A.20 Angular momentum of vector particles

This addendum is concerned with vector particles, particles whose wave
functions are vectors. To be sure, the wave function of an electron
can also be written as a vector, chapters 3.1 and
5.5.1:

But that is not a normal vector. It is a two-dimensional vector in three-dimensional space, and is known as a spinor. This addendum is concerned with wave functions that are normal three-dimensional vectors. That is of importance for understanding, for example, the spin angular momentum of photons. A photon is a vector particle, though a special one. It will be shown in this addendum that the spin of a vector particle is 1. The parity of such a particle will also be discussed.

To really appreciate this addendum, you may want to read the previous
addendum {A.19} first. In any case, according to
that addendum angular momentum is related to what happens to the wave
function under rotation of the coordinate system. In particular, the
angular momentum in the

Consider first the simplest possible vector wave function:

Here

The factor

Now consider three very special vectors:

The vector

To get at square angular momentum, first the operator

Here

Plugging in the components of

So the operator

So the square operator just drops the

Of course, the operators

Since each of the operators in the right hand side drops a different component and adds a factor

So the square spin operator always produces a simple multiple of the original vector. That makes any vector an eigenvector of square spin angular momentum. Also, the azimuthal quantum number

That then means that the spin

You can write the most general vector wave function in the form

Then you can put the coefficients in a vector much like the wave function of the electron, but now three-dimensional:

Like the electron, the vector particle can of course also have orbital
angular momentum. That is due to the coefficients

(To check this, take a dot, or rather inner, product of the wave function with itself. Then integrate over all space using spherical coordinates and the orthonormality of the spherical harmonics.)

The wave function (A.84) above has orbital angular momentum
in the

In general, then, the magnetic quantum number

However, the situation for the azimuthal quantum number

That is a complicated issue best left to chapter 12.
But a couple of special cases are worth mentioning already. First, if

The other special case is that there is zero net angular momentum.
Zero net angular momentum means that the wave function is exactly the
same regardless how the coordinate system is rotated. And that only
happens for a vector wave function if it is purely radial:

Here

The above state has zero net angular momentum. The question of
interest is what can be said about its spin and orbital angular
momentum. To answer that, it must be rewritten in terms of Cartesian
components. Now the unit vector

The spatial factors in this expression can be written in terms of the spherical harmonics

To check this, just plug in the expressions for the

The bottom line is that by combining states of unit spin

The above relation may be written more neatly in terms of “ket
notation.” In ket notation, an angular momentum state with
azimuthal quantum number

Here the subscripts

There is a quicker way to get this result than going through the above
algebraic mess. You can simply read off the coefficients in the
appropriate column of the bottom-right tabulation in figure
12.6. (In this figure take

The parity of vector wave functions is also important. Parity is what
happens to a wave function if you invert the positive direction of all
three Cartesian axes. What happens to a vector wave function under
such an inversion can vary. A normal, or “polar,” vector changes sign when you invert the axes. For
example, a position vector

But now consider an example like a classical angular momentum vector,

Note however that the orbital angular momentum of the particle also
has an effect on the net parity. In particular, if the quantum number
of orbital angular momentum

Particles of all types often have definite parity. Such a particle
may still have uncertainty in