Elementary particles have intrinsic properties. Some of these properties are the same properties we associate with macroscopic objects, such as mass and charge. Some are purely quantum-mechanical and have no macroscopic analog.
Protons, neutrons, electrons, and photons all have spin. Spin is intrinsic angular momentum associated with elementary particles. It is a purely quantum mechanical phenomenon without any analog in classical physics. Spin is not associated with any rotating internal parts of elementary particles, it is intrinsic to the particle itself. An electron has spin, even though it is believed to be a point particle, possessing no internal structure. The concept of spin was introduced in 1925 by Ralph Kronig, and independently by George Uhlenbeck and Samuel Goudsmit.
Spin is quantized, and can only take on discrete values. Electrons are spin 1/2 particles. The spin angular momentum of an electron, measured along any particular direction, can only take on the values h/2 or -h/2. (h (hbar) = 1.054*10-34 Js) For example, if we measure Sz, the spin angular momentum of an electron along the z-axis, we will either measure Sz = h/2 or Sz = -h/2. If we measure Sz = h/2, then we know that after the measurement the particle is in an eigenstate of Sz with eigenvalue h/2. The measurement changes the information we have about the electron. Before the measurement we may have only been able to give the probability of measuring h/2, but after the measurement we can predict with certainty, that a second measurement will again yield h/2, if we do not measure anything else before we make the second measurement.
How do we measure spin?
Most particles with spin possess a magnetic moment, they act like small magnets. The magnetic moment of an elementary particle is proportional to its spin. The magnetic moment can be measured, by passing the particle through an inhomogeneous magnetic field, where the magnitude and alignment of the magnetic moment will determine the deflection angle, or by measuring the magnetic fields generated by the particle itself. Ferromagnetism arises from the alignment of the spins of the atomic electrons in a solid.
An experiment:
Assume an electron gun produces a beam of electrons. Assume the electrons are moving into the y-direction and pass through a magnet with an inhomogeneous field pointing into the -z-direction. In the figure below the magnetic field points from the north pole to the south pole. This apparatus measures Sz. When electrons move through this apparatus, an electron with Sz = h/2 is deflected upward. An electron with Sz = -h/2 is deflected downward. The amount of deflection up or down is exactly the same.
Since the spins of the electrons from the electron gun are randomly oriented, 50% of the electrons in the beam are deflected upward and the other 50% are deflected downward. Before the magnet we cannot predict whether an individual electron will be deflected upward or downward. But after an electron has passed through the magnet, we know the z-component of its spin. We have constructed a spin analyzer.
Now consider an arrangement of three magnets in series with the polarity of the middle, longer magnet reversed. The paths of spin up or spin down electrons through the arrangement are different, but both spin orientations emerge undeflected. If we insert a beam block, only spin up particles can pass.
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With the beam block in place, one half of the incident electrons will be stopped inside the apparatus, while the other half will emerge undeflected. We have constructed a spin filter for spin-up electrons.
Let us schematically represent the spin filter by a box as shown in the figure below. The arrow on the front side of the box to indicates what direction is "up." Only spin-up electrons can pass.
If we change the orientation of the box, we measure the component of an electron's spin along a different axis. We always observe that 50% of the incident electrons from the gun emerge from the filter box, while the 50% do not, independent of the orientation. For all the orientations shown below 50%of the incident electrons emerge, while the other 50% do not.
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Now let us put two filters in series. If we put a second filter behind the first filter with the same orientation as the first filter, then the second filter has no effect. 50% of the electrons from the electron gun emerge from the first filter, and all of those electrons pass through the second filter. Once we have measured that an electron is in the eigenstate of Sz with eigenvalue h/2, a second measurement will yield the same result.
If we reverse the orientation of the second filter, then the second filter will stop all electrons. 50% of the electrons from the electron gun emerge from the first filter, and none of those electrons will pass through the second filter. The first filters definition of "up" is the second filters definition of "down".
Now let us orient the second filter at 90° relative to
the first one. Once again, 50% of the electrons from the electron gun
emerge from the first filter. 50% of those electrons will 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
that satisfied the first definition will then satisfy the second
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 emerging from the first filter which will pass the second
filter changes continuously from 100% to 0%.
If the relative angle is q, the percentage of
electrons passing the second filter after having passed the first filter
is 100% * cos2(q/2).
Photons are spin 1 particles. The spin of a photon is measure by making polarization measurement. If we measure the linear polarization of a single photon along any axis, we we can only find it aligned with the axis or perpendicular to this axis. If we measure the linear polarization of photons along any axis, there are only two possible results. To measure the polarization we can use a polaroid filter. It has a transmission axis and only passes photons whose polarization is aligned with this axis. Photons whose polarization is perpendicular to the transmission axis are stopped. The polaroid filter becomes a "spin filter".
Assume we use a polaroid filter. One half of the photons from an incandescent lamp will pass through such a filter. If a second filter is placed behind the first one and the axes of both filters have the same orientations, all the photons will pass the second filter (at least in the case of perfect polaroid filters). If the relative orientation of the axes of the two polaroid filters is 90°, then no light will emerges from the second filter. If the relative orientation of the axes of the two polaroid filters is 45°, one half of the photons that pass the first filter will also emerge from the second filter. The only difference between the actions of two electron spin filters and two polaroid filters for light is a factor of 2 in the relative orientations.
Now let us consider one final combination of polaroid or electron spin filters.
Consider the combination of electron filters shown below.
One-half of the beam of electron from the gun emerges from the first filter. 50% of those electrons emerge from the second filter. And 50% of those electrons will make it through the third upside-down filter. Note that if the second filter is not present, no electrons emerge from the upside-down filter.
We denote the eigenstates of the Sz operator by |+> and |->. The Sz operator is associated with a measurement of the z-component of the spin. For an electron, this measurement can have two outcomes.
Sz|+> = h/2|+>, Sz |-> = -h/2|->.
We denote the eigenstates of the Su operator by |+>u and |->u. The Su operator is associated with a measurement of the component of the spin along some other axis u defined by two angles angles (q, f)u. For an electron, this measurement can have two outcomes.
Su|+>u = h/2|+>u, Su |->u = -h/2|->u.
The eigenstates of the operator Su associated with measuring the projections of the spin along an axis defined by the angles (q, f)u are linear combinations of |+> and |-> .
|+>u = a|+> + b |->, |->u = c|+> + d |->.
If u denote the x-direction, for example, then |+>x
= 2-1/2(|+> + |->) and |->x = 2-1/2(|+>
- |->).
If we measure Su and find Su = h/2,
we know that after the measurement the electron is in the eigenstate |+>u.
If we measure Su again, we again will find Su =
h/2. But if for our second measurement we
measure Sz, the probability of measuring h/2
is |a|2 and the probability of measuring -h/2
is |b|2. We can only know the components of the spin along
one axis at any one time. If we have measured the component of the spin
along one axis, we only have probabilistic information about the the outcome
of a measurement along a second axis. If we measure the component
along the second axis, we loose the information about the component along
the first axis.
Because the squares of the coefficients a and b give us the
probabilities of obtaining one or the other of two possible results, and
because the total probability of measuring something is equal to 1, we need
|a|2 + |b|2 = 1. Similarly, we need |c|2
+ |d|2 = 1.
Problem:
The figure below shows a sequence of measurements.
What percentage of electrons from an electron gun will leave the spin analyzer in the spin up state?
Solution:
50% of the electrons from the gun will pass the spin filter. They will be in the spin up state. They all will leave the analyzer in the spin up state. So 50% of the electrons from the gun will leave the analyzer in the spin up state.