Next: Eigenvalues for the Orbital Up: Angular Momentum Previous: Introduction
Next: Eigenvalues
for the Orbital Up: Angular
Momentum Previous: Introduction
In classical mechanics, the AM of a particle (with respect to the origin) is given by the formula:
so that,
where 
and 
are the position and linear momentum vectors, respectively.
The mathematics of AM comes about through the quantum rule of replacing
the linear momentum 
of a classical point particle, located at point r, by 
, thus replacing the classical angular momentum of a point particle (about
the origin of some chosen Cartesian coordinate system), by the AM operator:
This implies:
Equation (6) can be written more compactly as
where 
component, 
component, 
component and we have used the Einstein summation convention (i.e. the
expression is summed over repeated indices). 
is the completely antisymmetric tensor of the third rank (also called the
Levi-Civita tensor)
Some useful identities that the
satisfy are
The antisymmetric nature of the
means that
The product of two of these tensors, when summed over a pair of indices (i.e. the indices are contracted) has the following useful properties
which can be proved by expanding out the terms. We will use these results later to prove some AM operator identities, but familiarity with them will be useful in your future study of Special/General Relativity, Classical/Quantum Electrodynamics and Quantum Field Theory.
The quantal AM properties of a simple one-particle system are then to be inferred from the properties of these operators and their actions in the associated Hilbert space. In the following sections, we will deduce the eigenvalues and eigenfunctions of these operators. The components of the quantum mechanical AM operator satisfy the following commutation relations:
or
Further identities include
truein
Partial Solution Here is the proof for (15):
We use the following operator identity to expand out the right hand side of (18)
Making the substitutions
into (19) we have
We now make use of the following operator identities
so that (18) reduces to
Since 
etc. we find that
Substituting (21) into (20) we have
Note that in the last term of the above equation we have relabelled
the indices 
and 
since these are dummy indices (remember, due to the summation convention,
summing over repeated indices 
is the same as summing over repeated indices 
). Equation (22) reduces to
Since 
and 
we finally have the result we want to prove i.e.
Actually I have done the hardest part. The proof for the other identities is much easier! truein
From these fundamental commutation relations, the entire theory of AM
can be deduced. Evidently 
are incompatible observables so it will be futile to look for states that
are simultaneous eigenfunctions of 
and 
. From the Generalised Uncertainty Principle we deduce that:
On the other hand, the square of the total angular momentum, 
,
does commute with 
and 
. truein
or more compactly,
truein
truein
and 
.
commutes with all three components of 
, provided that V depends only on r. (Thus 
and
form a CSCO.) truein
