Non-Relativistic Limits

When the mass is non-zero and the group velocity is much less than the speed of light, it is useful to compute approximate forms of the above equations valid in this limit. Using the approximation $(1 + \epsilon )^x \approx 1 + x \epsilon$, we find that the dispersion relation becomes

$$\omega = \mu + \frac{k^2 c^2}{2 \mu} ~~~~~ E = mc^2 + \frac{\Pi^2}{2m} ,$$ (8.11)

and the group velocity equation takes the approximate form

$$u_g = \frac{c^2 k}{\mu} ~~~~~ u_g = \frac{\Pi}{m} .$$ (8.12)

The non-relativistic limits for equations (7.8) and (7.9) become

$$\omega = \mu + \frac{\mu u_g^2}{2c^2} ~~~~~ E = mc^2 + \frac{m u_g^2}{2}$$ (8.13)

and

$$k = \mu u_g /c^2 ~~~~~ \Pi = m u_g ,$$ (8.14)

while the approximate kinetic energy equation is

$$\kappa = \frac{\mu u_g^2}{2c^2} ~~~~~ K = \frac{m u_g^2}{2} .$$ (8.15)

Just a reminder -- the equations in this section are not valid for massless particles!