Addition of Angular Momenta in Quantum Mechanics

j

1

2

1

3

2

m

1

-

3

2

-

1

2

1

2

3

2

j

2

1

2

1

3

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.