J = L +
S
This animation
shows how orbital angular momentum L
and intrinsic/spin angular momentum S
add to produce the total angular momentum J and how the corresponding
magnetic dipole moment vectors combine. Gray
is used for the orbital angular momentum vector L
and its corresponding orbital magnetic dipole moment vector µL
,
which is by nature opposite L. Blue
is used for the intrinsic angular momentum vector S
and its corresponding intrinsic magnetic dipole moment vector µS
,
also opposite S by nature. L
and S add (though only at certain angles
allowed by the rules of quantum mechanics) to produce J, shown in
black.
µL
and
µS also add, producing
total magnetic moment vector µJ. But µJ
is not required to be opposite J. In fact, since the ratio of µS
to S is by nature essentially twice
as great as the ratio of µL
to
L, as the diagram bears out, µJ
is in general not opposite J.
An external magnetic field, shown in red,
establishes a "special" axis, and the total angular momentum vector J
precesses with fixed component along that axis (determined by the quantized
value of mJ), tracing the upper dashed circle. L
and S also precess, but as a unit about
J
and at a much faster rate. It follows that µL,
µS
and µJ likewise precess as a unit at this fast
rate about a line through J. Though µJ "wobbles"
about that line, it has an obvious average value, shown by the thin black
line, and a corresponding component along the special (red) axis as it
precesses about it, tracing the lower dashed circle. It is this average
component that determines the magnetic orientation energy of the entire
system in the external magnetic field, the heart of the Zeeman effect.
ST = S1
+ S2
This animation shows all the ways two s = 1/2 intrinsic
angular momentum vectors, S1
and S2, can add to produce
a total spin angular momentum ST for which sT
= 1. Certain things are common to all three animations: (1) To give an
ST
of the same magnitude, the angle between S1
and
S2 is the same in all;
(2) S1 and S2
may point in an infinite number of directions while producing the same
ST;
and (3) this ST can point in an infinite number of directions,
tracing the dashed circle, while maintaining a fixed component along a
given
axis. The three animations differ in the value of that component.
The rules of quantum mechanics allow only three: one positive, one zero,
and one negative. These three demonstrably different states for the case
sT = 1 are known as a "triplet" of states.
