for the following question:
Consider a pion ($π^\pm$) scattering on a target. What pion momenta are needed to observe structures of the size of an atom ($10^{-10}$ m), an atomic nucleus ($10^{−14}$ m), or the proton ($10^{−15}$ m)? For each case, consider if you need a relativistic treatment and explain your reasoning.
I know the Broglie wavelength is $h/p$. But for the $10^{-14}$ m and $10^{-15}$ m I need to treat it relativistically, how do I do this?
The de Broglie relation $$\lambda=\frac{h}{p} \tag{1}$$ is already relativistically correct. So you only need simple math to find the momentum $p$.
May be you confused it with the relation between wavelength $\lambda$, velocity $v$ and rest mass $m$ $$\lambda=\frac{h\sqrt{1-\frac{v^2}{c^2}}}{mv}. \tag{2}$$ You can get this equation by combining the de Broglie relation (1) with the relativistic momentum $p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}$ To solve (2) for the velocity $v$ you would indeed need some more complicated math.
