Space-Quantization of Angular Momentum

j

1

2

1

3

2

m

-

1

2

1

2

show precession

The commutation relations for angular momentum in quantum mechanics are given by

[

J

x

,

J

y

]=iℏ

J

z

,



2

J

,

J

z

=0

, with cyclic permutations. From these, the allowed values of quantized angular momentum can be derived, namely,

2

J

=j(j+1)

2

ℏ

and

J

z

=mℏ

, with

m=-j,-j+1,…,j

,

j=0,

1

2

,1,

3

2

,2,…

. Customarily, the

z

component is singled out, with the other two components retaining indefinite or fluctuating values (except when

j=0)

. The definite magnitude and direction of one component of angular momentum is known as "space quantization". Restriction of

m

to integer values was exploited in Bohr's model of the hydrogen atom. When spin is involved,

m

and

j

can also take half-integer values.

The vector model of angular momentum pictures the total angular momentum vector as precessing about its constant

z

component. This is also consistent with the fluctuating values of

J

x

and

J

y

.

The fact that the quantized value of

2

J

equals

j(j+1)

, rather than

2

j

, can be rationalized by the fact that the average value of the sum of the squares of the three

J

components is given by

3

2j+1

j

∑

m=-j

2

m

=j(j+1)

.