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Potential Momentum

For a free, non-relativistic particle of mass , the total energy equals the kinetic energy and is related to the momentum of the particle by

(14.1)

In the non-relativistic case, the momentum is , where is the particle velocity.

If the particle is not free, but is subject to forces associated with a
potential energy , then equation (13.1)
must be modified to account for the contribution of to the total energy:

(14.2)

The force on the particle is related to the potential energy by

(14.3)

For a free, relativistic particle, we have

(14.4)

The obvious way to add forces to the relativistic case is by rewriting
equation (13.4)
with a potential energy, in analogy with equation (13.2):

(14.5)

Unfortunately, equation (13.5)
is incomplete, because we have subtracted something () from the energy without subtracting something from the momentum as well. However, is a four-vector, so an equation with something
subtracted from just one of the components of this four-vector is not
relativistically invariant. In other words, equation (13.5)
doesn't obey the principle of relativity, and therefore cannot be correct!

How can we fix this problem? One way is to define a new four-vector with being its timelike part and some new vector being its spacelike part:

(14.6)

We then subtract from the momentum
. When we do this, equation (13.5)
becomes

(14.7)

The quantity is called the potential momentum and is the potential four-momentum.

Some additional terminology is useful. We define

(14.8)

as the kinetic momentum for reasons discussed below. In order to
avoid confusion, we rename the total momentum.14.1 Thus, the total momentum equals the kinetic
plus the potential momentum, in analogy with energy.

So far, we have shown that the introduction of a potential momentum
complements the potential energy so as to make the energy-momentum relationship
for a particle relativistically invariant. However, we as yet have no idea what
causes potential momentum and what it does to the affected particle. We shall
put off answering the former question and address only the latter at this point.
A hint comes from the corresponding behavior of energy. The total energy
of a particle is related to the quantum mechanical frequency of the particle, and the total momentum is related
to its wave vector :

(14.9)

However, the kinetic energy14.2and the kinetic momentum are related
to the particle's velocity :

(14.10)

where .

The relationship between kinetic momentum and velocity can be proven by
dividing equation (13.7)
by to obtain a dispersion relation and then computing the
group velocity, which we equate to the particle velocity. However, we will not
do this here.

D. J. Raymond 2003-05-14