The Stern–Gerlach setup of Figure 2.1 with the direction of inhomogeneity
of the magnetic field defined as the z-axis is going to be the basis for defining
2
The standard unit for atomic magnetic moment is the Bohr Magneton, given by
e~
2m
e
c
,
numerically equal to about 5.8×10
−5
eV/T, where m
e
is the mass of the electron, c is the
speed of light, and ~ = h/2π is the (reduced) Planck constant.
A Simple Quantum System 19
and measuring electron spin. Let’s put it in a box and abbreviate as SG
z
. The
incident beam consists of unpolarized electrons. The machine SG
z
produces
as output two beams, one above the z = 0 axis with electrons of “up” spin
and the other below z = 0 with electrons of “down” spin. The SG
z
machine
thus manufactures definite quantum states out of an arbitrary beam. If we
isolate or “‘ilter out” either of these two states by blocking the other beam,
the surviving beam is said to be polarized, and each electron in that beam is in
a specific quantum state, called a basis state. A quantum state is represented
as an angular bracket with a descriptive label inside: a very versatile and
useful notation due to Dirac [29]. The two basis states are represented as |↑i
and |↓i. (Note that these states are defined with respect to a direction of
inhomogeneity of an applied magnetic field.) We can thus use the SG
z
filter
to prepare electrons in a predefined quantum state.
|↑i |↓i
FIGURE 2.3: The SG
z
filters (the paths of the beams are bent back to z = 0
using suitable magnets).
The two SG
z
filters, producing the two basis states, are illustrated in Figure
2.3. (The paths of the beams can be bent back to the z = 0 axis by using
appropriate magnets.)
We thus not only use the SG
z
as a measuring tool for determining the
state of an electron, but also as a factory for preparing a known state. This
state will be labelled by the spin component along the z-direction.
Suppose a beam of electrons in an unknown state is analyzed using an SG
machine. The intensity of a particular output beam can be thought of as the
number of electrons in the input beam that are in the corresponding output
state. However there are subtleties here. A particular electron in the input
beam randomly chooses the up or down output port of the machine. From
the fraction of the total number of electrons that exit from a particular port,
we can deduce the probability of the incident electron being in that particular
state. This is how quantum mechanics works. We collect a set of statistics of
probabilities from measurements and then infer the properties of the system
and its state. This is the reason quantum mechanics is often described as a
probabilistic theory.
A system could be in a purely quantum mechanical state, with quantum
probabilities, and is said to be a pure quantum state. However, classical un-
certainties could also be present in a given system, in which case the system
is said to be in a mixed quantum state. For example, the unpolarized beam
of silver atoms from the oven in the Stern–Gerlach experiment is actually in
a mixed state. We will see more of this distinction in later chapters. For our
present introduction, however, we will assume that our systems are always in
pure quantum states only
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