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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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ẑ
ŷ
x̂
| "z i
SG(ẑ)
| "x i
SG(x̂)
| #x i
| #z i
| "z i
SG(ẑ)
| #z i
Figure 12: The third SG variation.
but rather can orient the SG apparatus at an arbitrary angle in the plane. We consider an
apparatus oriented along the unit vector,
n̂ = ẑ cos ✓ + x̂ sin ✓
(2.13)
This apparatus would also split a beam into two components, which we denote as,
| "n i and | #n i
(2.14)
These are states with their magnetic dipole moment polarized parallel and anti-parallel to
n̂. Using this general direction, we can perform the fourth experiment shown in Fig. 13. As
we have seen before, an initial beam is split with an SG device along the ẑ-direction, each
of the outgoing beams has 50% of the original intensity. Then one of the beams is blocked,
while the other with | "z i is passed through a second SG device that is oriented along the
n̂-direction. The pure | "z i beam is observed to split into two beams, each carrying an
intensity related to the angle of rotation, ✓, as,
✓ ◆
✓
2
intensity of | "n i beam is = cos
(2.15)
2
✓ ◆
✓
2
intensity of | #n i beam is = sin
(2.16)
2
Here I am referring to the intensity relative to the incoming intensity, not that intensity of
the original beam (which was reduced by half by the first SG device).
The SG device in the x̂-direction is a special case of this result with ✓ = ⇡/2 (90 ). Using
Eqs. (2.15) with ✓ = ⇡/2 we find that the intensity of the spin-up beam is equal to the
intensity of the spin-down beam and both are exactly 1/2 of the incoming intensity (50%).
The SG device is acting on spins analogously to the way a beam-splitting polarizer acts on
light. Just like the SG device, a beam-splitting polarizer oriented in a particular direction
splits the incoming light into two perpendicular polarizations, parallel to the polarizer || and
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ẑ
ŷ
x̂
50%| "z i
SG(ẑ)
cos2
⇣ ⌘
| "n i
sin2
⇣ ⌘
| #n i
✓
2
SG(n̂)
50%| #z i
✓
2
Figure 13: The fourth SG variation.
transverse to the polarizer ?,
Stern-Gerlach device
,
beam-splitting polarizer
| "z i
,
transverse polarization
| #z i
,
parallel polarization
But, there is also a notable di↵erence. If we replace all the SG devices in Fig. 13 by beamsplitting polarizers, we would find that the intensity of the two polarizations after the second
polarizer goes like,
Stern-Gerlach device
,
✓ ◆
✓
cos
,
2
✓ ◆
✓
2
sin
,
2
2
beam-splitting polarizer
cos2 (✓)
sin2 (✓)
Thus, with a polarizer we can completely block a polarized beam of light by simply rotating
the second polarizer by 90 (✓ = ⇡/2). In contrast, for the silver atoms we needed to rotate
by 180 (✓ = ⇡) to completely block an incoming polarized beam. This di↵erence is stemming
from the fact that photons, the particles that make up light, have spin-1 whereas the silver
atoms carry spin- 12 .
3.5
Quantum Spin
The Stern-Gerlach apparatus and its variations have revealed a behavior that is strikingly
di↵erent than that expected from classical magnetic dipoles. In the classical case, a magnetic
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dipole is a vector
µ
~ = x̂µx + ŷµy + ẑµz
(2.17)
p
with the constraint that |~µ| = µ2x + µ2y + µ2z . In other words, I should be able to independently measure the size of the dipole |~µ| and its value along the x̂- and ẑ- directions.
Moreover, the value of µx,z varies continuously between |~µ| and |~µ|. Equivalently, since the
magnetic moment is proportional to the angular moment through the gyromagnetic ratio,
Eq. (2.19), we understand the classical angular momentum to be a continuous vector7 ,
~s = x̂sx + ŷsy + ẑsz
(2.18)
with sx and sy to be independently determined and in the range |~s|  sx,z  |~s|. The
classical evolution of a spin is described by the time dependence of sx (t), sy (t), and sz (t).
Each component evolves continuously in time and represent the magnitude of the spin in a
particular direction.
In sharp contrast to the classical expectations, the quantum spin exhibited very di↵erent
characteristics:
1. The components of the spin take only discrete values, sz = ± 12 ~, and similarly for sx
or any other orientation.
2. It is not possible to simultaneously measure the z-component as well as the x-component.
We can only measure the magnitude of the spin |~s| and its value along one axis, say sz .
A measurement along the x-direction completely erases any knowledge of the value of
the z-component as we saw in the third experiment, Fig. 12.
3. If an incoming beam of atoms all in the | "z i state is passed through a SG apparatus
along an arbitrary direction n̂ given by Eq. (2.13) the beam is split into two. The
intensity of the | "n i beam is cos2 (✓/2) and the intensity of | #n i beam is sin2 (✓/2).
So unlike the classical spin, it is clear that the system is not described by some vector ~s
whose components sx (t), sy (t), and sz (t) are all continuous functions of time and can all be
measured simultaneously. An entirely di↵erent description is needed, we need the quantum
theory of spin!
In quantum mechanics the state of a spin- 12 system is described by a two-dimensional abstract
vector, denoted by | i. Just like a regular vector, its components are defined with respect
7
In what follows I will mostly discuss things in terms of the angular moment, or spin. But it should be
clear that the spin and the magnetic dipole moment are entirely interchangeable since they are proportional
to each other. I prefer to discuss the spin because it is in some sense the more fundamental quantity.
Nevertheless, understanding what is a magnetic dipole moment is crucial since practically it is through
this dipole that spins are most commonly manipulated in actual experiments, as we saw in the case of the
Stern-Gerlach experiment.
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to some basis. The ẑ-basis is one where the basis vectors are | "z i and | #z i and the state
vector can be written as,
| i = ↵z | "z i +
z|
#z i
(2.19)
You can think of this in the same way you think about a real vector in real space, e.g.
~s = sx x̂ + sy ŷ + sz ẑ for the classical spin vector. The ~s notation is an abstract notation
denoting a vector, x̂, ŷ, ẑ are abstract basis vectors, and sx , sy and sz are their components.
Except that I caution you that | i is not a vector in real space, it is a vector in Hilbert space.
3.6
Bra and Ket states and inner products
The Hilbert space we are dealing with is very simple, it is a two dimensional vector space
with a linear basis {| "z i, | #z i}. The state | i is a general vector in this space. These are
known as Ket states for reasons that will become obvious soon, and this way of denoting
a quantum state is known as Dirac notation after its inventor. Just like any linear algebra
space, we can also discuss the dual vector space. The basis describing the dual space is
written as,
h"z |
and
h#z |
(2.20)
The state | i has a dual, which is given by
h | = ↵z⇤ h"z | +
⇤
z h#z
|
(2.21)
Note the complex conjugation of the coefficients ↵z and z . As you are familiar from linear
algebra, we can define an inner product between vectors (states) and their duals. The inner
product between a state | i and another state, say | 0 i is written as,
inner product = h | 0 i
with
| i = ↵z0 | "z i +
0
z|
#z i
(2.22)
where ↵z0 and z0 are some di↵erent complex number. This inner product is known as a
bra-ket or bracket, hence the funny names for |·i (ket) and h·| (bra)! The result of the inner
product is itself some complex number. But, how do you actually evaluate something like
that? As always in linear algebra, it suffices for me to tell you how the basis vectors behave.
The behavior is very simple, the basis vectors are normalized to unity,
h"z | "z i = 1
h#z | #z i = 1
(2.23)
(2.24)
but, the overlap between the two states is zero (they are orthogonal ),
h"z | #z i = 0
h#z | "z i = 0
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(2.25)
(2.26)
So, now I can evaluate Eq. (2.22),
✓
◆✓
0
⇤
⇤
h | i =
↵z h"z | + z h#z | ↵z0 | "z i +
= ↵z⇤ ↵z0 h"z | "z i + ↵z⇤
= ↵z⇤ ↵z0 +
0
z
0
z|
h"z | #z i +
#z i
⇤ 0
z ↵z
◆
h#z | "z i +
(2.27)
⇤ 0
z z
h#z | #z i
⇤ 0
z z
Notice that,
h 0| i
3.7
⇤
= h | 0i
(2.28)
Interpretation
The real question is what is the meaning of this description. According to the Born interpretation, which we will develop more generally in the next section, the probability of finding
the atom in a state of spin-up along the ẑ direction is given by the square of the overlap of
the state | i with the bra-state h"z |
probability of finding the atom with up-spin in ẑ-direction = |h"z | i|2
(2.29)
Similarly, the probability of finding the atom in a state of spin-down along the ẑ direction
is given by the square of the overlap of the state | i with the bra-state h#z |,
probability of finding the atom with down-spin in ẑ-direction = |h#z | i|2 (2.30)
But, using the orthogonality of the basis, we immediately see that h"z | i = ↵z and h#z | i =
z . Therefore, the square of the coefficient of each basis vector is telling you the probability
of finding it in any one of these states,
probability of finding the atom with up-spin in ẑ-direction
probability of finding the atom with down-spin in ẑ-direction = | z |2
(2.31)
(2.32)
It’s important to note that I am using the square modulus, because in general the coefficients
are complex numbers! This relation between the coefficients of the basis vector and the
probability of finding the system in a particular state is known as the Born rule. Since
the system must be found with either up-spin or down-spin, the probability interpretation
implies that we must have,
|↵z |2 + | z |2 = 1
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(2.33)
3.8
Rotation to another basis
Just as with regular vectors, we can choose to describe the state of the system with a di↵erent
but equivalent basis to that formed by the states | "z i and | #z i. In particular, as we have
seen above we can orient the SG device in a general direction n̂ with,
n̂ = ẑ cos ✓ + x̂ sin ✓
(2.34)
Then the basis states are | "n i and | #n i. The question is how does this basis relates to the
ẑ-basis? We saw that experimentally if we start with an up-spin state along the ẑ direction
| "z i and measure its spin along the n̂ direction we get the result described in Eq. (2.15).
Since the intensity is directly proportional to the probability of finding a single silver atom
in a particular state we can conclude that,
| "z i = cos( 2✓ ) | "n i
sin( 2✓ ) | #n i
(2.35)
The choice of sign on the second term is not dictated by the measurement since the measurement is only sensitive to the square of this coefficient. It is merely a matter of convention.
To form an orthonormal basis, the other basis state must therefore be
| #z i = sin( 2✓ ) | "n i + cos( 2✓ ) | #n i
(2.36)
It is straightforward to prove that these are orthonormal states,
h"n | "n i = 1
h#n | #n i = 1
(2.37)
(2.38)
h"n | #n i = 0
h#n | "n i = 0
(2.39)
(2.40)
and
If and only if the ẑ basis forms an orthonormal basis. We can now invert these relations to
find the new basis states in terms of the ẑ-basis,
| "n i =
cos( 2✓ ) | "z i + sin( 2✓ ) | #z i
(2.41)
| #n i =
sin( 2✓ ) | "z i + cos( 2✓ ) | #z i
(2.42)
Thus, if we start with a general state in the ẑ-basis such as in Eq. (2.19) we can use Eqs. (2.35)
and (2.36) above to express it in the n̂-basis,
| i = ↵n | "n i +
26
n
| #n i
(2.43)
with
↵z cos( 2✓ ) +
↵n =
n
=
↵z sin( 2✓ ) +
sin( 2✓ )
(2.44)
cos( 2✓ )
(2.45)
z
z
This is an equivalent description of the state of the system in terms of a di↵erent basis, the
n̂-basis. Just as in regular vectors, it is completely up to us to decide which basis to use. It
is straightforward to verify that probability is conserved and that
|↵n |2 + |
n|
2
=1
(2.46)
if and only if |↵z |2 + | z |2 = 1. Such rotations between two equivalent basis that preserve
the probability norm are known as Unitary transformations. They play a central role in
quantum mechanics.
3.9
A special case, rotation by ⇡/2
A special case of the general rotation is the rotation to the x̂ basis with ✓ = ⇡/2 since then
we have from Eq. (2.34),
n̂ = ẑ cos(⇡/2) + x̂ sin(⇡/2) = x̂
(2.47)
In this case we see from the above relation, Eq. (2.35), that if we prepare a beam in the state
| "z i then we can write that state as,
| "z i =
p1
2
p1
2
| "x i
| #x i
(2.48)
According to our probability rule above, Eq. (2.31), we will then find that,
1
p
2
2
=
1
= probability of finding the atom with up-spin in x̂-direction
2
2
1
1
p
=
= probability of finding the atom with down-spin in x̂-direction
2
2
This exactly agrees with our observations in the second SG variation above.
3.10
Fermions and Bosons
There is something a bit odd about the rotation rule, Eq. (2.41). Note that under a full
rotation ✓ = 2⇡, we have that,
| "n i =
| "z i
27
(2.49)
So the spin state returns to itself, as we would expect since we rotated the system by 360 , but
surprisingly it acquired a minus sign. Quantum systems that under a full 2⇡ rotation return
to themselves times a minus sign are said to obey Fermi-Dirac statistics and such particles
are known as fermions. All spin- 12 particles are therefore fermions. This has profound e↵ect
on the structure of matter and is the basis for the Pauli exclusion principle.
If we did this exercise with atoms with integer spin, e.g. s = 1 we would have found that
under a full 2⇡ rotation the state of the system returns to itself with no extra minus signs.
Such systems are said to obey Bose-Einstein statistics and such particles are known as
bosons.
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