- A.21.1 The wave function
- A.21.2 Simplifying the wave function
- A.21.3 Photon spin
- A.21.4 Energy eigenstates
- A.21.5 Normalization of the wave function
- A.21.6 States of definite linear momentum
- A.21.7 States of definite angular momentum

A.21 Photon type 2 wave function

In quantum mechanics, photons are the particles of the electromagnetic field. To actually use photons, something like a wave function for them is needed. But that is not quite trivial for a purely relativistic particle with zero rest mass like the photon. That is the primary topic of this addendum. It will be assumed throughout that the photon is in empty space.

A.21.1 The wave function

To see the problem with a photon wave function, a review of the wave
function of the nonrelativistic electron is useful, chapters
3.1 and 5.5.1. The electron wave function
can be written as a vector with two components:

This wave function takes on two different meanings

- 1.
- It gives the probability per unit volume of finding the electron
at a given position with a given spin. For example,
gives the probability of finding the electron with spin-up in an vicinity of infinitesimal volume around position . That is the Born statistical interpretation. - 2.
- It is the unobservable function that nature seems to use to do
its quantum
computations

of how physics behaves.

Now a wave function of type 1 is not really meaningful for a photon. What would it mean, find a photon? Since the photon has no rest mass, you cannot bring them to a halt: there would be nothing left. And anything you do to try to localize the electromagnetic field is likely to just produce new photons. (To be sure, with some effort something can be done towards a meaningful wave function of type 1, e.g. [Sype, J.E. 1995 Phys. Rev. A 52, 1875]. It would have two components like the electron, since the photon has two independent spin states. But wave functions of that type are not widely accepted, nor useful for the purposes here.)

So what? A wave function of type 1 is not that great anyway. For
one, it only defines the magnitudes of the components of the wave
function. If you only define the magnitude of a complex function, you
define only half of it. True, even as a type 2 wave function the
classical electron wave function is not quite unique. You can still
multiply either component by a factor

Furthermore, relativistic quantum mechanics has discovered that what we call an electron is something cloaked in a cloud of virtual particles. It is anybody’s guess what is inside that cloak, but it will not be anything resembling what we would call an electron. So what does it really mean, finding an electron within an infinitesimal volume around a point? What happens to that cloak? And to really locate an electron in an infinitesimal volume requires infinite energy. If you try to locate the electron in a region that is small enough, you are likely to just create additional electron-positron pairs much like for photons.

For most practical purposes, classical physics understands the particle behavior of electrons very well, but not their wave behavior. Conversely, it understands the wave behavior of photons very well, but not their particle behavior. But when you go to high enough energies, that distinction becomes much less obvious.

The photon most definitely has a wave function of type 2 above. In quantum electrodynamics, it may simply be called the photon wave function, [24, p. 240]. However, since the term already seems to be used for type 1 wave functions, this book will use the term “photon type 2 wave function.” It may not tell you where to find that elusive photon, but you will definitely need it to figure out how that photon interacts with, say, an electron.

What the type 2 wave function of the photon is can be guessed readily
from classical electromagnetics. After all, the photon is supposed to
be the particle of the electromagnetic field. So, consider first
electrostatics. In classical electrostatics the forces on charged
particles are described by an electric force per unit charge

But quantum mechanics uses potentials, not forces. For example, the
solution of the hydrogen atom of chapter 4.3 used a
potential energy of the electron

Clearly, an unobservable function vector potential

The following relationships give the electric and magnetic fields in
terms of these potentials:

is called nabla or del. As an example, for the

When both potentials are allowed for, the nonuniqueness becomes much
larger. In particular, for any arbitrary function

Finally, it turns out that classical relativistic mechanics likes to
combine the four scalar potentials in a four-dimensional vector, or
four-vector, chapter 1.3.2:

That is the one. Quantum mechanics takes a four-vector potential of
this form to be the type 2 wave function of the photon

Wave functions are in general complex. The classical four-potential, and especially its physically observable derivatives, the electric and magnetic fields, must be real. Indeed, according to quantum mechanics, observable quantities correspond to eigenvalues of Hermitian operators, not to wave functions. What the operators of the observable electric and magnetic fields are will be discussed in addendum {A.23}.The photon wave functionshould not be confused with the classical four-potential .

A.21.2 Simplifying the wave function

To use the photon wave function in practical applications, it is
essential to simplify it. That can be done by choosing a clever gauge
function

One very helpful simplification is to choose

To achieve the Lorenz condition, assume an initial wave function

This equation for

There is another reason why you want to satisfy the Lorenz condition.
The photon is a purely relativistic particle with zero rest mass.
Then following the usual ideas of quantum mechanics, in empty space
its wave function should satisfy the homogeneous Klein-Gordon
equation, {A.14} (A.43):

normalphoton wave functions, the ones that do satisfy the Klein-Gordon equation, should be exactly the ones that also satisfy the Lorenz condition.

Maxwell’s classical electromagnetics provides additional support for that idea. There the Klein-Gordon equation for the potentials also requires that the Lorenz condition is satisfied, {A.37}.

Since the inhomogeneous Klein-Gordon equation for the gauge function

And so the fully simplified photon wave function becomes:

solenoidal.A gauge function that makes

It should be noted that the Coulomb gauge is not Lorentz invariant. A
moving observer will not agree that the potential

A.21.3 Photon spin

Now that the photon wave function has been simplified the photon spin
can be determined. Recall that for the electron, the two components
of the wave function correspond to its two possible values of the spin
angular momentum

The simplified wave function (A.90) has only three
nontrivial components. And the gauge property requires that this
simplified wave function still describes all the physics. Since the
only nontrivial part left is the three-dimensional vector

However, that is not quite the end of the story. There is still that
additional condition

Still, the additional constraint does limit the angular momentum of the photon. In particular, a photon does not have independent spin and orbital angular momentum. The two are intrinsically linked. What that means for the net angular momentum of photons is worked out in subsection A.21.7.

For now it may already be noted that the photon has no state of zero
net angular momentum. A state of zero angular momentum needs to look
the same from all directions. That is a consequence of the
relationship between angular momentum and symmetry, chapter
7.3. Now the only vector wave functions that look the
same from all directions are of the form

A.21.4 Energy eigenstates

Following the rules of quantum mechanics, {A.14},
photon states of definite energy

Here

Substitution in the Klein-Gordon equation and cleaning up shows that
this eigenfunction needs to satisfy the eigenvalue problem,
{A.14},

A.21.5 Normalization of the wave function

A classical wave function for a particle is normalized by demanding that the square integral of the wave function is 1. That does not work for a relativistic particle like the photon, since the Klein-Gordon equation does not preserve the square integral of the wave function, {A.14}.

However, the Klein-Gordon equation does preserve the following
integral, {D.36.1},

Reasonably speaking, you would expect this integral to be related to the energy in the electromagnetic field. After all, what other scalar physical quantity is there to be preserved?

Consider for a second the case that

Now classical physics does not have photons of energy

Of course, there needs to be an additional constant; the integral
above does not have units of energy. If you check, you find that the
permittivity of space

Now the photon wave function is not physically observable and does not
have to conform to the rules of classical physics. But if you have to
choose a normalization constant anway? Why not choose it so that what
classical physics would take to be the energy is in fact the correct
energy

So, the photon wave function normalization that will be used in this
book is:

(To be sure, classical physics would take

Assume that you start with an unnormalized energy eigenfunction

A.21.6 States of definite linear momentum

The simplest quantum states for photons are states of definite linear
momentum

In that case, the photon wave function takes the form

Here

The vector

The bottom line is that there are only two independent states, even though the wave function is a three-dimensional vector. The wave function cannot have a component in the direction of motion. It may be noted that the first term in the right hand side above is called a wave that is “linearly polarized” in the

There is another useful way to write the wave function:

where

There are still only two independent states. But another way of
thinking about that is that the spin angular momentum in the direction
of motion cannot be zero. The relative spin in the direction of
motion,

Note further that the angular momenta in the

For later use, it is necessary to normalize the wave function using
the procedure described in the previous subsection. To do so, it must
be assumed that the photon is in a periodic box of volume

In order to compare to the classical electromagnetic wave in chapter
7.7.1, another example is needed. This photon wave function
has its linear momentum in the

The normalized eigenfunction and unobservable fields are in that case

Note that

For a general direction of the wave motion and its linear
polarization, the above expession becomes

For convenience, the density of states as needed for Fermi’s
golden rule will be listed here. It was given earlier in chapter
6.3 (6.7) and 6.19:

A.21.7 States of definite angular momentum

It is often convenient to describe photons in terms of states of definite net angular momentum. That makes it much easier to apply angular momentum conservation in the emission of radiation by atoms or atomic nuclei. Unfortunately, angular momentum states are a bit of a mess compared to the linear momentum states of the previous subsection. Fortunately, engineers are brave.

Before diving in, it is a good idea to look first at a spinless
particle. Assume that this hypothetical particle is in an energy
eigenstate. Also assume that this state has square orbital angular
momentum

Here

Now the photon is a particle with spin 1. Its wave function is
essentially a vector

The two types of photon energy eigenfunctions with definite net
angular momentum are, {D.36.2} and with drums please,

The azimuthal quantum number

The parity of the electric multipole wave functions is negative if

Atomic or nuclear transitions in which a photon in a state

In particular, for net angular momentum

Such transitions may be accomodated by transitions in which photons in
different states are emitted or absorbed, using the photon angular
momenta and parities as noted above. Electric multipole transitions
with

Similarly, transitions in which photons in a state

Like the states of definite linear momentum in the previous
subsection, the states of definite angular momentum cannot be
normalized in infinite space. To deal with that, it will be assumed
that the photon is confined inside a sphere of a very large radius
boundary condition

on the
sphere, it will be assumed that the Bessel function is zero. In terms
of the wave functions, that works out to mean that the magnetic ones
are zero on the sphere, but only the radial component of the electric
ones is.

The normalized wave function and unobservable fields for electric
multipole photons are then, subsection A.21.5 and
{D.36},

The normalized wave function and unobservable fields for magnetic
multipole photons are

Assume now that there is an atom or atomic nucleus at the origin that
interacts with the photon. An atom or nucleus is typically very small
compared to the wave length of the photon that it interacts with.
Phrased more appropriately, if

Now the wave functions

A glance at the unobservable fields of electric multipole photons
above then shows that for these photons, the field is primarily
electric at the atom or nucleus. And even the electric field will be
small unless

For the magnetic multipole photons, it is the magnetic field that
dominates at the atom or nucleus. And even that will be small unless

For later reference, the density of states as needed for Fermi’s
golden rule will be listed here, {D.36.2.6}: