*Michael Fowler
10/29/07*

As a warm up to analyzing how a wave function transforms
under rotation, we review the effect of *linear translation* on a single
particle wave function _{}. We have already seen
an example of this: the coherent states of a simple harmonic oscillator
discussed earlier were (at *t* = 0) identical to the ground state except
that they were centered at some point displaced from the origin. In fact, the
operator creating such a state from the ground state is a translation operator.

The *translation operator* *T*(*a*) is *defined*
at that operator which when it acts on a wave function ket _{} gives the ket
corresponding to that wave function moved over by *a*, that is,

_{}

so, for example, if _{}is a wave function centered at the origin, *T*(*a*)
moves it to be centered at the point *a*.

We have written the wave function as a ket here to emphasize
the parallels between this operation and some later ones, but it is simpler at
this point to just work with the wave function as a function, so we will drop
the ket bracket for now. The form of *T*(*a*)
as an operator on a function is made evident by rewriting the

_{}

Now for the quantum connection: the differential operator appearing
in the exponential is in quantum mechanics proportional to the momentum
operator (_{}) so the translation operator

_{}

An important special case is that of an infinitesimal translation,

_{}

The momentum operator _{}is said to be the *generator* of the translation.

(*A note on possibly confusing notation*: Shankar
writes (page 281) _{} Here _{} denotes a delta-function
type wave function centered at *x*. It might be better if he had written _{}, then we would see right away that this translates into the
wave function transformation _{}, the sign of _{} now obviously
consistent with our usage above.)

It is important to be clear about whether the *system*
is being translated by *a*, as we have done above or whether, alternately,
the *coordinate axes* are being translated by *a, *that latter would
result in the *opposite* change in the wave function. Translating the
coordinate axes, along with the apparatus and any external fields by -*a*
relative to the wave function would of course give the same physics as
translating the wave function by +*a*.
In fact, these two equivalent operations are analogous to the time
development of a wave function being described either by a Schrödinger picture,
in which the bras and kets change in time, but not the operators, and the
Heisenberg picture in which the operators develop but the bras and kets do not
change. To pursue this analogy a little
further, in the “Heisenberg” case

_{}

and _{}is unchanged since it commutes with the operator. So there are two possible ways to deal with
translations: transform the bras and kets, *or* transform the operators.
We shall almost always leave the operators alone, and transform the bras and
kets.

We have established that *the momentum operator is the
generator of spatial translations* (the generalization to three dimensions
is trivial). We know from earlier work
that the Hamiltonian is the generator of *time* translations, by which we
mean

_{}

It is tempting to conclude that the *angular momentum*
must be the operator generating *rotations* of the system, and, in fact,
it is easy to check that this is correct.
Let us consider an infinitesimal rotation _{}about some axis through the origin (the infinitesimal vector
being in the direction of the axis). A
wavefunction _{}^{ }initially
localized at _{}will shift to be localized at _{}, where _{} So, how does a wave
function transform under this small rotation?
Just as for the translation case, _{}. If you don’t
understand the minus sign, reread the discussion on translations and the sign
of _{}.

Thus

_{}

to first order in the infinitesimal quantity, so the rotation operator

_{}

If we write this as

_{}

it is clear that a finite rotation is given by multiplying together
a large number of these operators, which just amounts to replacing _{}by _{} in the
exponential. Another way of going from
the infinitesimal rotation to a full rotation is to use the identity

_{}

which is clearly valid even if *A* is an operator.

We have therefore established that the orbital angular
momentum operator _{} is the generator of
spatial rotations, by which we mean that if we rotate our apparatus, and the
wave function with it, the appropriately transformed wave function is generated
by the action of _{} on the original wave
function. It is perhaps worth giving an explicit example: suppose we rotate the
system, and therefore the wave function, through an infinitesimal angle _{}_{ }about the *z*-axis.
Denote the rotated wave function by _{}. Then

_{}

That is to say, the value of the new wave function at (*x*,*y*)
is the value of the old wave function at the point which was rotated into (*x*,*y*).

However, it has long been known that in quantum mechanics,
orbital angular momentum is *not* the whole story. Particles like the electron are found
experimentally to have an internal angular momentum, called spin. In contrast to the spin of an ordinary
macroscopic object like a spinning top, the electron’s spin is *not* just
the sum of orbital angular momenta of internal parts, and any attempt to
understand it in that way leads to contradictions.

To take account of this new kind of angular momentum, we
generalize the orbital angular momentum _{} to an operator _{} which is *defined*
as the generator of rotations on *any* wave function, including possible
spin components, so

_{}

This is of course identical to the equation we found for *L*,
but there we derived if from the quantum angular momentum operator including
the momentum components written as differentials. But up to this point _{}has just been a complex valued function of position. From now
on, the wave function at a point can have several components, so it is in some vector
space, and the rotation operator will operate in this space as well as being a
differential operator with respect to position.
For example, the wave function could be a vector at each point, so
rotation of the system could rotate this vector as well as moving it to a
different _{}.

To summarize: _{} is in general an *n*-component
function at each point in space, _{} is an _{} matrix in the
component space, and the above equation is the *definition* of *J*. Starting from this definition, we will find *J*’s
properties.

The first point to make is that in contrast to translations,
rotations do not commute even for a classical system. Rotating a book through _{} first about the *z*-axis
then about the *x*-axis leaves it in a different orientation from that
obtained by rotating from the same starting position first _{} about the *x*-axis
then _{}about the *z*-axis.
Even small rotations do not commute, although the commutator is second
order. Since the *R*-operators are
representations of rotations, they will reflect this commutativity structure,
and we can see just how they do that by considering ordinary classical
rotations of a real vector in three-dimensional space.

The matrices rotating a vector by _{} about the *x*, *y*
and *z* axes are

_{}

In the limit of rotations about infinitesimal angles (ignoring higher order terms),

_{}

It is easy to check that

_{}

The rotation operators on quantum mechanical kets must, like all rotations, follow this same pattern, that is, we must have

_{}

where we have used the definition of the infinitesimal
rotation operator on kets, _{}. The zeroth and
first-order terms in *e* all cancel, the second-order term gives _{}. The general statement is:

_{}

This is one of the most important formulas in quantum mechanics.

The commutation formula _{}which is, after all, a straightforward extension of the
result for ordinary classical rotations, has surprisingly far-reaching
consequences: it leads directly to the directional quantization of spin and
angular momentum observed in atoms subject to a magnetic field.

It is by now very clear that in quantum mechanical systems such
as atoms the total angular momentum, and also the component of angular momentum
in a given direction, can only take certain values. Let us try to construct a basis set of
angular momentum states for a given system: a complete set of kets corresponding
to all allowed values of the angular momentum.
Now, angular momentum is a *vector *quantity: it has magnitude and
direction. Let’s begin with the
magnitude, the natural parameter is the length squared:

_{}.

Now we must specify direction—but here we run into a
problem. *J _{x}*,

The bottom line, then, is that in attempting to construct
eigenkets describing the different possible angular momentum states of a
quantum system, the best we can do is to find the common eigenkets of _{}and *one* direction, say *J _{z}*. The commutation relations do not allow us to
be more precise about direction, analogous to the Uncertainty Principle for
position and momentum, which also comes from noncommutativity of the relevant
operators.

We conclude that the appropriate angular momentum basis is
the set of common eigenkets of the commuting Hermitian matrices _{} :

_{}

Our next task is to find the allowed values of *a* and *b*.

The sets of allowed eigenvalues *a*, *b* can be
found using the “ladder operator” trick previously discussed for the simple
harmonic oscillator. It turns out

_{}

are closely analogous to the simple harmonic oscillator
raising and lowering operators _{} and *a*.

_{} and _{} have commutation
relations with *J _{z}*:

_{}

and they of course *commute * with _{}, as do *J _{z}*,

Therefore, _{}operating on _{} cannot affect the
value of *a*. But they *do*
change the value of *b*:

_{}

so if _{} is an eigenket of *J _{z}*
with eigenvalue

The squared norm of _{}

_{}

and

_{}

from which

_{}

recalling that _{}

Now *a*, being the eigenvalue of a sum of squares of
Hermitian operators, is necessarily nonnegative, and *b* is real. Hence for a given *a*, *b* is *bounded*:
there must be a *b*_{max} and a (negative or zero) *b*_{min}. But this must mean that

_{}

Note that for a given *a*, *b*_{max} and *b*_{min}
are determined uniquely—there cannot be two kets with the same *a* but
different *b* annihilated by *J*_{+}. It also follows immediately that _{} Furthermore, we know
that if we keep operating on _{} with *J*_{+},
we generate a sequence of kets with *J _{z} *eigenvalues

At this point, we switch to the standard notation. We have established that the eigenvalues of *J _{z}*
form a finite ladder, spacing

The
operators _{} have a common set of
orthonormal eigenkets _{},

_{}

where* j*, *m* are integers or half integers. The
allowed quantum numbers *m* form a
ladder with step spacing unity, the maximum value of *m* is *j*, the minimum value
is -*j*.

It is now straightforward to compute the normalization factors needed to find matrix elements:

_{}

and _{}, so

_{}

With
these formulas, and the base set of normalized eigenkets _{}, we are in a position to construct explicit matrix
representations of the angular momentum algebra for any integer or half integer
value of angular momentum *j*.

*Historical note*: the use of *m* to denote the
component of angular momentum in one direction came about because a Bohr-type
electron in orbit is a current loop, with a magnetic moment parallel to its
angular momentum, so the *m* measured the component of magnetic moment in
a chosen direction, usually along an external magnetic field, and *m* is
often called the magnetic quantum number.