# Addition of Angular Momenta in Quantum Mechanics

Addition of Angular Momenta in Quantum Mechanics

Angular momentum in quantum mechanics is a quantized vector with magnitude and component =mℏ in any direction, conventionally chosen as the axis. The quantum numbers are restricted to integer or half-integer values: , with . Vector addition of two angular momenta is restricted by a triangle inequality with . 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 and precess about , which itself precesses about a axis. The amplitude for addition of and to give with component =mℏ can be expressed in terms of Clebsch–Gordan coefficients as with the sum restricted by . You can set the precessions into motion with the trigger control. To choose a new set of and values, pause and reset the trigger.

J=

j(j+1)

ℏJ

z

z

j=0,,1,,2,…

1

2

3

2

m=-j,-j+1,…,j

J=+

J

1

J

2

|-|⩽j⩽+

j

1

j

2

j

1

j

2

m=+

m

1

m

2

J

1

J

2

J

z

J

1

J

2

J

J

z

jm>=<jm>>

j

1

j

2

∑

m

1

m

2

j

1

m

1

j

2

m

2

j

1

j

2

j

1

m

1

j

2

m

2

m=+

m

1

m

2

j

m