfy both definitions. 

If
we slowly rotate the orientation of the second filter with respect to the first
one from zero degrees to 180 degrees, the fraction of the electrons that passed
the first filter that get through the second filter goes continuously from 100%
to 0%.
Technical
note: if the relative angle is A, the percentage is 100 cos^{2}(A/2).
All
of the above may remind you of polaroid filters for light. One half of a beam
of light from, say, an incandescent lamp will pass through such a filter. If a
second filter is placed behind the first one with the same orientation, all the
light from the first filter passes through the second (at least in the case of
perfect polaroid filters). A brief summary of light polarisation appears here.
If
the relative orientation of the two polaroid filters for light is 90°, then no
light emerges from the second filter. This corresponds to the case above for electron
filters when the relative orientation is 180°.
If
the relative orientation of the two polaroid filters for light is 45°, one half
of the light from the first filter will emerge from the second. This
corresponds to the case above for electron filters when the relative
orientation is 90°.
We
conclude that the only difference between electron and light filters is a
factor of 2 in the relative orientations. Thus, often we call the electron
filters "polarisers."
Here is a final example of combining
electron filters. Onehalf of the beam from the electron gun
emerges from the first polariser; onehalf of those electrons emerge from the
second filter. And onehalf of those electrons will make it through the third
upsidedown filter! Note that is the second filter were not present, no
electrons will emerge from the upsidedown filter. So we see that the middle
filter actually changes the definition of "up" for the electrons.
This is yet another manifestation of the Heisenberg Uncertainty Principle. 

We imagine a radioactive substance that
emits a pair of electrons in each decay. These two electrons go in opposite
directions, and are emitted nearly simultaneously. When another nucleus in
the sample decays, another pair of electrons are emitted nearly
simultaneously and in opposite directions. So we can have a sample emitting
these pairs of electrons. To the right we show such a sample, represented as
an octahedron, and electron filters measuring the spin of each member of the
pair: 

For
the radioactive substance we will be considering here, onehalf of the
electrons incident on the right hand filter emerge and onehalf do not.
Similarly, onehalf of the electrons incident on the left hand filter emerge
and onehalf do not.
But
if we look at the correlation between these electrons, we find that if,
say, the right hand electron does pass through the filter, then its left hand
companion does not pass its filters. Similarly, if the right hand electron does
not pass through the filter, then its left hand companion always emerges from
its filter.
We
say that each radioactive decay has a total spin of zero: if one electron is
spin up its companion is spin down. Of course, this is provided that both
filters have the same definition of up.
To the right is a case where the two
filters have opposite definitions of up. Again, onehalf of the right hand
electrons pass through their filter and onehalf of the left hand electrons
pass through their filter. But this time if a particular right hand electron
passes its filter, then its companion left hand electron always passes its
filter. Similarly, if the right hand electron does not pass its filter, its
companion electron doesn't pass through its filter either. 

Now we consider yet another example. The two filters define "up" to
be in perpendicular directions to each other. If you are still following this
business with electron filters, you will not be surprised to learn that:


These
sorts of measurements are called correlation experiments. We show an
arbitrary relative orientation of the two filters. 

We
summarise all of the above by saying that when the two filters have the same
orientation, the correlation is zero: if the right hand electron passes its
companion does not. When the two filters have opposite orientations, the
correlation is 100%: if the right hand electron passes, so does its companion,
while if the right hand electron does not pass, neither does its companion.
When the two filters have perpendicular orientations, the correlation is 50%.
It turns out that the correlation goes smoothly from zero to 100% as the
relative orientation goes from 0° to 180°. For the mathphilic student, the
actual formula is that the correlation is sin(a/2) squared, where a
is the relative angle between the filters.
There
are radioactive substances that emits pairs of photons similar the the pairs of
electrons we have been consider so far. Some such substances have similar
correlations to the electron source we have been considering, except that there
is a difference of a factor of two in the relative orientations of the
polarisers. If the light polarisers have the same orientation, the correlation
is zero; this is the same as for electrons.
If
the light polarisers have a relative orientation of 90°, the correlation is
100%: if the right hand photon passes through its polariser it companion photon
will pass its polarisers, while if the right hand photon does not pass, neither
does its companion. This corresponds to the case for electrons where the
relative orientation of the filters was 180°.
Similarly,
if you are still following all this, the correlation when the relative
orientation of the light polarisers is 45° is 50%, just the correlation for
electron with relative filter orientations of 90°.
As
we shall see these correlation experiments, both for electrons and photons,
have been performed and turn out to give us important information about the way
the world is put together. This is the thrust of Bell's Theorem, also
sometimes known as the EinsteinPololskyRosen (EPR) paradox.
This
document was written by David M. Harrison, Department of Physics, University of
Toronto, mailto:harrison@physics.utoronto.ca
in March 1998. This is version 1.20, date (m/d/y) 09/02/04.
This
document is Copyright © 1998  2004 David M. Harrison.
This work is
licensed under a Creative Commons License. 