Welcome to quantum mechanics 8, spin.
In many ways this video is about the number two, how it crops up in atomic behavior, and
its seemingly impossible explanation.
We've already seen in video 6 that the number of hydrogen orbitals per energy level, 1,
4, 9 & 16, when multiplied by TWO gives a sequence of numbers corresponding to the periods
of the periodic table: 2 [pause] 8 [pause] 18 [pause] and 32.
In 1887, the same year their famous ether drift experiment help lay the foundations
for relativity theory, Michelson and Morley discovered the so-called fine structure of
the hydrogen spectrum. In video 4 we discussed the central role that the hydrogen spectrum
played in the development of the quantum mechanical theory of the hydrogen atom. One of the most
notable hydrogen lines is the so-called H alpha line in the red region of the visible
spectrum.
In normal resolution this looks like a single line of wavelength 656 nm. But by constructing
an extremely precise optical system, Michelson and Morley were able to show that it actually
consists of two lines about 0.02 nm apart. That's a split of only about 30 parts in a
million. Further precise observations of other atomic spectra showed more examples of this
fine structure of spectral lines.
With the development of the quantum theory of the hydrogen atom it became clear that
this implied a tiny splitting of hydrogen energy levels. In video 6 we saw that the
Schrewdinger solution predicts that the different orbital types for a given energy level have
exactly the same energy. But in fact precise measurement shows that the s orbitals have
single energy values, while the p and the d orbitals are split into TWO closely spaced
energy levels. Why single energies for the s orbitals and double energies for the others?
Recall from video 7, s orbitals have quantum number L equals 0. These are states of zero
orbital angular momentum. In these states the electron does not on average �go around�
the nucleus. The other orbitals have quantum number L equal to 1, 2, 3 and so on. In these
orbitals the electron does have angular momentum. So apparently in orbitals with angular momentum
the energy level splits in two, but not in the s orbitals with no angular momentum. Why?
Also in video 7 we saw how an atom in a magnetic field can produce the Zeeman effect where
the electron orbital's energy changes for different z components of angular momentum.
Since there are always an odd number of possible components the Zeeman effect show up as a
splitting into an odd number of lines. But in some cases, the so-called anomalous Zeeman
effect, we see a splitting into an even number of lines, that is, a multiple of the number
two.
One of the most striking experiments characterized by the number two was the Stern-Gerlach experiment
of 1922 using silver atoms. A similar experiment 5 years later by Phipps and Taylor used hydrogren
atoms. These experiments employed a non-uniform magnetic field created by a magnet with different
sized north and south poles. This causes the magnetic field to spread out, hence get weaker
as we move from top to bottom.
Suppose we now put a little magnetic dipole inside with its south pole upward. Opposite
poles attract, so the dipole feels an upward force at its south pole and a downward force
at its north pole. If the magnetic field was uniform these would cancel and there would
be no net force on the dipole. But the magnetic field is stronger at the top, so the upward
force is a little stronger than the downward force, and there's a net upward force on the
entire dipole.
If we flip the dipole over, like poles repel, so the north pole feels a downward force and
the south pole an upward force. Again the top force is stronger due to the non-uniform
magnetic field and the dipole feels a net downward force.
Now consider a side view of our magnetic field. We put a screen at right and shoot uncharged
atoms through the non-uniform magnetic field. If an atom has a magnetic dipole with south
pole up, the atom will curve upward. If the dipole is horizontal there will be no net
force and the atom will travel straight. If the dipole has its north pole upward the atom
will curve downward. Each dipole orientation will produce a spot on the screen.
If these dipoles are due to orbital motion of an electron then, as we've seen, there
should be an odd number of spots. For silver and hydrogen we should expect to see a single
spot. However, both experiments found two spots.
Here Stern and Gerlach's results for silver atoms. Their beam was more of a ribbon. Along
its center where the magnetic field had the greatest non-uniformity we clearly see the
splitting into two. Phipps and Taylor found similar results for hydrogen, although the
experiment was more difficult and the effect was less pronounced.
In 1924 Wolfgang Pauli proposed that these experimental results are evidence that the
electron has a two-valuedness that is, quote, not describable classically. He was being
careful to avoid any analogy with classical phenomena.
A year later graduate students George Uhlenbeck and Samuel Goudsmit proposed that the two-valuedness
was due to the electron spinning. Classically a spinning charged particle should have angular
momentum and produce a magnetic dipole.
As we've already discussed, the motion of a particle around an orbit results in orbital
angular momentum. If the particle is charged, as the electron is, this will also produce
a small magnetic dipole.
We might also envision the electron spinning about an axis analogous to the way a planet
rotates. This would create another component of angular momentum that we'll label S, for
spin angular momentum. Since the electron's charge would also be spinning, this electric
current should also produce another little magnetic dipole.
The quantum mechanical theory of angular momentum, which we briefly described at the end of video
7, tells us how a two-valuedness of angular momentum can arise. If lowercase s is the
spin quantum number, then the magnitude of spin angular momentum is uppercase s equals
square-root of s times s plus 1 times h bar (h bar is Planck's constant over 2 pi). The
z, or up and down component is quantum number m sub s h bar. m sub s takes on values minus
s, minus s plus 1, up to s. s can be zero, a positive integer or a half integer. If it's
one half then m sub s takes on two values, minus one half and plus one half.
A classical picture of this situation would be the electron as a spinning sphere with
mass and charge, and angular momentum of square-root of 3 over 2 times h bar. The angular momentum
has a z component of minus h bar over 2, or plus h bar over 2, which we can refer to as
spin down and spin up. So it seems there is a classical phenomenon, the spinning of a
body, that when placed in a quantum context can explain the two-valuedness that appears
in several experiments.
However, Pauli was justified in avoiding reference to classical phenomenon. If we imagine an
electron with mass m, traveling with velocity v around a circle of radius r equal to the
size of an atom, and we set the orbital angular momentum m v r equal to h bar, we find a velocity
v that is about 1 percent the speed of light. That's fast, but physical plausible.
However, if a spherical electron of radius r and mass m spins such that the velocity
of the surface is v and we require the angular momentum to be h bar, then for even the largest
plausible radius we find a surface velocity that is hundreds of times the speed of light.
There is no classical model of a spinning particle that can come close to explaining
the angular momentum of the electron. Electron, quote, spin, is truly not describable classically.
Still, the term �two-valuedness not describable classically� is an abstract mouthful. It's
so much easier to just say �spin.� And that's the term people have settled on, while
offering the caveat that quantum mechanical spin does not correspond to any classical
phenomenon. Previously we discussed Bohr's correspondence principle where for ever larger
quantum numbers quantum mechanics corresponds ever more closely to classical mechanics.
Although small quantum number orbitals don't look anything like a classical orbit, for
large enough quantum numbers we can start to see a correspondence. However, the spin
quantum number s is fixed at �. That in itself is weird. In both the classical and
quantum cases we can change orbital angular momentum. But spin angular momentum cannot
be changed. It's an intrinsic property of the particle, like its mass and electric charge.
It never gets big, so we can't observe quantum mechanical spin transitioning into a corresponding
classical phenomenon. We'll continue to use the word spin � it at least gives us something
visual to relate to � but it's actually a strictly quantum mechanical phenomenon associate
with the intrinsic angular momentum of a quantum particle.