WOLFRAM|DEMONSTRATIONS PROJECT

Addition of Angular Momenta in Quantum Mechanics

​
j
1
1
2
1
3
2
2
m
1
-
3
2
-
1
2
1
2
3
2
j
2
1
2
1
3
2
2
m
2
-1
0
1
j
5
2
show precession
Angular momentum in quantum mechanics is a quantized vector with magnitude
J=
j(j+1)
ℏ
and component
J
z
=mℏ
in any direction, conventionally chosen as the
z
axis. The quantum numbers are restricted to integer or half-integer values:
j=0,
1
2
,1,
3
2
,2,…
, with
m=-j,-j+1,…,j
. Vector addition of two angular momenta
J=
J
1
+
J
2
is restricted by a triangle inequality
|
j
1
-
j
2
|⩽j⩽
j
1
+
j
2
with
m=
m
1
+
m
2
. Although quantum formalism is indifferent to such interpretations, the addition of angular momentum in the absence of any electric or magnetic field can be pictured by a vector model in which
J
1
and
J
2
precess about
J
, which itself precesses about a
z
axis. The amplitude for addition of
J
1
and
J
2
to give
J
with component
J
z
=mℏ
can be expressed in terms of Clebsch–Gordan coefficients as

j
1
j
2
jm>=
∑
m
1
m
2
<
j
1
m
1
j
2
m
2

j
1
j
2
jm>
j
1
m
1
j
2
m
2
>
with the sum restricted by
m=
m
1
+
m
2
. You can set the precessions into motion with the trigger control. To choose a new set of
j
and
m
values, pause and reset the trigger.