Is there a significant error in using De Broglie's equation for an electron at really high speed? I was wondering if using the De Broglie equation 
$ \lambda = \frac{h}{p}$
for object traveling at really high speeds would result in a significant error. For example if an object travelled at $0.02c$ would the error be negligible? How can I calculate the uncertainty in the result?
 A: The formula is still correct for speeds close to $c$, provided you use the relativistic definition of momentum for $\mathbf{p}$.
Using natural units ($\hbar=c=1$), the wave function of a particle of momentum $\mathbf{p}$ is given by
$$ \psi(\mathbf{x},t) = \psi_0 \exp(-iEt + i\mathbf{p}\cdot\mathbf{x}).$$
The formula is the same, both in non-relativistic and relativistic mechanics. The only difference is the definition of energy, which in relativity is
$$ E = \sqrt{m^2+\mathbf{p}^2}.$$
Also the dependence of momentum on speed changes. Since you are interested in a particle traveling slower than $c$, it must be massive, and the momentum is
$$ \mathbf{p} = m\gamma \mathbf{v}.$$
The ratio of the relativistic and nonrelativistic wavelengths is just
$$ \frac{\lambda_{rel}}{\lambda_{nr}} = \frac{p_{nr}}{p_{rel}} = \frac{1}{\gamma} = \sqrt{1-v^2} \approx 1 -\frac{v^2}{2}$$
So, for $v=0.02$, the relativistic wavelength would be around $0.04\%$ smaller than the non-relativistic one.
A: In addition to the answer of @FelipeLopes. The formula is valid for light as well, so in the ultra relativistic case of speed c. In fact it was inspired by the relation for light.
