The Stern-Gerlach Experiment, Electron Spin, and Correlation
Experiments

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Author

This document was written by David M. Harrison, Department of Physics,
University of Toronto, mailto:[email-masked]
in March 1998. This is version 1.18, date (m/d/y) 02/28/02.

This document is Copyright © 1998 by David M. Harrison. This material may be
distributed only subject to the terms and conditions set forth in the Open
Content License, v1.0 or later (the latest version is presently available at http://opencontent.org/opl.shtml).

Introduction

This page summarises summarises the classic Stern-Gerlach 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 spin-states. We also
mention photons, which also have two spin-states. 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.

Classical Charged Spinning Objects

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:

The weird shape of the magnet pole pieces.

The fact that there will be horizontal deflections. These can be
cancelled by putting an electric field perpendicular to the plane of the
ball's trajectory.

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

The total amount and distribution of electric charge on the ball.

The orientation and rate of spin. As the rate of spin increases, so
does the deflection. As the axis of the spin becomes more vertical, that
amount of deflection also increases.

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.

The Spin of the Electron

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."

Building a Spin Filter

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, one-half 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, one-half of the incident beam, the spin-down electrons, will be
stopped inside the apparatus, while all the spin-up 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 spin-up 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 one-half 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.

Using the Spin Filter

Note that one-half 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 one-half 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 one-half 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
cos2(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.

One-half of the beam from the electron gun emerges from the first
polariser; one-half of those electrons emerge from the second filter. And
one-half of those electrons will make it through the third upside-down
filter! Note that is the second filter were not present, no
electrons will emerge from the upside-down 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.

Correlation Measurements

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, one-half of the
electrons incident on the right hand filter emerge and one-half do not.
Similarly, one-half of the electrons incident on the left hand filter emerge and
one-half 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, one-half of the right hand electrons pass through their filter
and one-half 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:

One-half of the right hand electrons emerge from their filter.

One-half of the left hand electrons emerge from their filter.

If a particular right hand electron passes its filter, one-half of
the time its companion left hand electron will emerge from its filter,
one-half of the time it will not.

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 Einstein-Pololsky-Rosen (EPR)
paradox.