Angular momentum and its quantization

The angular momentum **L**, as we know from classical mechanics, is given by

*x*, *y*, *z* being the position coordinates and
*p*_{x}, *p*_{y}, *p*_{z} being the corresponding linear momenta.

The three angular momentum components are therefore

In quantum mechanics, we work with the corresponding operators, obtained in the position representation by replacing the position coordinates *q*_{i} by themselves and the momentum coordinates *p*_{i} by
- *i* as follows

We already know what is meant by commutation relations for operators. Let us check out the commutation relations for the angular momentum operators:

It may also be shown that where

We have discussed earlier
that any pair of complementary observables whose operators do not commute cannot be determined simultaneously. Hence from the above commutation relations we find that only one component of the angular momentum can be specified precisely. As per convention, we take this component to be *L*_{z}, but we could as well have chosen *L*_{x} or *L*_{y}.

Notice also that *L*^{2} commutes with each of the components, hence the magnitude of the angular momentum || can be specified simultaneously with each of its components.

Thus the two fundamental observables are *L*^{2} and *L*_{z}.

We would soon encounter another angular momentum observable called
spin, denoted by *S*. Hence it would be better if we use a general notation for angular momentum, viz., *J*. The two
fundamental observables, in this notation, are *J*^{2} and *J*_{z}

Since *J*^{2} and *J*_{z} commute, they have the same eigenstate or eigenfunction which is represented in terms of two quantum numbers
*j* and *m*_{j} as follows:

The corresponding eigenvalues
are *j*(*j* + 1) and *m*_{j}. *j* can have integer or half-integer values and
*m*_{j} = - *j*... + *j* in integral steps.

*OQ* = || = and
*OP* = *J*_{z} = *m*_{j}. Note that if *J*_{z} is specified or fixed we may simultaneously specify only ||. The direction of is arbitrary.
In the vector model of angular momentum shown through the figure above,
may be represented by the sides of a cone ( any one among the many shortest lines joining the vertex of the cone to the periphery of the bottom of the cone, but subject to the quantization condition on *J*_{z}), the height of the cone being fixed at *J*_{z}. In the process *J*_{x} and *J*_{y} also become arbitrary.

- Angular momentum of coupled systems
- Normal Zeeman effect
- Electron spin
- Spin-orbit coupling and fine-structure
- Anomalous Zeeman effect

2007-09-27