Wave and Particle Quantities

Let us now recapitulate what we know about relativistic waves, and how this knowledge translates into knowledge about the mass, energy, and momentum of particles. In the following equations, the left form is expressed in wave terms, i. e., in terms of frequency, wavenumber, and rest frequency. The right form is the identical equation expressed in terms of energy, momentum, and mass. Since the latter variables are just scaled forms of the former, the two forms of each equation are equivalent.

We begin with the dispersion relation for relativistic waves:

$$\omega^2 = k^2 c^2 + \mu^2 ~~~~~ E^2 = \Pi^2 c^2 + m^2 c^4 .$$ (8.6)

Calculation of the group velocity, $u_g = d \omega /dk$, from the dispersion relation yields

$$u_g = \frac{c^2 k}{\omega} ~~~~~ u_g = \frac{c^2 \Pi}{E} .$$ (8.7)

These two sets of equations represent what we know about relativistic waves, and what this knowledge tells us about the relationships between the mass, energy, and momentum of relativistic particles. When in doubt, refer back to these equations, as they work in all cases, including for particles with zero mass!

It is useful to turn equations (7.6) and (7.7) around so as to express the frequency as a function of rest frequency and group velocity,

$$\omega = \frac{\mu}{(1 - u_g^2 /c^2 )^{1/2}} ~~~~~ E = \frac{mc^2}{(1 - u_g^2 /c^2 )^{1/2}} ,$$ (8.8)

and the wavenumber as a similar function of these quantities:

$$k = \frac{\mu u_g /c^2}{(1 - u_g^2 /c^2 )^{1/2}} ~~~~~ \Pi = \frac{m u_g}{(1 - u_g^2 /c^2 )^{1/2}} .$$ (8.9)

Note that equations (7.8) and (7.9) work only for particles with non-zero mass! For zero mass particles you need to use equations (7.6) and (7.7) with $m = 0$ and $\mu = 0$.

The quantity $\omega - \mu$ indicates how much the frequency exceeds the rest frequency. Notice that if $\omega = \mu$, then from equation (7.6) $k = 0$. Thus, positive values of $\kappa \equiv \omega - \mu$ indicate $\vert k \vert > 0$, which means that the particle is moving according to equation (7.7). Let us call $\kappa$ the kinetic frequency:

$$\kappa = \left[ \frac{1}{(1 - u_g^2 /c^2 )^{1/2}} - 1 \righ... ...= \left[ \frac{1}{(1 - u_g^2 /c^2 )^{1/2}} - 1 \right] mc^2 .$$ (8.10)

We call $K$ the kinetic energy for similar reasons. Again, equation (7.10) only works for particles with non-zero mass. For zero mass particles the kinetic energy equals the total energy.