9.
SPIN1/2 PARTICLES
1.
Spinors. Eigenvalues and Eigenstates
2.
The Polarization Vector
3.
Magnetic Moments and Magnetic Fields
4.
Time Dependence. Precessing the Polarization
5.
Magnetic Resonance. Flipping the Polarization
6.
SternGerlach Experiments (not yet written)
Problems
The simplest system with nonzero angular momentum is a spin1/2 particle.
Quarks, the building blocks of protons, neutrons, and the other baryons, as well as
of the mesons, are spin1/2 particles. Protons and neutrons, the building blocks of
nuclei, are spin1/2 particles. Electrons, which with nuclei are the building blocks
of atoms, are spin1/2 particles. And so are muons and neutrinos. Thus a particle
with spin 1/2 is an important system. The spin of a particle is as intrinsic to it,
and as invariable, as is its mass and charge—the electron
always
has spin 1/2.
A particle will also have other properties such as momentum and orbital angular
momentum, but here we are only concerned with the spin of a spin1/2 particle.
Not all particles have spin 1/2. The “carriers” of the fundamental forces—
the photon, gluons,
W
and
Z
bosons, and graviton—have spin 1 or (the graviton)
spin 2. Particles made of an even number of spin1/2 particles (the mesons are
such) have integer spins. And particles—nuclei and atoms, for example—can have
higher angular momenta. In the ground states of carbon, nitrogen, oxygen, iron,
silver, and platinum, the six, seven, eight, 26, 47, and 78 electrons conspire to
have total angular momenta of 0, 3/2, 2, 4, 1/2, and 3.
We start with the twocomponent vectors and 2by2 operators for s=1/2.
We learn how to express any spin state in terms of the “up” and “down” states
with respect to any direction, and how to associate a polarization vector with the
spin.
An angular momentum usually has an associated magnetic moment, and
magnetic moments interact with magnetic fields.
In a static magnetic field, the
spin precesses; in a properly arranged timedependent field, the spin can be flipped
up and down; in an inhomogeneous field, the spinup and down states can be
separated into two beams.
Such arrangements are the basis for many scientific
experiments and for magneticresonance imaging.
c
⃝
2011, Charles G. Wohl, work in progress.
1
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9
·
1. SPINORS. EIGENVALUES AND EIGENSTATES
States as vectors
—When
s
= 1
/
2 (we shall use
s
, for spin, instead of
j
here), there are
just two states that are simultaneously eigenstates of the angularmomentum operators ˆ
s
2
and ˆ
s
z
:

s, m
⟩
=

1
/
2
,
+1
/
2
⟩
and

1
/
2
,
−
1
/
2
⟩
. These states are represented in the literature
in a number of shorthand ways, among them

+
z
⟩
=

+
⟩
=

α
⟩
=

1
⟩
=
↑
and
−
z
⟩
=
−⟩
=

β
⟩
=

2
⟩
=
↓
.
Here we shall usually use

+
z
⟩
and
−
z
⟩
in order later to distinguish them from
−
x
⟩
,

+
y
⟩
,
and other such states. The

+
z
⟩
and
−
z
⟩
states are orthogonal because they are eigenstates
of ˆ
s
z
with different eigenvalues, and we shall of course take them to be normalized.
We
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 Spring '09
 Lee
 mechanics, Angular Momentum, Polarization, spinor, b0, polarization vector

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