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3
Quantum Spin
I begin this section with a description of the surprising result of the experiment performed
by Stern and Gerlach in 1922. I will then use several variations of the basic apparatus to
gain further insight about the system and its proper description, namely the quantum theory
of spin- 12 systems.
3.1
The Stern-Gerlach experiment
The Rutherford’s model of the atom with electrons revolving around a very small nucleus
one naively expects atoms to possess a magnetic dipole moment. Bohr’s model with its
quantized orbits suggests that the value of the magnetic moment should be quantized. In
1922 Stern and Gerlach conducted an experiment to look for this e↵ect with silver atoms
and discovered that indeed the magnetic moment of silver atoms seems quantized. We now
know that this magnetic moment is associated with the spin of the outer shell electron rather
than the angular momentum of its rotation around the nucleus (the ground state of silver
atoms is in L = 0 state), but importantly the result is in sharp contrast to the classical
expectation3 . Let’s then forget about the internal structure of the silver atom, simply think
about it as potentially possessing some magnetic moment, and try to understand the result
of the experiment.
The schematic experimental apparatus is shown in Fig. (7). The atoms emerge from the over
unpolarized and so we expect the magnetic moment to point in a random direction. They
then pass through a small region where an inhomogeneous magnetic field is present having
some gradient along the ẑ direction4 ,
~ = ẑB(z)
B
(2.1)
This results in a force on the dipoles that will tend to deflect them. The force can be
~ (see Eq. (2.30)),
calculated using the potential energy V = µ
~ ·B
✓
◆
@B
F~ = rV = µz
ẑ ,
(2.2)
@z
where I have used the fact that the only non-zero spatial derivative is that of the magnetic
field with respect to z. Here µz is the component of the magnetic moment along the ẑ
direction. Let’s then work out what is the expected deflection along the ẑ direction.
3
You can read more about this interesting history and all the confusion here: http://plato.stanford.
edu/entries/physics-experiment/app5.html .
4
~ = 0. This can be
This actually cannot be the entire story since Maxwell’s equations demand that r · B
~
satisfied with a component of the magnetic field along the ŷ direction so that B = ẑBz (z) + ŷBy (y) with
@z Bz = @y By . This extra component will result in a force along the ŷ direction and so does not result in
deflection and is inconsequential for what follows.
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ẑ
ŷ
x̂
to detector
S
~
B
Oven
N
collimator collimator
Magnet
Figure 7: A schematic of the Stern-Gerlach experiment. Silver atoms emanate unpolarized
from an oven on the left. They move through a series of collimators to produce a beam,
that is then passed through a region with a magnetic field with a uniform gradient. They
get deflect slightly and continue on to the right towards the detector.
A silver atom is moving in the positive ŷ direction with some velocity, say v. Over a small
region associated with the magnet it feels a force as in Eq. (2.2) that causes it to accelerate
in the ẑ direction, either positive or negative depending on the orientation of µ
~ . The velocity
gained is,
vz = at =
Fd
mv
(2.3)
Here m is the mass of the silver atom, d is the length of region over which the magnetic field
is present. After being deflected by the magnetic field the silver atom travels some further
distance D to the screen, and so its displacement away from the centre along the ẑ direction
is
z = vz
D
dD
@B 2dD
=F
=
µ
.
z
v
mv 2
@z K.E.
(2.4)
In the last equality I introduced K.E. = 12 mv 2 as the kinetic energy of the atom. This result
is telling us something very simple and important: if the experimentalist is careful enough
to prepare the beam of atoms with only a small spread in kinetic energy, a stable magnetic
field so @B/@z is constant, and only a small spread in the direction of motion (so that the
e↵ective distances d and D are the same for each atom), then the displacement away from
the centre is entirely due to the orientation of the magnetic dipole,
z / µz
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(2.5)
Classically, since the atoms leave the source unpolarized we expect the angular momentum to
be pointing in a random direction and for the magnetic dipole to then be evenly distributed,
µ < µz < +µ
classical expectation
(2.6)
This would result in an even uniform spread of the displacement along the ẑ about the origin.
Observed experimentally
Expectation from
classical physics
S
S
~
B
~
B
N
N
Magnet
Magnet
detector screen
detector screen
Figure 8: On the left is the expectation for the result of the Stern-Gerlach experiment where
we expect the magnetic dipole moment along the ẑ direction to be uniformly distributed and
lead to deflection in all angles. Instead what is observed is more like the figure on the right
where the beam is seen to be split into two components.
3.2
Spin- 12
Experimentally what is observed is rather di↵erent: the beam of atoms is split in two, half
of the atoms are deflected upwards whereas the other half deflect downwards. Contrary
to our expectations, this result seem to imply that the ẑ-component of the magnetic dipole
moment has only two values. While it is not obvious from this particular experiment, further
investigations over the years have made it clear that it is the magnetic moment of the outer
shell electron that is responsible for the overall magnetic dipole of the silver atom. Moreover,
in this particular case the magnetic moment is not associated with the motion of the electron
in the atom, but rather it is an intrinsic quantity of the electron itself. It is related to the
intrinsic angular momentum carried by the electron, a quantity called spin thorugh
ge
µ
~=
~s
(2.7)
2m
This formula is the result we found in Eq. (2.19) with the spin ~s playing the role of the
angular momentum, m being the mass of the electron, and q = e is the charge of the
electron. The g-factor for the electron is found to be very close to g = 2. In fact, it is one of
the best measured quantities in nature,
g = 2.002 319 304 361 53(53)
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(2.8)
Figure 9: The actual result of the SG experiment was sent on a postcard to Niels Bohr. On
the left is the result without the magnetic field turned on. The figure on the right shows the
splitting of the beam by the magnetic field. Compare this result to the schematics shown in
Fig. 8. Note that the figure is rotated by 90 compared with Fig. 8. (Source: Physics Today,
Courtesy AIP Emilio Segre Visual Archives)
where the number in brackets is the uncertainty on the last two digits5 . Stern-Gerlach type
experiments reveal that that the intrinsic spin of the electron along the ẑ-direction takes
only one of two values,
sz = ± 12 ~
(2.9)
Here ~ is the reduced Planck constant, which is known to some nine significant digits ~ =
1.054 571 726(47) ⇥ 10 34 J · sec. We say that the electron therefore carries spin 12 .
So it seems that we can describe the atom as being in one of two states6 ,
| "z i or | #z i
5
(2.10)
The g-factor for the electron is often quoted with the minus sign instead of the minus sign in Eq. (2.7).
In what follows I will ignore all the other degrees of freedom of the atom, such as its internal state,
atomic level, position, etc. The Stern-Gerlach apparatus allows us to simplify the system and isolate and
study only its magnetic dipole moment.
6
18
The upward pointing arrow represents the sz = +~/2 state whereas the downward pointing
arrow represents the sz = ~/2 state. The | i notation is known as Dirac notation to
represent quantum states.
3.3
Variations on a theme
Let’s now take the basic Stern-Gerlach apparatus and consider some more complicated “experiments” we can do with it and what it teaches us about the system. I am placing the word
experiments in quotes because I am not going to worry about the technical implementation
of such setups, and how precisely they can be realized. Instead I want to use them to build
the appropriate quantum description of the system.
ẑ
ŷ
x̂
| "z i
SG(ẑ)
| "z i
SG(ẑ)
| #z i
Figure 10: The first SG variation.
The first experiment is shown in Fig. 10. The original beam is first split into two beams
composed of atoms in | "z i and | #z i states, as in the original SG experiment. Then while
the | #z i beam is blocked, the | "z i beam is passed through another SG apparatus pointing
along the ẑ-direction. No further splitting of the beam is observed. The | "z i beam now
behaves as if composed of atoms all in the | "z i state and subsequent measurement of the
spin along the ẑ-direction continues to register this same value.
Since the atoms are moving in the ŷ direction, we can also orient the SG magnet in the
x̂-direction and measure that component of the spin in that direction. This is what is done
in the second experiment shown in Fig. 11. The initial beam is split into two beams as
in the previous experiment with an SG device along ẑ-direction. Then, the | #z i beam is
blocked while the | "z i is passed through another SG device, but this time pointing along
the x̂-direction. Such a device can separate the beam depending on the orientation of the
spin along the x̂-direction. It is observed that the pure | "z i beam is split into two beams,
one with | "x i and the other with | #x i. The x̂-component of the spin is therefore also seen
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to take only one of two values,
sx = ± 12 ~
(2.11)
ẑ
ŷ
x̂
| "z i
SG(ẑ)
| "x i
SG(x̂)
| #z i
| #x i
Figure 11: The second SG variation.
We might be tempted to conclude that the state of the system can therefore be described
by the value of sz as well as the value of sx , such as,
| "z , "x i or | "z , #x i
(2.12)
But, surprisingly, this turns out to be false!
Consider the third experiment shown in Fig. 12. The first part of the experiment is the same
as the second experiment, Fig. 11. Then the beam with | #x i is blocked and the beam with
| "x i is allowed to pass through a third SG apparatus, but oriented in the ẑ-direction. From
the first part of the experiment we expect this part of the beam to be made of pure | "z i
states and therefore to pass through without splitting as in the first experiment, Fig. 10.
But, this is not what is observed. Instead we observe that the pure | "x i beam is split again
in half as if it was composed of equal mixture of | "z i and | #z i states! The measurement
of the x̂-component of spin has strongly a↵ected the system and it is incommensurate with
the measurement of the ẑ-component of spin. After measuring the x̂-component, we
seem to have lost all the information on the state in the ẑ-component.
3.4
Stern-Gerlach at arbitrary angle
So far we have only considered SG devices oriented along the ẑ- and x̂-directions and found
some interesting surprises. Experimentally we are not restricted to only these two directions,
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