The Density Matrix
Michael Fowler 11/19/07
Pure States and Mixed States
Our treatment here more or less follows that of Sakurai, beginning with two imagined Stern-
Gerlach experiments.
In that experiment, a stream of (non-ionized) silver atoms from an oven is
directed through an inhomogeneous vertical magnetic field, and the stream splits into two.
The
silver atoms have nonzero magnetic moments, and a magnetic moment in an inhomogeneous
magnetic field experiences a nonzero force, causing the atom to veer from its straight line path,
the magnitude of the deflection being proportional to the component of the atom’s magnetic
moment in the vertical (field) direction.
The observation of the beam splitting into two, and no
more, means that the vertical component of the magnetic moment, and therefore the associated
angular momentum, can only have
two
different values.
From the basic analysis of rotation
operators and the properties of angular momentum that follow, this observation forces us to the
conclusion that the total angular momentum of a silver atom is
1
2
=
.
Ordinary orbital angular
momenta cannot have half-integer values; this experiment was one of the first indications that the
electron has a spin degree of freedom, an angular momentum that cannot be interpreted as orbital
angular momentum of constituent parts.
The silver atom has 47 electrons, 46 of them have total
spin and orbital momenta that separately cancel, the 47
th
has no orbital angular momentum, and
its spin is the entire angular momentum of the atom.
Here we shall use the Stern-Gerlach stream as an example of a large collection of quantum
systems (the atoms) to clarify just how to describe such a collection, often called an
ensemble
.
To avoid unnecessary complications, we only consider the
spin
degrees of freedom.
We begin
by examining two different streams:
Suppose experimentalist
A
prepares a stream of silver atoms such that each atom is in the spin
state
A
ψ
:
(
)
1
2
A
ψ
=
↑ + ↓
.
Meanwhile, experimentalist
B
prepares a stream of silver atoms which is a
mixture
: half the
atoms are in state
↑
and half are in the
state
↓
: call this mix
B
.
Question
: can we distinguish the
A
stream from the
B
stream?
Evidently, not by measuring the spin in the
z
-direction!
Both will give up 50% of the time, down
50%.
But: we
can
distinguish them by measuring the spin in the
x
-direction: the
A
ψ
quantum state is
in fact just that of a spin in the
x
-direction, so it will give “up” in the
x
-direction every time—

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