Chapter 5
ANGULAR MOMENTUM AND ROTATIONS
In classical mechanics the total angular momentum
~
L
of an isolated system about any
…xed point is conserved. The existence of a conserved vector
~
L
associated with such a
system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian)
is invariant under rotations, i.e., if the coordinates and momenta of the entire system are
rotated “rigidly” about some point, the energy of the system is unchanged and, more
importantly, is the same function of the dynamical variables as it was before the rotation.
Such a circumstance would not apply, e.g., to a system lying in an externally imposed
gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated
system under rotations ultimately arises from the fact that, in the absence of external
…elds of this sort, space is isotropic; it behaves the same way in all directions.
Not surprisingly, therefore, in quantum mechanics the individual Cartesian com-
ponents
L
i
of the total angular momentum operator
~
L
of an isolated system are also
constants of the motion. The di¤erent components of
~
L
are not, however, compatible
quantum observables. Indeed, as we will see the operators representing the components
of angular momentum along di¤erent directions do not generally commute with one an-
other. Thus, the vector operator
~
L
is not, strictly speaking, an observable, since it does
not have a complete basis of eigenstates (which would have to be simultaneous eigenstates
of all of its non-commuting components). This lack of commutivity often seems, at …rst
encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying
structure of the three dimensional space in which we are immersed, and has its source
in the fact that rotations in three dimensions about di¤erent axes do not commute with
one another. Indeed, it is this lack of commutivity that imparts to angular momentum
observables their rich characteristic structure and makes them quite useful, e.g., in classi-
fying the bound states of atomic, molecular, and nuclear systems containing one or more
particles, and in decomposing the scattering states of such systems into components as-
sociated with di¤erent angular momenta. Just as importantly, the existence of internal
“spin” degrees of freedom, i.e., intrinsic angular momenta associated with the internal
structure of fundamental particles, provides additional motivation for the study of angu-
lar momentum and to the general properties exhibited by dynamical quantum systems
under rotations.
5.1 Orbital Angular Momentum of One or More Particles
The classical orbital angular momentum of a single particle about a given origin is given
by the cross product
~
`
=
~
r
£
~p
(5.1)
of its position and momentum vectors. The total angular momentum of a system of such
structureless point particles is then the vector sum
~
L
=
X
®
~
`
®
=
X
®
~
r
®
£
~
p
®
(5.2)